diff --git a/README.md b/README.md
index acbf8ab..60b650a 100644
--- a/README.md
+++ b/README.md
@@ -1 +1,8 @@
 # MyZKP: Introduction to Zero Knowledge Protocol
+
+## Book
+
+```bash
+cd docs/book
+mdbook serve --open --port 3001
+```
\ No newline at end of file
diff --git a/docs/book/book.toml b/docs/book/book.toml
index 822fc92..bc276bd 100644
--- a/docs/book/book.toml
+++ b/docs/book/book.toml
@@ -7,3 +7,4 @@ title = "MyZKP: Building Zero Knowledge Proof from Scratch"
 
 [output.html]
 mathjax-support = true
+additional-css = ["theme/css/custom.css"]
diff --git a/docs/book/book/404.html b/docs/book/book/404.html
index 308657f..51e868d 100644
--- a/docs/book/book/404.html
+++ b/docs/book/book/404.html
@@ -30,6 +30,7 @@
         <link rel="stylesheet" href="ayu-highlight.css">
 
         <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="theme/css/custom.css">
 
         <!-- MathJax -->
         <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
@@ -91,7 +92,7 @@
 
         <nav id="sidebar" class="sidebar" aria-label="Table of contents">
             <div class="sidebar-scrollbox">
-                <ol class="chapter"><li class="chapter-item expanded "><a href="number_theory/index.html"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="number_theory/subsec1.html"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="zksnark/index.html"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="zkstark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="zkvm/index.html"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
             </div>
             <div id="sidebar-resize-handle" class="sidebar-resize-handle">
                 <div class="sidebar-resize-indicator"></div>
diff --git a/docs/book/book/index.html b/docs/book/book/index.html
index 7348f60..2ec9de1 100644
--- a/docs/book/book/index.html
+++ b/docs/book/book/index.html
@@ -3,7 +3,7 @@
     <head>
         <!-- Book generated using mdBook -->
         <meta charset="UTF-8">
-        <title>Basics of Number Theory - MyZKP: Building Zero Knowledge Proof from Scratch</title>
+        <title>Introduction - MyZKP: Building Zero Knowledge Proof from Scratch</title>
 
 
         <!-- Custom HTML head -->
@@ -29,6 +29,7 @@
         <link rel="stylesheet" href="ayu-highlight.css">
 
         <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="theme/css/custom.css">
 
         <!-- MathJax -->
         <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
@@ -90,7 +91,7 @@
 
         <nav id="sidebar" class="sidebar" aria-label="Table of contents">
             <div class="sidebar-scrollbox">
-                <ol class="chapter"><li class="chapter-item expanded "><a href="number_theory/index.html" class="active"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="number_theory/subsec1.html"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="zksnark/index.html"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="zkstark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="zkvm/index.html"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="index.html" class="active"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
             </div>
             <div id="sidebar-resize-handle" class="sidebar-resize-handle">
                 <div class="sidebar-resize-indicator"></div>
@@ -175,14 +176,13 @@ <h1 class="menu-title">MyZKP: Building Zero Knowledge Proof from Scratch</h1>
 
                 <div id="content" class="content">
                     <main>
-                        <h1 id="basics-of-number-theory"><a class="header" href="#basics-of-number-theory">Basics of Number Theory</a></h1>
-
+                        
                     </main>
 
                     <nav class="nav-wrapper" aria-label="Page navigation">
                         <!-- Mobile navigation buttons -->
 
-                            <a rel="next prefetch" href="number_theory/subsec1.html" class="mobile-nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
+                            <a rel="next prefetch" href="number_theory/index.html" class="mobile-nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
                                 <i class="fa fa-angle-right"></i>
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@@ -193,7 +193,7 @@ <h1 id="basics-of-number-theory"><a class="header" href="#basics-of-number-theor
 
             <nav class="nav-wide-wrapper" aria-label="Page navigation">
 
-                    <a rel="next prefetch" href="number_theory/subsec1.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
+                    <a rel="next prefetch" href="number_theory/index.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
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diff --git a/docs/book/book/number_theory/index.html b/docs/book/book/number_theory/index.html
index ecdcf6f..e0d2695 100644
--- a/docs/book/book/number_theory/index.html
+++ b/docs/book/book/number_theory/index.html
@@ -29,6 +29,7 @@
         <link rel="stylesheet" href="../ayu-highlight.css">
 
         <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="../theme/css/custom.css">
 
         <!-- MathJax -->
         <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
@@ -90,7 +91,7 @@
 
         <nav id="sidebar" class="sidebar" aria-label="Table of contents">
             <div class="sidebar-scrollbox">
-                <ol class="chapter"><li class="chapter-item expanded "><a href="../number_theory/index.html" class="active"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html" class="active"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
             </div>
             <div id="sidebar-resize-handle" class="sidebar-resize-handle">
                 <div class="sidebar-resize-indicator"></div>
@@ -181,6 +182,9 @@ <h1 id="basics-of-number-theory"><a class="header" href="#basics-of-number-theor
 
                     <nav class="nav-wrapper" aria-label="Page navigation">
                         <!-- Mobile navigation buttons -->
+                            <a rel="prev" href="../index.html" class="mobile-nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
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+                            </a>
 
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                                 <i class="fa fa-angle-right"></i>
@@ -192,6 +196,9 @@ <h1 id="basics-of-number-theory"><a class="header" href="#basics-of-number-theor
             </div>
 
             <nav class="nav-wide-wrapper" aria-label="Page navigation">
+                    <a rel="prev" href="../index.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
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+                    </a>
 
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diff --git a/docs/book/book/number_theory/subsec1.html b/docs/book/book/number_theory/subsec1.html
index efba9cc..9854f93 100644
--- a/docs/book/book/number_theory/subsec1.html
+++ b/docs/book/book/number_theory/subsec1.html
@@ -29,6 +29,7 @@
         <link rel="stylesheet" href="../ayu-highlight.css">
 
         <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="../theme/css/custom.css">
 
         <!-- MathJax -->
         <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
@@ -90,7 +91,7 @@
 
         <nav id="sidebar" class="sidebar" aria-label="Table of contents">
             <div class="sidebar-scrollbox">
-                <ol class="chapter"><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html" class="active"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html" class="active"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
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             <div id="sidebar-resize-handle" class="sidebar-resize-handle">
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@@ -175,20 +176,22 @@ <h1 class="menu-title">MyZKP: Building Zero Knowledge Proof from Scratch</h1>
 
                 <div id="content" class="content">
                     <main>
-                        <h2 id="computation-rule-and-properties"><a class="header" href="#computation-rule-and-properties">Computation Rule and Properties</a></h2>
+                        <h1 id="computation-rule-and-properties"><a class="header" href="#computation-rule-and-properties">Computation Rule and Properties</a></h1>
+<h3 id="definition-binary-operation"><a class="header" href="#definition-binary-operation">Definition: Binary Operation</a></h3>
+<hr />
+<p><em>A mapping \(\circ: X \times X \rightarrow X\) is a binary operation on \(X\) if for any pair of elements \((x_1, x_2)\) in \(X\), \(x_1 \circ x_2\) is also in \(X\).</em></p>
 <hr />
-<p><em><strong>Definition: Binary Operation</strong></em></p>
-<p>A mapping \(\circ: X \times X \rightarrow X\) is a binary operation on \(X\) if for any pair of elements \((x_1, x_2)\) in \(X\), \(x_1 \circ x_2\) is also in \(X\).</p>
 <p><strong>Example</strong>: Addition (+) on the set of integers is a binary operation. For example, \(5 + 3 = 8\), and both \(5, 3, 8\) are integers, staying within the set of integers.</p>
+<h3 id="definition-associative-property"><a class="header" href="#definition-associative-property">Definition: Associative Property</a></h3>
+<hr />
+<p><em>A binary operation \(\circ\) is associative if \((a \circ b) \circ c = a \circ (b \circ c)\) for all \(a, b, c \in X\).</em></p>
 <hr />
-<p><em><strong>Definition: Associative Property</strong></em></p>
-<p>A binary operation \(\circ\) is associative if \((a \circ b) \circ c = a \circ (b \circ c)\) for all \(a, b, c \in X\).</p>
 <p><strong>Example</strong>: Multiplication of real numbers is associative: \((2 \times 3) \times 4 = 2 \times (3 \times 4) = 24\). In a modular context, we also have addition modulo \(n\) being associative. For example, for \(n = 5\), \((2 + 3) \bmod 5 + 4 \bmod 5 = 2 + (3 \bmod 5 + 4) \bmod 5 = 4\).</p>
+<h3 id="definition-commutative-property"><a class="header" href="#definition-commutative-property">Definition: Commutative Property</a></h3>
 <hr />
-<p><em><strong>Definition: Commutative Property</strong></em></p>
-<p>A binary operation \(\circ\) is commutative if \(a \circ b = b \circ a\) for all \(a, b \in X\).</p>
-<p><strong>Example</strong>: Addition modulo \(n\) is also commutative. For \(n = 7\), \(5 + 3 \bmod 7 = 3 + 5 \bmod 7 = 1\).</p>
+<p><em>A binary operation \(\circ\) is commutative if \(a \circ b = b \circ a\) for all \(a, b \in X\).</em></p>
 <hr />
+<p><strong>Example</strong>: Addition modulo \(n\) is also commutative. For \(n = 7\), \(5 + 3 \bmod 7 = 3 + 5 \bmod 7 = 1\).</p>
 
                     </main>
 
@@ -198,7 +201,7 @@ <h2 id="computation-rule-and-properties"><a class="header" href="#computation-ru
                                 <i class="fa fa-angle-left"></i>
                             </a>
 
-                            <a rel="next prefetch" href="../zksnark/index.html" class="mobile-nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
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@@ -212,7 +215,7 @@ <h2 id="computation-rule-and-properties"><a class="header" href="#computation-ru
                         <i class="fa fa-angle-left"></i>
                     </a>
 
-                    <a rel="next prefetch" href="../zksnark/index.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
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diff --git a/docs/book/book/number_theory/subsec2.html b/docs/book/book/number_theory/subsec2.html
new file mode 100644
index 0000000..e7cfe13
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+++ b/docs/book/book/number_theory/subsec2.html
@@ -0,0 +1,455 @@
+<!DOCTYPE HTML>
+<html lang="en" class="light" dir="ltr">
+    <head>
+        <!-- Book generated using mdBook -->
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html" class="active"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
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+                        <h1 id="semigroup-group-ring"><a class="header" href="#semigroup-group-ring">Semigroup, Group, Ring</a></h1>
+<h3 id="definition-semigroup"><a class="header" href="#definition-semigroup">Definition: Semigroup</a></h3>
+<hr />
+<p><em>A pair \((H, \circ)\), where \(H\) is a non-empty set and \(\circ\) is an associative binary operation on \(H\), is called a semigroup.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of positive integers under multiplication modulo \(n\) forms a semigroup. For instance, with \(n = 6\), the elements \(\{1, 2, 3, 4, 5\}\) under multiplication modulo 6 form a semigroup, since multiplication modulo 6 is associative.</p>
+<h3 id="definition-abelian-semigroup"><a class="header" href="#definition-abelian-semigroup">Definition: Abelian Semigroup</a></h3>
+<hr />
+<p><em>A semigroup whose operation is commutative is called an abelian semigroup.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of natural numbers under addition modulo \(n\) forms an abelian semigroup. For \(n = 7\), addition modulo 7 is both associative and commutative, so it is an abelian semigroup.</p>
+<h3 id="definition-identity-element"><a class="header" href="#definition-identity-element">Definition: Identity Element</a></h3>
+<hr />
+<p><em>An element \(e \in H\) is an identity element of \(H\) if it satisfies \(e \circ a = a \circ e = a\) for any \(a \in H\).</em></p>
+<hr />
+<p><strong>Example</strong>: 0 is the identity element for addition modulo \(n\). For example, \(0 + a \bmod 5 = a + 0 \bmod 5 = a\). Similarly, 1 is the identity element for multiplication modulo \(n\). For example, \(1 \times a \bmod 7 = a \times 1 \bmod 7 = a\).</p>
+<h3 id="definition-monoid"><a class="header" href="#definition-monoid">Definition: Monoid</a></h3>
+<hr />
+<p><em>A semigroup with an identity element is called a monoid.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of non-negative integers under addition modulo \(n\) forms a monoid. For \(n = 5\), the set \(\{0, 1, 2, 3, 4\}\) under addition modulo 5 forms a monoid with 0 as the identity element.</p>
+<h3 id="definition-inverse"><a class="header" href="#definition-inverse">Definition: Inverse</a></h3>
+<hr />
+<p><em>For an element \(a \in H\), an element \(b \in H\) is an inverse of \(a\) if \(a \circ b = b \circ a = e\), where \(e\) is the identity element.</em></p>
+<hr />
+<p><strong>Example</strong>: In modulo \(n\) arithmetic (addition), the inverse of an element exists if it can cancel itself out to yield the identity element. In the set of integers modulo 7, the inverse of 3 is 5, because \(3 \times 5 \bmod 7 = 1\), where 1 is the identity element for multiplication.</p>
+<h3 id="definition-group"><a class="header" href="#definition-group">Definition: Group</a></h3>
+<hr />
+<p><em>A monoid in which every element has an inverse is called a group.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of integers modulo a prime \(p\) under multiplication forms a group (Can you prove it?). For instance, in \(\mathbb{Z}/5\mathbb{Z}\), every non-zero element \(\{1 + 5\mathbb{Z}, 2 + 5\mathbb{Z}, 3 + 5\mathbb{Z}, 4 + 5\mathbb{Z}\}\) has an inverse, making it a group.</p>
+<h3 id="definition-order-of-a-group"><a class="header" href="#definition-order-of-a-group">Definition: Order of a Group</a></h3>
+<hr />
+<p><em>The order of a group is the number of elements in the group.</em></p>
+<hr />
+<p><strong>Example</strong>: The group of integers modulo 4 under addition has order 4, because the set of elements is \(\{0, 1, 2, 3\}\).</p>
+<h3 id="definition-ring"><a class="header" href="#definition-ring">Definition: Ring</a></h3>
+<hr />
+<p><em>A triple \((R, +, \cdot)\) is a ring if \((R, +)\) is an abelian group, \((R, \cdot)\) is a semigroup, and the distributive property holds: \(x \cdot (y + z) = (x \cdot y) + (x \cdot z)\) and \((x + y) \cdot z = (x \cdot z) + (y \cdot z)\) for all \(x, y, z \in R\).</em></p>
+<hr />
+<p><strong>Example</strong>: The set of integers with usual addition and multiplication modulo \(n\) forms a ring. For example, in \(\mathbb{Z}/6\mathbb{Z}\), addition and multiplication modulo 6 form a ring.</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>use std::fmt;
+use std::ops::{Add, Mul, Neg, Sub};
+
+pub trait Ring:
+    Sized
+    + Clone
+    + PartialEq
+    + fmt::Display
+    + Add&lt;Output = Self&gt;
+    + Sub&lt;Output = Self&gt;
+    + Mul&lt;Output = Self&gt;
+    + Mul&lt;i64, Output = Self&gt;
+    + Neg&lt;Output = Self&gt;
+{
+    // A ring is an algebraic structure with addition and multiplication
+    fn zero() -&gt; Self;
+    fn one() -&gt; Self;
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-communicative-ring"><a class="header" href="#definition-communicative-ring">Definition: Communicative Ring</a></h3>
+<hr />
+<p><em>A ring is called a commutative ring if its multiplication operation is commutative.</em></p>
+<hr />
+<p><strong>Example</strong> The set of real numbers under usual addition and multiplication forms a commutative ring.</p>
+<h3 id="definition-field"><a class="header" href="#definition-field">Definition: Field</a></h3>
+<hr />
+<p><em>A commutative ring with a multiplicative identity element where every non-zero element has a multiplicative inverse is called a field.</em></p>
+<hr />
+<p><strong>Example</strong> The set of rational numbers under usual addition and multiplication forms a field.</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>use crate::modules::ring::Ring;
+
+pub trait Field: Ring + Div&lt;Output = Self&gt; {
+    fn inverse(&amp;self) -&gt; Self;
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-residue-class"><a class="header" href="#definition-residue-class">Definition: Residue Class</a></h3>
+<hr />
+<p><em>The residue class of \( a \) modulo \( m \), denoted as \( a + m\mathbb{Z} \), is the set \( \{b : b \equiv a \pmod{m}\} \).</em></p>
+<hr />
+<p><strong>Example:</strong> For \( m = 3 \), the residue class of 2 is \( 2 + 3\mathbb{Z} = \{\ldots, -4, -1, 2, 5, 8, \ldots\} \).</p>
+<h3 id="definition-inverse-of-residue-class"><a class="header" href="#definition-inverse-of-residue-class">Definition: Inverse of Residue Class</a></h3>
+<hr />
+<p><em>We denote the set of all residue classes modulo \( m \) as \( \mathbb{Z} / m\mathbb{Z} \). We say that \( a + m\mathbb{Z} \) is invertible in \( \mathbb{Z} / m\mathbb{Z} \) if and only if there exists a solution for \( ax \equiv 1 \pmod{m} \).</em></p>
+<hr />
+<p><strong>Example:</strong> In \( \mathbb{Z}/5\mathbb{Z} \), \( 3 + 5\mathbb{Z} \) is invertible because \( \gcd(3, 5) = 1 \) (since \( 3\cdot2 \equiv 1 \pmod{5} \)). However, in \( \mathbb{Z}/6\mathbb{Z} \), \( 3 + 6\mathbb{Z} \) is not invertible because \( \gcd(3, 6) = 3 \neq 1 \).</p>
+<h3 id="lemma-121"><a class="header" href="#lemma-121">Lemma 1.2.1</a></h3>
+<hr />
+<p><em>If \( a \) and \( b \) are coprime, the residues of \( a \), \( 2a \), \( 3a \), ..., \( (b-1)a \) modulo \( b \) are all distinct.</em></p>
+<hr />
+<p><strong>Proof:</strong> Suppose, for contradiction, that there exist \( x, y \in \{1, 2, \dots, b-1\} \) with \( x \neq y \) such that \( xa \equiv ya \pmod{b} \). Then \( (x-y)a \equiv 0 \pmod{b} \), implying \( b \mid (x-y)a \). Since \( a \) and \( b \) are coprime, we must have \( b \mid (x-y) \). However, \( |x-y| &lt; b \), so this is only possible if \( x = y \), contradicting our assumption. Therefore, all residues must be distinct.</p>
+<h3 id="theorem-122"><a class="header" href="#theorem-122">Theorem 1.2.2</a></h3>
+<hr />
+<p><em>For any integers \( a \) and \( b \), the equation \( ax + by = 1 \) has a solution in integers \( x \) and \( y \) if and only if \( a \) and \( b \) are coprime.</em></p>
+<hr />
+<p><strong>Proof:</strong></p>
+<ul>
+<li><strong>(\( \Rightarrow \))</strong> Suppose \( a \) and \( b \) are not coprime, let \( d = \gcd(a,b) &gt; 1 \). Then \( d \mid a \) and \( d \mid b \), so \( d \mid (ax + by) \) for any integers \( x \) and \( y \). Thus, \( ax + by \neq 1 \) for any \( x \) and \( y \).</li>
+<li><strong>(\( \Leftarrow \))</strong> Suppose \( a \) and \( b \) are coprime. By the previous lemma, the residues of \( a \), \( 2a \), ..., \( (b-1)a \) modulo \( b \) are all distinct. Therefore, there exists an \( m \in \{1, 2, ..., b-1\} \) such that \( ma \equiv 1 \pmod{b} \). This means there exists an integer \( n \) such that \( ma = bn + 1 \), or equivalently, \( ma - bn = 1 \).</li>
+</ul>
+<h3 id="theorem-bézouts-identity"><a class="header" href="#theorem-bézouts-identity">Theorem: Bézout's Identity</a></h3>
+<hr />
+<p><em>For any integers \( a \) and \( b \), we have:
+\(
+a\mathbb{Z} + b \mathbb{Z} = \gcd(a, b)\mathbb{Z}
+\)</em></p>
+<hr />
+<p><strong>Proof:</strong></p>
+<p>This statement is equivalent to proving that \( ax + by = c \) has an integer solution if and only if \( c \) is a multiple of \( \gcd(a,b) \).</p>
+<ul>
+<li><strong>(\( \Rightarrow \))</strong> If \( ax + by = c \) for some integers \( x \) and \( y \), then \( \gcd(a,b) \mid a \) and \( \gcd(a,b) \mid b \), so \( \gcd(a,b) \mid (ax + by) = c \).</li>
+<li><strong>(\( \Leftarrow \))</strong> Let \( c = k\gcd(a,b) \) for some integer \( k \). We can write \( a = p\gcd(a,b) \) and \( b = q\gcd(a,b) \), where \( p \) and \( q \) are coprime. By the previous theorem, there exist integers \( m \) and \( n \) such that \( pm + qn = 1 \). Multiplying both sides by \( k\gcd(a,b) \), we get:
+[
+akm + bkn = c
+]
+Thus, \( x = km \) and \( y = kn \) are integer solutions to \( ax + by = c \).</li>
+</ul>
+<p>This theorem implies that for any integers \( a \), \( b \), and \( n \), the equation \( ax + by = n \) has an integer solution if and only if \( \gcd(a,b) \mid n \).</p>
+<h3 id="theorem-123"><a class="header" href="#theorem-123">Theorem 1.2.3</a></h3>
+<hr />
+<p><em>\( a + m\mathbb{Z} \) is invertible in \( \mathbb{Z}/m\mathbb{Z} \) if and only if \( \gcd(a,m) = 1 \).</em></p>
+<hr />
+<p><strong>Proof:</strong></p>
+<ul>
+<li><strong>(\( \Rightarrow \))</strong> Suppose \( a + m\mathbb{Z} \) is invertible in \( \mathbb{Z}/m\mathbb{Z} \). Then there exists an integer \( x \) such that \( ax \equiv 1 \pmod{m} \). Let \( g = \gcd(a,m) \). Since \( g \mid ax \) and \( g \mid (ax - 1) \), we must have \( g \mid 1 \), so \( g = 1 \).</li>
+<li><strong>(\( \Leftarrow \))</strong> Suppose \( \gcd(a,m) = 1 \). By Bézout's identity, there exist integers \( x \) and \( y \) such that \( ax + my = 1 \). Thus, \( x + m\mathbb{Z} \) is the multiplicative inverse of \( a + m\mathbb{Z} \) in \( \mathbb{Z}/m\mathbb{Z} \).</li>
+</ul>
+<h3 id="definition-residue-class-ring"><a class="header" href="#definition-residue-class-ring">Definition: Residue Class Ring</a></h3>
+<p><em>\( (\mathbb{Z} / m \mathbb{Z}, +, \cdot) \) is a commutative ring where \( 1 + m \mathbb{Z} \) is the multiplicative identity element. This ring is called the residue class ring modulo \( m \).</em></p>
+<p><strong>Example:</strong> \( \mathbb{Z}/4\mathbb{Z} = \{0 + 4\mathbb{Z}, 1 + 4\mathbb{Z}, 2 + 4\mathbb{Z}, 3 + 4\mathbb{Z}\} \).</p>
+<h3 id="definition-primitive-residue-class"><a class="header" href="#definition-primitive-residue-class">Definition: Primitive Residue Class</a></h3>
+<hr />
+<p><em>A residue class \( a + m\mathbb{Z} \) is called primitive if \( \gcd(a, m) = 1 \).</em></p>
+<hr />
+<p><strong>Example:</strong> In \( \mathbb{Z}/6\mathbb{Z} \), the primitive residue classes are \( 1 + 6\mathbb{Z} \) and \( 5 + 6\mathbb{Z} \).</p>
+<h3 id="theorem-124"><a class="header" href="#theorem-124">Theorem 1.2.4</a></h3>
+<hr />
+<p><em>A residue ring \( \mathbb{Z} / m\mathbb{Z} \) is a field if and only if \( m \) is a prime number.</em></p>
+<hr />
+<p><strong>Proof:</strong> TBD</p>
+<p><strong>Example:</strong> For \( m = 5 \), \( \mathbb{Z}/5\mathbb{Z} = \{0 + 5\mathbb{Z}, 1 + 5\mathbb{Z}, 2 + 5\mathbb{Z}, 3 + 5\mathbb{Z}, 4 + 5\mathbb{Z}\} \) forms a field because 5 is a prime number, and every non-zero element has a multiplicative inverse.</p>
+<h3 id="definition-primitive-residue-class-group"><a class="header" href="#definition-primitive-residue-class-group">Definition: Primitive Residue Class Group</a></h3>
+<hr />
+<p><em>The group of all primitive residue classes modulo \( m \) is called the primitive residue class group, denoted by \( (\mathbb{Z}/m\mathbb{Z})^{\times} \).</em></p>
+<hr />
+<p><strong>Example:</strong> For \( m = 8 \), the set of all primitive residue classes is \( (\mathbb{Z}/8\mathbb{Z})^{\times} = \{1 + 8\mathbb{Z}, 3 + 8\mathbb{Z}, 5 + 8\mathbb{Z}, 7 + 8\mathbb{Z}\} \). These are the integers less than 8 that are coprime to 8 (i.e., \(\gcd(a, 8) = 1\)).</p>
+<p>Contrast this with \( m = 9 \). The primitive residue class group is \((\mathbb{Z}/9\mathbb{Z})^{\times} = \{1 + 9\mathbb{Z}, 2 + 9\mathbb{Z}, 4 + 9\mathbb{Z}, 5 + 9\mathbb{Z}, 7 + 9\mathbb{Z}, 8 + 9\mathbb{Z}\}\), as these are the integers less than 9 that are coprime to 9.</p>
+<h3 id="definition-eulers-totient-function"><a class="header" href="#definition-eulers-totient-function">Definition: Euler's Totient Function</a></h3>
+<hr />
+<p><em>Euler's totient function \(\phi(m)\) is equal to the order of the primitive residue class group modulo \(m\), which is the number of integers less than \(m\) and coprime to \(m\).</em></p>
+<hr />
+<p><strong>Example</strong>: For \(m = 12\), \(\phi(12) = 4\) because there are 4 integers less than 12 that are coprime to 12: \({1, 5, 7, 11}\).</p>
+<p>For \(m = 10\), \(\phi(10) = 4\), as there are also 4 integers less than 10 that are coprime to 10: \({1, 3, 7, 9}\).</p>
+<h3 id="definition-order-of-an-element-within-a-group"><a class="header" href="#definition-order-of-an-element-within-a-group">Definition: Order of an Element within a Group</a></h3>
+<hr />
+<p><em>The order of \(g \in G\) is the smallest natural number \(e\) such that \(g^{e} = 1\). We denote it as \(\text{order}_g(g)\)or \(\text{order } g\).</em></p>
+<hr />
+<p><strong>Example</strong>: In \((\mathbb{Z}/7\mathbb{Z})^{\times}\), the element 3 has order 6 because \(3^6 \bmod 7 = 1\). In other words, \(3 \times 3 \times 3 \times 3 \times 3 \times 3 \bmod 7 = 1\), and 6 is the smallest such exponent.</p>
+<h3 id="definition-subgroup"><a class="header" href="#definition-subgroup">Definition: Subgroup</a></h3>
+<hr />
+<p><em>A subset \(U \subseteq G\) is a subgroup of \(G\) if \(U\) itself is a group under the operation of \(G\).</em></p>
+<hr />
+<p><strong>Example</strong>: Consider \((\mathbb{Z}/8\mathbb{Z})^{\times} = \{1 + 8\mathbb{Z}, 3+ 8\mathbb{Z}, 5+ 8\mathbb{Z}, 7+ 8\mathbb{Z}\}\) under multiplication modulo 8. The subset \({1+ 8\mathbb{Z}, 7+ 8\mathbb{Z}}\) forms a subgroup because it satisfies the group properties: closed under multiplication, contains the identity element (1), and every element has an inverse (\(7 \times 7 \equiv 1 \bmod 8\)).</p>
+<h3 id="definition-subgroup-generated-by-g"><a class="header" href="#definition-subgroup-generated-by-g">Definition: Subgroup Generated by \(g\)</a></h3>
+<hr />
+<p><em>The set \(\{g^{k} : k \in \mathbb{Z}\}\), for some element \(g \in G\), forms a subgroup of \(G\) and is called the subgroup generated by \(g\), denoted by \(\langle g \rangle\).</em></p>
+<hr />
+<p><strong>Example</strong>: Consider the group \((\mathbb{Z}/7\mathbb{Z})^{\times} = \{1+ 7\mathbb{Z}, 2+ 7\mathbb{Z}, 3+ 7\mathbb{Z}, 4+ 7\mathbb{Z}, 5+ 7\mathbb{Z}, 6+ 7\mathbb{Z}\}\) under multiplication modulo 7. If we take \(g = 3\), then \(\langle 3 +7\mathbb{Z} \rangle =\) \(\{3^1+7\mathbb{Z}, 3^2+7\mathbb{Z}, 3^3+7\mathbb{Z}, 3^4+7\mathbb{Z}, 3^5+7\mathbb{Z}, 3^6+7\mathbb{Z}\} \bmod 7 =\) \(\{3+7\mathbb{Z}, 2+7\mathbb{Z}, 6+7\mathbb{Z}, 4+7\mathbb{Z}, 5+7\mathbb{Z}, 1+7\mathbb{Z}\}\), which forms a subgroup generated by 3. This subgroup contains all elements of \((\mathbb{Z}/7\mathbb{Z})^{\times}\), making 3 a generator of the entire group.</p>
+<p>If \(g\) has a finite order \(e\), we have that \(\langle g \rangle = \{g^{k}: 0 \leq k \leq e\}\), meaning \(e\) is the order of \(\langle g \rangle\).</p>
+<h3 id="definition-cyclic-group"><a class="header" href="#definition-cyclic-group">Definition: Cyclic Group</a></h3>
+<hr />
+<p><em>A group \(G\) is called a cyclic group if there exists an element \(g \in G\) such that \(G = \langle g \rangle\). In this case, \(g\) is called a generator of \(G\).</em></p>
+<hr />
+<p><strong>Example</strong>: The group \((\mathbb{Z}/6\mathbb{Z})^{\times} = \{1+6\mathbb{Z}, 5+6\mathbb{Z}\}\) under multiplication modulo 6 is a cyclic group. In this case, both 1 and 5 are generators of the group because \(\langle 5 +6\mathbb{Z} \rangle = \{(5^1 \bmod 6)+6\mathbb{Z} = 5 +6\mathbb{Z}, (5^2 \bmod 6)+6\mathbb{Z} = 1+6\mathbb{Z}\}\). Since 5 generates all the elements of the group, \(G\) is cyclic.</p>
+<h3 id="theorem-125"><a class="header" href="#theorem-125">Theorem 1.2.5</a></h3>
+<hr />
+<p><em>If \(G\) is a finite cyclic group, it has \(\phi(|G|)\)generators, and each generator has order \(|G|\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: Consider the group \((\mathbb{Z}/8\mathbb{Z})^{\times} = \{1+8\mathbb{Z}, 3+8\mathbb{Z}, 5+8\mathbb{Z}, 7+8\mathbb{Z}\}\). This group is cyclic, and \(\phi(8) = 4\). The generators of this group are \(\{1+8\mathbb{Z}, 3+8\mathbb{Z}, 5+8\mathbb{Z}, 7+8\mathbb{Z}\}\), each of which generates the entire group when raised to successive powers modulo 8. Each generator has the same order, which is \(|G| = 4\).</p>
+<h3 id="theorem-126"><a class="header" href="#theorem-126">Theorem 1.2.6</a></h3>
+<hr />
+<p><em>If \(G\) is a finite cyclic group, the order of any subgroup of \(G\) divides the order of \(G\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: Consider the cyclic group \((\mathbb{Z}/6\mathbb{Z})^{\times} = \{1+6\mathbb{Z}, 5+6\mathbb{Z}\}\) under multiplication modulo 6. If we take the subgroup \(\langle 5+6\mathbb{Z} \rangle = \{1+6\mathbb{Z}, 5+6\mathbb{Z}\}\), this is a subgroup of order 2, and 2 divides the order of the original group, which is 6. This theorem generalizes this property: for any subgroup of a cyclic group, its order divides the order of the group.</p>
+<h3 id="theorem-fermats-little-theorem"><a class="header" href="#theorem-fermats-little-theorem">Theorem: Fermat's Little Theorem</a></h3>
+<hr />
+<p><em>If \(\gcd(a, m) = 1\), then \(a^{\phi(m)} \equiv 1 \pmod{m}\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: Take \(a = 2\) and \(m = 5\). Since \(\gcd(2, 5) = 1\), Fermat's Little Theorem tells us that \(2^{\phi(5)} = 2^4 \equiv 1 \bmod 5\). Indeed, \(2^4 = 16\) and \(16 \bmod 5 = 1\).</p>
+<p>This theorem suggests that \(a^{\phi(m) - 1} + m \mathbb{Z}\) is the inverse residue class of \(a + m \mathbb{Z}\).</p>
+<h3 id="theorem-127"><a class="header" href="#theorem-127">Theorem 1.2.7</a></h3>
+<hr />
+<p><em>The order of any element in a group divides the order of the group.</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: In the group \((\mathbb{Z}/7\mathbb{Z})^{\times}\), consider the element \(3 + 7\mathbb{Z}\). The order of \(3 + 7\mathbb{Z}\) is 6, as \(3^6 \equiv 1 \bmod 7\). The order of the group itself is also 6, and indeed, the order of the element divides the order of the group.</p>
+<h3 id="theorem-generalization-of-fermats-little-theorem"><a class="header" href="#theorem-generalization-of-fermats-little-theorem">Theorem: Generalization of Fermat's Little Theorem</a></h3>
+<hr />
+<p><em>For any element \(g \in G\), we have \(g^{|G|} = 1\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: In the group \((\mathbb{Z}/7\mathbb{Z})^{\times}\), for any element \(g\), such as \(g = 3 + 7\mathbb{Z}\), we have \(3^6 \equiv 1 \bmod 7\). This holds for any \(g \in (\mathbb{Z}/7\mathbb{Z})^{\times}\) because the order of the group is 6. Thus, \(g^{|G|} = 1\) is satisfied.</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html" class="active"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
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+                        <h1 id="polynomials"><a class="header" href="#polynomials">Polynomials</a></h1>
+<p>Let \(K\) be a field.</p>
+<h3 id="definition-polynomial"><a class="header" href="#definition-polynomial">Definition: Polynomial</a></h3>
+<p><em>A univariate polynomial over a commutative ring \(R\) with unity \(1\) is an expression of the form \(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where \(x\) is a variable and coefficients \(a_0, \ldots, a_n\) belong to \(R\). The set of all polynomials over \(R\) in the variable \(x\) is denoted as \(R[x]\).</em></p>
+<p><strong>Example:</strong> In \(\mathbb{Z}[x]\), we have polynomials such as \(2x^3 + x + 1\), \(x\), and \(1\). In \(\mathbb{R}[x]\), we have polynomials like \(\pi x^2 - \sqrt{2}x + e\).</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>/// Polynomial struct representing a polynomial over Field.
+#[derive(Debug, Clone, PartialEq)]
+pub struct Polynomial&lt;F: Field&gt; {
+    pub poly: Vec&lt;F&gt;,
+    pub var: String,
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-degree"><a class="header" href="#definition-degree">Definition: Degree</a></h3>
+<p><em>The degree of a non-zero polynomial \(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), denoted as \(\deg f\), is the largest integer \(n\) such that \(a_n \neq 0\). The zero polynomial is defined to have degree \(-1\).</em></p>
+<p><strong>Example</strong></p>
+<ul>
+<li>\(\deg(2x^3 + x + 1) = 3\)</li>
+<li>\(\deg(x) = 1\)</li>
+<li>\(\deg(1) = 0\)</li>
+<li>\(\deg(0) = -1\)</li>
+</ul>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Polynomial&lt;F&gt; {
+    /// Removes trailing zeroes from a polynomial's coefficients.
+    fn trim_trailing_zeros(poly: Vec&lt;F&gt;) -&gt; Vec&lt;F&gt; {
+        let mut trimmed = poly;
+        while trimmed.last() == Some(&amp;F::zero(None)) {
+            trimmed.pop();
+        }
+        trimmed
+    }
+
+    /// Returns the degree of the polynomial.
+    pub fn degree(&amp;self) -&gt; isize {
+        let trimmed = Self::trim_trailing_zeros(self.poly.clone());
+        if trimmed.is_empty() {
+            -1
+        } else {
+            (trimmed.len() - 1) as isize
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="sum-of-polynomials"><a class="header" href="#sum-of-polynomials">Sum of Polynomials</a></h3>
+<p><em>For polynomials \(f(x) = \sum_{i=0}^n a_i x^i\) and \(g(x) = \sum_{i=0}^m b_i x^i\), their sum is defined as: \((f + g)(x) = \sum_{i=0}^{\max(n,m)} (a_i + b_i) x^i\), where we set \(a_i = 0\) for \(i &gt; n\) and \(b_i = 0\) for \(i &gt; m\).</em></p>
+<p><strong>Example:</strong> Let \(f(x) = 2x^2 + 3x + 1\) and \(g(x) = x^3 - x + 4\). Then, \((f + g)(x) = x^3 + 2x^2 + 2x + 5\)</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Add for Polynomial&lt;F&gt; {
+    type Output = Self;
+
+    fn add(self, other: Self) -&gt; Polynomial&lt;F&gt; {
+        let max_len = std::cmp::max(self.poly.len(), other.poly.len());
+        let mut result = Vec::with_capacity(max_len);
+
+        let zero = F::zero(None);
+
+        for i in 0..max_len {
+            let a = self.poly.get(i).unwrap_or(&amp;zero);
+            let b = other.poly.get(i).unwrap_or(&amp;zero);
+            result.push(a.clone() + b.clone());
+        }
+        Polynomial {
+            poly: Self::trim_trailing_zeros(result),
+            var: self.var.clone(),
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="product-of-polynomials"><a class="header" href="#product-of-polynomials">Product of polynomials</a></h3>
+<p><em>For polynomials \(f(x) = \sum_{i=0}^n a_i x^i\) and \(g(x) = \sum_{j=0}^m b_j x^j\), their product is defined as: \((fg)(x) = \sum_{k=0}^{n+m} c_k x^k\), where \(c_k = \sum_{i+j=k} a_i b_j\)</em></p>
+<p><strong>Example:</strong> Let \(f(x) = x + 1\) and \(g(x) = x^2 - 1\). Then, \((fg)(x) = x^3 + x^2 - x - 1\)</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Mul for Polynomial&lt;F&gt; {
+    type Output = Self;
+
+    /// Multiplication of two polynomials.
+    fn mul(self, other: Polynomial&lt;F&gt;) -&gt; Polynomial&lt;F&gt; {
+        let mut result = vec![F::zero(None); self.degree() as usize + other.degree() as usize + 1];
+
+        for (i, a) in self.poly.iter().enumerate() {
+            for (j, b) in other.poly.iter().enumerate() {
+                result[i + j] = result[i + j].clone() + (a.clone() * b.clone());
+            }
+        }
+        Polynomial {
+            poly: Self::trim_trailing_zeros(result),
+            var: self.var.clone(),
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="lemma"><a class="header" href="#lemma">Lemma</a></h3>
+<p><em>Let \(f, g \in K[x]\) be non-zero polynomials. Then, \(\deg(fg) = \deg f + \deg g\).</em></p>
+<p><strong>Example:</strong> Let \(f(x) = x^2 + 1\) and \(g(x) = x^3 - x\) in \(\mathbb{R}[x]\). Then, \(\deg(fg) = \deg(x^5 - x^3 + x^2 + 1) = 5 = 2 + 3 = \deg f + \deg g\)</p>
+<p>We can also define division in the polynomial ring \(K[x]\).</p>
+<h3 id="theorem"><a class="header" href="#theorem">Theorem</a></h3>
+<p><em>Let \(f, g \in K[x]\), with \(g \neq 0\). There exist unique polynomials \(q, r \in K[x]\) that satisfy \(f = qg + r\) and either \(\deg r &lt; \deg g\) or \(r = 0\).</em></p>
+<p><strong>Proof</strong>
+TBD</p>
+<p>\(q\) is called the quotient of \(f\) divided by \(g\), and \(r\) is called the remainder; we write \(r = f \bmod g\).</p>
+<p><strong>Example:</strong> In \(\mathbb{R}[x]\), let \(f(x) = x^3 + 2x^2 - x + 3\) and \(g(x) = x^2 + 1\).  Then \(f = qg + r\) where \(q(x) = x + 2\) and \(r(x) = -3x + 1\).</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Div for Polynomial&lt;F&gt; {
+    type Output = Self;
+
+    /// Division of two polynomials, returns quotient.
+    fn div(self, other: Polynomial&lt;F&gt;) -&gt; Polynomial&lt;F&gt; {
+        let mut remainder_coeffs = Self::trim_trailing_zeros(self.poly.clone());
+        let divisor_coeffs = Self::trim_trailing_zeros(other.poly.clone());
+        let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse();
+
+        let mut quotient =
+            vec![F::zero(None); self.degree() as usize - other.degree() as usize + 1];
+
+        let mut i = 0_i32;
+        while remainder_coeffs.len() &gt;= divisor_coeffs.len() {
+            let lead_term = remainder_coeffs.last().unwrap().clone() * divisor_lead_inv.clone();
+            let deg_diff = remainder_coeffs.len() - divisor_coeffs.len();
+            quotient[deg_diff] = lead_term.clone();
+
+            for i in 0..divisor_coeffs.len() {
+                remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i].clone()
+                    - (lead_term.clone() * divisor_coeffs[i].clone());
+            }
+            remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs);
+            i += 1;
+        }
+
+        Polynomial {
+            poly: Self::trim_trailing_zeros(quotient),
+            var: self.var.clone(),
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="corollary"><a class="header" href="#corollary">Corollary</a></h3>
+<p><em>Let \(f \in K[x]\) be a non-zero polynomial, and \(a \in K\) such that \(f(a) = 0\). Then, there exists a polynomial \(q \in K[x]\) such that \(f(x) = (x - a)q(x)\). In other words, \((x - a)\) is a factor of \(f(x)\).</em></p>
+<p><strong>Example:</strong> Let \(f(x) = x^2 + 1 \in (\mathbb{Z}/2\mathbb{Z})[x]\). We have \(f(1) = 1^2 + 1 = 0\) in \(\mathbb{Z}/2\mathbb{Z}\), and indeed: \(x^2 + 1 = (x - 1)^2 = x^2 - 2x + 1 = x^2 + 1\) in \((\mathbb{Z}/2\mathbb{Z})[x]\)</p>
+<h3 id="theorem-lagrange-interpolation"><a class="header" href="#theorem-lagrange-interpolation">Theorem: Lagrange Interpolation</a></h3>
+<p><em>A \(n\)-degre polynomial \(P(x)\) that goes through different \(n + 1\) points \(\{(x_1, y_1), (x_2, y_2), \cdots (x_{n + 1}, y_{n + 1})\}\) is uniquely represented as follows:</em></p>
+<p>\begin{align*}
+P(x) = \sum^{n+1}_{i=1} y_i \frac{f_i(x)}{f_i(x_i)}
+\end{align*}</p>
+<p><em>, where \(f_i(x) = \prod_{k \neq i} (x - x_k)\)</em></p>
+<p><strong>Proof</strong> TBD</p>
+<p>For example, the quadratic polynomial that goes through \(\{(1, 0), (2, 3), (3, 8)\}\) is as follows:</p>
+<p>\begin{align*}
+P(x) = 0 \frac{(x - 2)(x - 3)}{(1 - 2) (1 - 3)} + 3 \frac{(x - 1)(x - 3)}{(2 - 1) (2 - 3)} + 8 \frac{(x - 1)(x - 2)}{(3 - 1) (3 - 2)} = x^{2} - 1
+\end{align*}</p>
+<p>Note that Lagrange interpolation finds the lowest degree of interpolating polynomial for the given vector.</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>pub fn interpolate(x_values: &amp;[F], y_values: &amp;[F]) -&gt; Polynomial&lt;F&gt; {
+    let mut lagrange_polys = vec![];
+    let numerators = Polynomial::from_monomials(x_values);
+
+    for j in 0..x_values.len() {
+        let mut denominator = F::one(None);
+        for i in 0..x_values.len() {
+            if i != j {
+                denominator = denominator * (x_values[j].clone() - x_values[i].clone());
+            }
+        }
+        let cur_poly = numerators
+            .clone()
+            .div(Polynomial::from_monomials(&amp;[x_values[j].clone()]).scalar_mul(&amp;denominator));
+        lagrange_polys.push(cur_poly);
+    }
+
+    let mut result = Polynomial {
+        poly: vec![],
+        var: "x".to_string(),
+    };
+    for (j, lagrange_poly) in lagrange_polys.iter().enumerate() {
+        result = result + lagrange_poly.scalar_mul(&amp;y_values[j]);
+    }
+    result
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="proposition-homomorphisms-of-lagrange-interpolation"><a class="header" href="#proposition-homomorphisms-of-lagrange-interpolation">Proposition: Homomorphisms of Lagrange Interpolation</a></h3>
+<p><em>Let \(L(v)\) and \(L(w)\) be the polynomial resulting from Lagrange Interpolation on the output (\(y\)) vector \(v\) and \(w\) for the same inputs (\(x\)). Then, the following properties hold:</em></p>
+<ul>
+<li><em>Additivity: \(L(v + w) = L(v) + L(w)\) for any vectors \(v\) and \(w\)</em></li>
+<li><em>Scalar multiplication: \(L(\gamma v) = \gamma L(v)\) for any scalar \(\gamma\) and vector \(v\)</em></li>
+</ul>
+<p><strong>Proof</strong></p>
+<p>Let \(v = (v_1, \ldots, v_n)\) and \(w = (w_1, \ldots, w_n)\) be vectors, and \(x_1, \ldots, x_n\) be the interpolation points.</p>
+<p>\begin{align*}
+L(v + w) &amp;= \sum_{i=1}^n (v_i + w_i) \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \sum_{i=1}^n v_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} + \sum_{i=1}^n w_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = L(v) + L(w) \\
+L(\gamma v) &amp;= \sum_{i=1}^n (\gamma v_i) \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \gamma \sum_{i=1}^n v_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \gamma L(v)
+\end{align*}</p>
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html" class="active"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
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+                        <h1 id="galois-field"><a class="header" href="#galois-field">Galois Field</a></h1>
+<p>We will now discuss the construction of finite fields with \(p^n\) elements, where \(p\) is a prime number and \(n\) is a positive integer. These fields are also known as Galois fields, denoted as \(GF(p^n)\). It is evident that \(\mathbb{Z}/p\mathbb{Z}\) is isomorphic to \(GF(p)\).</p>
+<h3 id="definition-irreducible-polynomial"><a class="header" href="#definition-irreducible-polynomial">Definition: Irreducible Polynomial</a></h3>
+<hr />
+<p><em>A polynomial \(f \in (\mathbb{Z}/p\mathbb{Z})[X]\) of degree \(n\) is called irreducible over \(\mathbb{Z}/p\mathbb{Z}\) if it cannot be factored as a product of two polynomials of lower degree in \((\mathbb{Z}/p\mathbb{Z})[X]\).</em></p>
+<hr />
+<p><strong>Example:</strong> In \((\mathbb{Z}/2\mathbb{Z})[X]\):</p>
+<ul>
+<li>\(X^2 + X + 1\) is irreducible</li>
+<li>\(X^2 + 1 = (X + 1)^2\) is reducible</li>
+</ul>
+<p>To construct \(GF(p^n)\), we use an irreducible polynomial of degree \(n\) over \(\mathbb{Z}/p\mathbb{Z}\).</p>
+<h3 id="definition-residue-class-modulo-a-polynomial"><a class="header" href="#definition-residue-class-modulo-a-polynomial">Definition: Residue Class modulo a Polynomial</a></h3>
+<hr />
+<p><em>For \(f, g \in (\mathbb{Z}/p\mathbb{Z})[X]\), the residue class of \(g \bmod f\) is the set of all polynomials \(h \in (\mathbb{Z}/p\mathbb{Z})[X]\) such that \(h \equiv g \pmod{f}\). This class is denoted as:</em></p>
+<p>\[ g + f(\mathbb{Z}/p\mathbb{Z})[X] = \{g + hf : h \in (\mathbb{Z}/p\mathbb{Z})[X]\} \]</p>
+<hr />
+<p><strong>Example:</strong> In \((\mathbb{Z}/2\mathbb{Z})[X]\), with \(f(X) = X^2 + X + 1\), the residue classes \(\bmod f\) are:</p>
+<ul>
+<li>\(0 + f(\mathbb{Z}/2\mathbb{Z})[X] = \{0, X^2 + X + 1, X^2 + X, X^2 + 1, X^2, X + 1, X, 1\}\)</li>
+<li>\(1 + f(\mathbb{Z}/2\mathbb{Z})[X] = \{1, X^2 + X, X^2 + 1, X^2, X + 1, X, 0, X^2 + X + 1\}\)</li>
+<li>\(X + f(\mathbb{Z}/2\mathbb{Z})[X] = \{X, X^2 + 1, X^2, X^2 + X + 1, X + 1, 1, X^2 + X, 0\}\)</li>
+<li>\((X + 1) + f(\mathbb{Z}/2\mathbb{Z})[X] = \{X + 1, X^2, X^2 + X + 1, X^2 + X, 1, 0, X^2 + 1, X\}\)</li>
+</ul>
+<p>These four residue classes form \(GF(4)\).</p>
+<h3 id="theorem"><a class="header" href="#theorem">Theorem:</a></h3>
+<hr />
+<p><em>If \(f \in (\mathbb{Z}/p\mathbb{Z})[X]\) is an irreducible polynomial of degree \(n\), then the residue ring \((\mathbb{Z}/p\mathbb{Z})[X]/(f)\) is a field with \(p^n\) elements, isomorphic to \(GF(p^n)\).</em></p>
+<hr />
+<p><strong>Proof</strong> (Outline) The irreducibility of \(f\) ensures that \((f)\) is a maximal ideal in \((\mathbb{Z}/p\mathbb{Z})[X]\), making the quotient ring a field. The number of elements is \(p^n\) because there are \(p^n\) polynomials of degree less than \(n\) over \(\mathbb{Z}/p\mathbb{Z}\).</p>
+<p>This construction allows us to represent elements of \(GF(p^n)\) as polynomials of degree less than \(n\) over \(\mathbb{Z}/p\mathbb{Z}\). Addition is performed coefficient-wise modulo \(p\), while multiplication is performed modulo the irreducible polynomial \(f\).</p>
+<p><strong>Example:</strong> To construct \(GF(8)\), we can use the irreducible polynomial \(f(X) = X^3 + X + 1\) over \(\mathbb{Z}/2\mathbb{Z}\). The elements of \(GF(8)\) are represented by:
+[ \{0, 1, X, X+1, X^2, X^2+1, X^2+X, X^2+X+1\} ]
+For instance, multiplication in \(GF(8)\):
+[ (X^2 + 1)(X + 1) = X^3 + X^2 + X + 1 \equiv X^2 \pmod{X^3 + X + 1} ]</p>
+<h3 id="lemma-schwartz---zippel-lemma"><a class="header" href="#lemma-schwartz---zippel-lemma">Lemma: Schwartz - Zippel Lemma</a></h3>
+<hr />
+<p><em>Let \(\mathbb{F}\) be a field and \(P: F^m \rightarrow \mathbb{F}\) and \(Q: \mathbb{F}^m \rightarrow \mathbb{F}\) be two distinct multivariate polynomials of total degree at most \(n\). For any finite subset \(\mathbb{S} \subseteq \mathbb{F}\), we have:</em></p>
+<p>\begin{align}
+Pr_{u \sim \mathbb{S}^{m}}[P(u) = Q(u)] \leq \frac{n}{|\mathbb{S}|}
+\end{align}</p>
+<p><em>, where \(u\) is drawn uniformly at random from \(\mathbb{S}^m\).</em></p>
+<hr />
+<p><strong>Proof</strong> TBD</p>
+<p>This lemma states that if \(\mathbb{S}\) is sufficiently large and \(n\) is relatively small, the probability that the two different polynomials return the same value for a randomly chosen input is negligibly small. In other words, if we observe \(P(u) = Q(u)\) for a random input \(u\), we can conclude with high probability that \(P\) and \(Q\) are identical polynomials.</p>
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+                        <h1 id="elliptic-curve"><a class="header" href="#elliptic-curve">Elliptic curve</a></h1>
+<h3 id="definition-elliptic-curve"><a class="header" href="#definition-elliptic-curve">Definition: Elliptic Curve</a></h3>
+<hr />
+<p><em>An elliptic curve \(E\) over a finite field \(\mathbb{F}_{p}\) is defined by the equation:</em></p>
+<p>\begin{equation}
+y^2 = x^3 + a x + b
+\end{equation}</p>
+<p><em>, where \(a, b \in \mathbb{F}_{p}\) and the discriminant \(\Delta_{E} = 4a^3 + 27b^2 \neq 0\).</em></p>
+<hr />
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>use crate::modules::field::Field;
+
+#[derive(Debug, Clone, PartialEq)]
+pub struct EllipticCurve&lt;F: Field&gt; {
+    pub a: F,
+    pub b: F,
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-mathbbf_p-rational-point"><a class="header" href="#definition-mathbbf_p-rational-point">Definition: \(\mathbb{F}_{p}\)-Rational Point</a></h3>
+<hr />
+<p><em>An \(\mathbb{F}_{p}\)-rational point on an elliptic curve \(E\) is a point \((x, y)\) where both \(x\) and \(y\) are elements of \(\mathbb{F}_{p}\) and satisfy the curve equation, or the point at infinity \(\mathcal{O}\).</em></p>
+<hr />
+<p>Example: For \(E: y^2 = x^3 + 3x + 4\) over \(\mathbb{F}_7\), the \(\mathbb{F}_{7}\)-rational points are:
+\(\{(0, 2), (0, 5), (1, 1), (1, 6), (2, 2), (2, 5), (5, 1), (5, 6), \mathcal{O}\}\)</p>
+<h3 id="definition-the-point-at-infinity"><a class="header" href="#definition-the-point-at-infinity">Definition: The point at infinity</a></h3>
+<hr />
+<p>The point at infinity denoted \(\mathcal{O}\), is a special point on the elliptic curve that serves as the identity element for the group operation. It can be visualized as the point where all vertical lines on the curve meet.</p>
+<hr />
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>#[derive(Debug, Clone, PartialEq)]
+pub struct EllipticCurvePoint&lt;F: Field&gt; {
+    pub x: Option&lt;F&gt;,
+    pub y: Option&lt;F&gt;,
+    pub curve: EllipticCurve&lt;F&gt;,
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-addition-on-elliptic-curve"><a class="header" href="#definition-addition-on-elliptic-curve">Definition: Addition on Elliptic Curve</a></h3>
+<hr />
+<p><em>For an elliptic curve \(E: y^2 = x^3 + ax + b\), the addition of points \(P\) and \(Q\) to get \(R = P + Q\) is defined as follows:</em></p>
+<ul>
+<li>
+<p><em>If \(P = \mathcal{O}\), \(R = Q\).</em></p>
+</li>
+<li>
+<p><em>If \(Q = \mathcal{O}\), \(R = P\).</em></p>
+</li>
+<li>
+<p><em>Otherwise, let \(P = (x_P, y_P), Q = (x_Q, y_Q)\). Then:</em></p>
+<ul>
+<li><em>If \(y_P = -y_Q\), \(R = \mathcal{O}\)</em></li>
+<li><em>If \(y_P \neq -y_Q\), \(R = (x_R = \lambda^2 - x_P - x_Q, y_R = \lambda(x_P - x_R) - y_P)\), where \(\lambda = \)</em> \( \begin{cases}
+\frac{y_P - y_Q}{x_P - x_Q} \quad &amp;\hbox{If } (x_P \neq x_Q) \\
+\frac{3^{2}_{P} + a}{2y_P} \quad &amp;\hbox{Otherwise}
+\end{cases}\)</li>
+</ul>
+</li>
+</ul>
+<hr />
+<p>Example: On \(E: y^2 = x^3 + 2x + 3\) over \(\mathbb{F}_{7}\), let \(P = (5, 1)\) and \(Q = (4, 4)\). Then, \(P + Q = (0, 5)\), where \(\lambda = \frac{1 - 4}{5 - 4} \equiv 4 \bmod 7\).</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; EllipticCurvePoint&lt;F&gt; {
+    fn new(x: F, y: F, curve: EllipticCurve&lt;F&gt;) -&gt; Self {
+        EllipticCurvePoint {
+            x: Some(x),
+            y: Some(y),
+            curve: curve,
+        }
+    }
+
+    pub fn point_at_infinity(curve: EllipticCurve&lt;F&gt;) -&gt; Self {
+        EllipticCurvePoint {
+            x: None,
+            y: None,
+            curve: curve,
+        }
+    }
+
+    pub fn is_point_at_infinity(&amp;self) -&gt; bool {
+        self.x.is_none() || self.y.is_none()
+    }
+
+    pub fn line_slope(&amp;self, other: Self) -&gt; F {
+        let x1 = self.x.clone().unwrap();
+        let y1 = self.y.clone().unwrap();
+        let x2 = other.x.clone().unwrap();
+        let y2 = other.y.clone().unwrap();
+
+        if self.x.clone() == other.x.clone() {
+            ((x1.clone() * x1.clone()) * (3_i64) + self.curve.a.clone()) / (y1.clone() * (2_i64))
+        } else {
+            (y2.clone() - y1.clone()) / (x2.clone() - x1.clone())
+        }
+    }
+}
+
+impl&lt;F: Field&gt; Add for EllipticCurvePoint&lt;F&gt; {
+    type Output = Self;
+
+    fn add(self, other: Self) -&gt; Self {
+        if self.is_point_at_infinity() {
+            return other;
+        }
+        if other.is_point_at_infinity() {
+            return self;
+        }
+
+        let m = self.line_slope(other.clone());
+
+        if self.x == other.x {
+            if self.y != other.y {
+                return EllipticCurvePoint::point_at_infinity(self.curve.clone());
+            } else {
+                let x1 = self.x.clone().unwrap();
+                let y1 = self.y.clone().unwrap();
+
+                let x3 = m.clone() * m.clone() - x1.clone() - x1.clone();
+                let y3 = m * (x1 - x3.clone()) - y1;
+
+                return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+            }
+        } else {
+            let x1 = self.x.clone().unwrap();
+            let y1 = self.y.clone().unwrap();
+            let x2 = other.x.clone().unwrap();
+            let x3 = m.clone() * m.clone() - x1.clone() - x2.clone();
+            let y3 = m * (x1 - x3.clone()) - y1;
+
+            return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-mordell-weil-group"><a class="header" href="#definition-mordell-weil-group">Definition: Mordell-Weil Group</a></h3>
+<hr />
+<p><em>The Mordell-Weil group of an elliptic curve \(E\) is the group of rational points on \(E\) under the addition operation defined above.</em></p>
+<hr />
+<p>Example: For \(E: y^2 = x^3 + x + 6\) over \(\mathbb{F}_{11}\), the Mordell-Weil group is the set of all \(\mathbb{F}_{11}\)-rational points on \(E\) with the elliptic curve addition operation.</p>
+<h3 id="definition-group-order"><a class="header" href="#definition-group-order">Definition: Group Order</a></h3>
+<hr />
+<p><em>The group order of an elliptic curve \(E\) over \(\mathbb{F}_{p}\), denoted \(\#E(\mathbb{F}_{p})\), is the number of \(\mathbb{F}_{p}\)-rational points on \(E\), including the point at infinity.</em></p>
+<hr />
+<p>Example: For \(E: y^2 = x^3 + x + 6\) over \(\mathbb{F}_{11}\), \(\#E(\mathbb{F}_{11}) = 13\).</p>
+<h3 id="theorem-hasse-weil"><a class="header" href="#theorem-hasse-weil">Theorem: Hasse-Weil</a></h3>
+<hr />
+<p><em>Let \(\#E(\mathbb{F}_{p})\) be the group order of the elliptic curve \(E\) over \(\mathbb{F}_{p}\). Then:</em></p>
+<p>\begin{equation}
+p + 1 - 2 \sqrt{p} \leq \#E \leq p + 1 + 2 \sqrt{p}
+\end{equation}</p>
+<hr />
+<p><strong>Example:</strong> For an elliptic curve over \(\mathbb{F}_{23}\), the Hasse-Weil theorem guarantees that:</p>
+<p>\begin{equation*}
+23 + 1 - 2 \sqrt{23} \simeq 14.42 \#E(\mathbb{F}_{23}) \geq 23 + 1 + 2 \sqrt{23} \simeq 33.58
+\end{equation*}</p>
+<h3 id="definition-point-order"><a class="header" href="#definition-point-order">Definition: Point Order</a></h3>
+<p>The order of a point \(P\) on an elliptic curve is the smallest positive integer \(n\) such that \(nP = \mathcal{O}\) (where \(nP\) denotes \(P\) added to itself \(n\) times). We also denote the set of points of order \(n\), also called \textit{torsion} group, by</p>
+<p>\begin{equation}
+E[n] = \{P \in E: [n]P = \mathcal{O}\}
+\end{equation}</p>
+<p>Example: On \(E: y^2 = x^3 + 2x + 2\) over \(\mathbb{F}_{17}\), the point \(P = (5, 1)\) has order 18 because \(18P = \mathcal{O}\), and no smaller positive multiple of \(P\) equals \(\mathcal{O}\).</p>
+<p>The intuitive view is that if you continue to add points to themselves (doubling, tripling, etc.), the lines drawn between the prior point and the next point will eventually become more vertical. When the line becomes vertical, it does not intersect the elliptic curve at any finite point. In elliptic curve geometry, a vertical line is considered to "intersect" the curve at a special place called the "point at infinity," This point is like a north pole in geographic terms: no matter which direction you go, if the line becomes vertical (reaching infinitely high), it converges to this point.</p>
+<h3 id="definition-field-extension"><a class="header" href="#definition-field-extension">Definition: Field Extension</a></h3>
+<hr />
+<p><em>Let \(F\) and \(L\) be fields. If \(F \subseteq L\) and the operations of \(F\) are the same as those of \(L\), we call \(L\) a field extension of \(F\). This is denoted as \(L/F\).</em></p>
+<hr />
+<p>A field extension \(L/F\) naturally gives \(L\) the structure of a vector space over \(F\). The dimension of this vector space is called the degree of the extension.</p>
+<p><strong>Examples:</strong></p>
+<ul>
+<li>\(\mathbb{C}/\mathbb{R}\) is a field extension with basis \(\{1, i\}\) and degree 2.</li>
+<li>\(\mathbb{R}/\mathbb{Q}\) is an infinite degree extension.</li>
+<li>\(\mathbb{Q}(\sqrt{2})/\mathbb{Q}\) is a degree 2 extension with basis \(\{1, \sqrt{2}\}\).</li>
+</ul>
+<h3 id="definition-algebraic-extension"><a class="header" href="#definition-algebraic-extension">Definition: Algebraic Extension</a></h3>
+<hr />
+<p><em>A field extension \(L/K\) is called algebraic if every element \(\alpha \in L\) is algebraic over \(K\), i.e., \(\alpha\) is the root of some non-zero polynomial with coefficients in \(K\).</em></p>
+<p><em>For an algebraic element \(\alpha \in L\) over \(K\), we denote by \(K(\alpha)\) the smallest field containing both \(K\) and \(\alpha\).</em></p>
+<hr />
+<p><strong>Example:</strong></p>
+<ul>
+<li>\(\mathbb{C}/\mathbb{R}\) is algebraic: any \(z = a + bi \in \mathbb{C}\) is a root of \(x^2 - 2ax + (a^2 + b^2) \in \mathbb{R}[x]\).</li>
+<li>\(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}\) is algebraic: \(\sqrt[3]{2}\) is a root of \(x^3 - 2 \in \mathbb{Q}[x]\).</li>
+<li>\(\mathbb{R}/\mathbb{Q}\) is not algebraic (a field extension that is not algebraic is called \textit{transcendental}).</li>
+</ul>
+<h3 id="definition-field-of-rational-functions"><a class="header" href="#definition-field-of-rational-functions">Definition: Field of Rational Functions</a></h3>
+<hr />
+<p><em>Let \(K\) be a field and \(X\) be indeterminate. The field of rational functions over \(K\), denoted \(K(X)\), is defined as:</em></p>
+<p>\begin{equation*}
+K(X) = \left\{ \frac{f(X)}{g(X)} \middle|\ f(X), g(X) \in K[X], g(X) \neq 0 \right\}
+\end{equation*}</p>
+<p><em>where \(K[X]\) is the ring of polynomials in \(X\) with coefficients in \(K\).</em></p>
+<hr />
+<p>\(K(X)\) can be viewed as the field obtained by adjoining a letter \(X\) to \(K\). This construction generalizes to multiple variables, e.g., \(K(X,Y)\).</p>
+<p>The concept of a function field naturally arises in the context of algebraic curves, particularly elliptic curves. Intuitively, the function field encapsulates the algebraic structure of rational functions on the curve.</p>
+<p>We first construct the coordinate ring for an elliptic curve \(E: y^2 = x^3 + ax + b\). Consider functions \(X: E \to K\) and \(Y: E \to K\) that extract the \(x\) and \(y\) coordinates, respectively, from an arbitrary point \(P \in E\). These functions generate the polynomial ring \(K[X,Y]\), subject to the relation \(Y^2 = X^3 + aX + b\).</p>
+<p>To put it simply, the function field is a field that consists of all functions that determine the value based on the point on the curve.</p>
+<h3 id="definition-coordinate-ring-of-an-elliptic-curve"><a class="header" href="#definition-coordinate-ring-of-an-elliptic-curve">Definition: Coordinate Ring of an Elliptic Curve</a></h3>
+<hr />
+<p><em>The <strong>coordinate ring</strong> of an elliptic curve \(E: y^2 = x^3 + ax + b\) over a field \(K\) is defined as:</em>
+\begin{equation}
+K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)
+\end{equation}</p>
+<hr />
+<p>In other words, we can view \(K[E]\) as a ring representing all polynomial functions on \(E\). Recall that \(K[X, Y]\) is the polynomial ring in two variables \(X\) and \(Y\) over the field K, meaning that it contains all polynomials in \(X\) and \(Y\) with coefficients from \(K\). Then, the notation \(K[X, Y]/(Y^2 - X^3 - aX - b)\) denotes the quotient ring obtained by taking \(K[X, Y]\) and "modding out" by the ideal \((Y^2 - X^3 - aX - b)\).</p>
+<p><strong>Example</strong> For example, for an elliptic curve \(E: y^2 = x^3 - x\) over \(\mathbb{Q}\), some elements of the coordinate ring \(\mathbb{Q}[E]\) include:</p>
+<ul>
+<li>Constants: \(3, -2, \frac{1}{7}, \ldots\)</li>
+<li>Linear functions: \(X, Y, 2X+3Y, \ldots\)</li>
+<li>Quadratic functions: \(X^2, XY, Y^2 (= X^3 - X), \ldots\)</li>
+<li>Higher-degree functions: \(X^3, X^2Y, XY^2 (= X^4 - X^2), \ldots\)</li>
+</ul>
+<p>Then, the function field is defined as follows:</p>
+<h3 id="definition-function-field-of-an-elliptic-curve"><a class="header" href="#definition-function-field-of-an-elliptic-curve">Definition: Function Field of an Elliptic Curve</a></h3>
+<hr />
+<p><em>Let \(E: y^2 = x^3 + ax + b\) be an elliptic curve over a field \(K\). The <strong>function field</strong> of \(E\), denoted \(K(E)\), is defined as:</em>
+\begin{equation}
+K(E) = \left\{\frac{f}{g} ,\middle|, f, g \in K[E], g \neq 0 \right\}
+\end{equation}
+<em>where \(K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)\) is the coordinate ring of \(E\).</em></p>
+<hr />
+<p>\(K(E)\) can be viewed as the field of all rational functions on \(E\).</p>
+<h3 id="definition-zero-of-a-function"><a class="header" href="#definition-zero-of-a-function">Definition: Zero of a Function</a></h3>
+<hr />
+<p><em>Let \(h \in K(E)\) be a non-zero function. A point \(P \in E\) is called a <strong>zero</strong> of \(h\) if \(h(P) = 0\).</em></p>
+<hr />
+<h3 id="definition-pole-of-a-function"><a class="header" href="#definition-pole-of-a-function">Definition: Pole of a Function</a></h3>
+<hr />
+<p><em>Let \(h \in K(E)\) be a non-zero function. A point \(P \in E\) is called a <strong>pole</strong> of \(h\) if \(h\) is not defined at \(P\) or, equivalently if \(1/h\) has a zero at \(P\).</em></p>
+<hr />
+<p><strong>Example</strong> Consider the elliptic curve \(E: Y^2 = X^3 - X\) over a field \(K\) of characteristic \(\neq 2, 3\). Let \(P_{-1} = (-1, 0)\), \(P_0 = (0, 0)\), and \(P_1 = (1, 0)\) be the points where \(Y = 0\).</p>
+<ul>
+<li>The function \(Y \in K(E)\) has three simple zeros: \(P_{-1}\), \(P_0\), and \(P_1\).</li>
+<li>The function \(X \in K(E)\) has a double zero at \(P_0\) (since \(P_0 = -P_0\)).</li>
+<li>The function \(X - 1 \in K(E)\) has a simple zero at \(P_1\).</li>
+<li>The function \(X^2 - 1 \in K(E)\) has two simple zeros at \(P_{-1}\) and \(P_1\).</li>
+<li>The function \(\frac{Y}{X} \in K(E)\) has a simple zero at \(P_{-1}\) and a simple pole at \(P_0\).</li>
+</ul>
+<p>An important property of functions in \(K(E)\) is that they have the same number of zeros and poles when counted with multiplicity. This is a consequence of a more general result known as the Degree-Genus Formula.</p>
+<h3 id="theorem-degree-genus-formula-for-elliptic-curves"><a class="header" href="#theorem-degree-genus-formula-for-elliptic-curves">Theorem: Degree-Genus Formula for Elliptic Curves</a></h3>
+<hr />
+<p><em>Let \(f \in K(E)\) be a non-zero rational function on an elliptic curve \(E\). Then:</em></p>
+<p>\begin{equation}
+\sum_{P \in E} ord_{P(f)} = 0
+\end{equation}</p>
+<p><em>, where \(ord_{P(f)}\) denotes the order of \(f\) at \(P\), which is positive for zeros and negative for poles.</em></p>
+<hr />
+<p>This theorem implies that the total number of zeros (counting multiplicity) equals the total number of poles for any non-zero function in \(K(E)\).</p>
+<p>We now introduce a powerful tool for analyzing functions on elliptic curves: the concept of divisors.</p>
+<h3 id="definition-divisor-of-a-function-on-an-elliptic-curve"><a class="header" href="#definition-divisor-of-a-function-on-an-elliptic-curve">Definition: Divisor of a Function on an Elliptic Curve</a></h3>
+<hr />
+<p><em>Let \(E: Y^2 = X^3 + AX + B\) be an elliptic curve over a field \(K\), and let \(f \in K(E)\) be a non-zero rational function on \(E\). The <strong>divisor</strong> of \(f\), denoted \(div(f)\), is defined as:</em>
+\begin{equation}
+div(f) = \sum_{P \in E} ord_P(f) [P]
+\end{equation}
+<em>, where \(ord_P(f)\) is the order of \(f\) at \(P\) (positive for zeros, negative for poles), and the sum is taken over all points \(P \in E\), including the point at infinity. This sum has only finitely many non-zero terms.</em></p>
+<hr />
+<p>Note that this \(\sum_{P \in E}\) is a symbolic summation, and we do not calculate the concrete value of a divisor.</p>
+<p><strong>Example</strong> Consider the elliptic curve \(E: Y^2 = X^3 - X\) over \(\mathbb{Q}\).</p>
+<ul>
+<li>For \(f = X\), we have \(div(X) = 2[(0,0)] - 2[\mathcal{O}]\).</li>
+<li>For \(g = Y\), we have \(div(Y) = [(1,0)] + [(0,0)] + [(-1,0)] - 3[\mathcal{O}]\).</li>
+<li>For \(h = \frac{X-1}{Y}\), we have \(div(h) = [(1,0)] - [(-1,0)]\).</li>
+</ul>
+<p>The concept of divisors can be extended to the elliptic curve itself:</p>
+<h3 id="definition-divisor-on-an-elliptic-curve"><a class="header" href="#definition-divisor-on-an-elliptic-curve">Definition: Divisor on an Elliptic Curve</a></h3>
+<hr />
+<p><em>A <strong>divisor</strong> \(D\) on an elliptic curve \(E\) is a formal sum</em>
+\begin{equation}
+D = \sum_{P \in E} n_P [P]
+\end{equation}
+<em>where \(n_P \in \mathbb{Z}\) and \(n_P = 0\) for all but finitely many \(P\).</em></p>
+<hr />
+<p><strong>Example</strong> On the curve \(E: Y^2 = X^3 - X\):</p>
+<ul>
+<li>\(D_1 = 3[(0,0)] - 2[(1,1)] - [(2,\sqrt{6})]\) is a divisor.</li>
+<li>\(D_2 = [(1,0)] + [(-1,0)] - 2[\mathcal{O}]\) is a divisor.</li>
+<li>\(D_3 = \sum_{P \in E[2]} [P] - 4[\mathcal{O}]\) is a divisor (where \(E[2]\) are the 2-torsion points).</li>
+</ul>
+<p>To quantify the properties of divisors, we introduce two important metrics:</p>
+<h3 id="definition-degree-of-a-divisor"><a class="header" href="#definition-degree-of-a-divisor">Definition: Degree of a Divisor</a></h3>
+<hr />
+<p><em>The <strong>degree</strong> of a divisor \(D = \sum_{P \in E} n_P [P]\) is defined as:</em>
+\begin{equation}
+deg(D) = \sum_{P \in E} n_P
+\end{equation}</p>
+<hr />
+<h3 id="definition-sum-of-a-divisor"><a class="header" href="#definition-sum-of-a-divisor">Definition: Sum of a Divisor</a></h3>
+<hr />
+<p><em>The <strong>sum</strong> of a divisor \(D = \sum_{P \in E} n_P [P]\) is defined as:</em>
+\begin{equation}
+Sum(D) = \sum_{P \in E} n_P P
+\end{equation}
+<em>where \(n_P P\) denotes the point addition of \(P\) to itself \(n_P\) times in the group law of \(E\).</em></p>
+<hr />
+<p><strong>Example</strong> For the divisors in the previous example:</p>
+<ul>
+<li>\(deg(D_1) = 3 - 2 - 1 = 0\)</li>
+<li>\(deg(D_2) = 1 + 1 - 2 = 0\)</li>
+<li>\(deg(D_3) = 4 - 4 = 0\)</li>
+<li>\(Sum(D_2) = (1,0) + (-1,0) - 2\mathcal{O} = \mathcal{O}\) (since \((1,0)\) and \((-1,0)\) are 2-torsion points)</li>
+</ul>
+<p>The following theorem characterizes divisors of functions and provides a criterion for when a divisor is the divisor of a function:</p>
+<h3 id="theorem"><a class="header" href="#theorem">Theorem</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve over a field \(K\).</em></p>
+<ul>
+<li><em>If \(f, f' \in K(E)\) are non-zero rational functions on \(E\) with \(div(f) = div(f')\), then there exists a non-zero constant \(c \in K^</em>\) such that \(f = cf'\).*</li>
+<li><em>A divisor \(D\) on \(E\) is the divisor of a rational function on \(E\) if and only if \(deg(D) = 0\) and \(Sum(D) = \mathcal{O}\).</em></li>
+</ul>
+<hr />
+<p><strong>Example</strong> On \(E: Y^2 = X^3 - X\):</p>
+<ul>
+<li>The function \(f = \frac{Y}{X}\) has \(div(f) = [(1,0)] + [(-1,0)] - 2[(0,0)]\). Note that \(deg(div(f)) = 0\) and \(Sum(div(f)) = (1,0) + (-1,0) - 2(0,0) = \mathcal{O}\).</li>
+<li>The divisor \(D = 2[(1,1)] - [(2,\sqrt{6})] - [(0,0)]\) has \(deg(D) = 0\), but \(Sum(D) \neq \mathcal{O}\). Therefore, \(D\) is not the divisor of any rational function on \(E\).</li>
+</ul>
+<h3 id="definition-pairing"><a class="header" href="#definition-pairing">Definition: Pairing</a></h3>
+<hr />
+<p><em>Let \(G_1\) and \(G_2\) be cyclic groups under addition, both of prime order \(p\), with generators \(P\) and \(Q\) respectively:</em></p>
+<p>\begin{align}
+G_1 &amp;= \{0, P, 2P, ..., (p-1)P\} \
+G_2 &amp;= \{0, Q, 2Q, ..., (p-1)Q\}
+\end{align}</p>
+<p><em>Let \(G_T\) be a cyclic group under multiplication, also of order \(p\). A pairing is a map \(e: G_1 \times G_2 \rightarrow G_T\) that satisfies the following bilinear property:</em></p>
+<p>\begin{equation}
+e(aP, bQ) = e(P, Q)^{ab}
+\end{equation} <em>for all \(a, b \in \mathbb{Z}_p\).</em></p>
+<hr />
+<p>Imagine \(G_1\) represents length, \(G_2\) represents width, and \(G_T\) represents area. The pairing function \(e\) is like calculating the area: If you double the length and triple the width, the area becomes six times larger: \(e(2P, 3Q) = e(P, Q)^{6}\)</p>
+<p>The Weil pairing is one of the bilinear pairings for elliptic curves. We begin with its formal definition.</p>
+<h3 id="definition-the-weil-pairing"><a class="header" href="#definition-the-weil-pairing">Definition: The Weil Pairing</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve and \(n\) be a positive integer. For points \(P, Q \in E[n]\), where \(E[n]\) denotes the \(n\)-torsion subgroup of \(E\), we define the Weil pairing \(e_n(P, Q)\) as follows:</em></p>
+<p>Let \(f_P\) and \(f_Q\) be rational functions on \(E\) satisfying:
+\begin{align}
+div(f_P) &amp;= n[P] - n[\mathcal{O}] \
+div(f_Q) &amp;= n[Q] - n[\mathcal{O}]
+\end{align}</p>
+<p><em>Then, for an arbitrary point \(S \in E\) such that \(S \notin \{\mathcal{O}, P, -Q, P-Q\}\), the Weil pairing is given by:</em></p>
+<p>\begin{equation}
+e_n(P, Q) = \frac{f_P(Q + S)}{f_P(S)} /\ \frac{f_Q(P - S)}{f_Q(-S)}
+\end{equation}</p>
+<hr />
+<p>We introduce a crucial theorem about a specific rational function on elliptic curves to construct the functions required for the Weil pairing.</p>
+<h3 id="theorem-1"><a class="header" href="#theorem-1">Theorem</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve over a field \(K\), and let \(P = (x_P, y_P)\) and \(Q = (x_Q, y_Q)\) be non-zero points on \(E\). Define \(\lambda\) as:</em></p>
+<p>\begin{equation}
+\lambda = \begin{cases}
+\hbox{slope of the line through \(P\) and \(Q\)} &amp;\quad \hbox{if \(P \neq Q\)} \\
+\hbox{slope of the tangent line to \(E\) at \(P\)} &amp;\quad \hbox{if \(P = Q\)} \\
+\infty &amp;\quad \hbox{if the line is vertical}
+\end{cases}
+\end{equation}</p>
+<p><em>Then, the function \(g_{P,Q}: E \to K\) defined by:</em>
+\begin{equation}
+g_{P,Q} = \begin{cases}
+\frac{y - y_P - \lambda(x - x_P)}{x + x_P + x_Q - \lambda^2} &amp;\quad \hbox{if } \lambda \neq \infty \\
+x - x_P &amp;\quad \hbox{if } \lambda = \infty
+\end{cases}
+\end{equation} <em>has the following divisor:</em></p>
+<p>\begin{equation}
+div(g_{P,Q}) = [P] + [Q] - [P + Q] - [\mathcal{O}]
+\end{equation}</p>
+<hr />
+<p><strong>Proof:</strong> We consider two cases based on the value of \(\lambda\).</p>
+<p>Case 1: \(\lambda \neq \infty\)</p>
+<p>Let \(y = \lambda x + v\) be the line through \(P\) and \(Q\) (or the tangent line at \(P\) if \(P = Q\)). This line intersects \(E\) at three points: \(P\), \(Q\), and \(-P-Q\). Thus,
+\begin{equation}
+div(y - \lambda x - v) = [P] + [Q] + [-P - Q] - 3[\mathcal{O}]
+\end{equation}
+Vertical lines intersect \(E\) at points and their negatives, so:
+\begin{equation}
+div(x - x_{P+Q}) = [P + Q] + [-P - Q] - 2[\mathcal{O}]
+\end{equation}
+It follows that \(g_{P,Q} = \frac{y - \lambda x - v}{x - x_{P+Q}}\) has the desired divisor.</p>
+<p>Case 2: \(\lambda = \infty\)</p>
+<p>In this case, \(P + Q = \mathcal{O}\), so we want \(g_{P,Q}\) to have divisor \([P] + [-P] - 2[\mathcal{O}]\). The function \(x - x_P\) has this divisor.
+\end{proof}</p>
+<h3 id="theorem-millers-algorithm"><a class="header" href="#theorem-millers-algorithm">Theorem: Miller's Algorithm</a></h3>
+<hr />
+<p><em>Let \(m \geq 1\) and write its binary expansion as:</em>
+\begin{equation}
+m = m_0 + m_1 \cdot 2 + m_2 \cdot 2^2 + \cdots + m_{n-1} \cdot 2^{n-1}
+\end{equation}
+<em>where \(m_i \in \{0, 1\}\) and \(m_{n-1} \neq 0\).</em></p>
+<p><em>The following algorithm, using the function \(g_{P,Q}\) defined in the previous theorem, returns a function \(f_P\) whose divisor satisfies:</em></p>
+<p>\begin{equation}
+div(f_P) = m[P] - m[P] - (m - 1)[\mathcal{O}]
+\end{equation}</p>
+<p>===========================</p>
+<p><strong>Miller's Algorithm</strong></p>
+<ol>
+<li>Set \(T = P\) and \(f = 1\)</li>
+<li><strong>For</strong> \(i \gets n-2 \cdots 0\) <strong>do</strong></li>
+<li>    Set \(f = f^2 \cdot g_{T, T}\)</li>
+<li>    Set \(T = 2T\)</li>
+<li>    <strong>If</strong> \(m_i = 1\) <strong>then</strong></li>
+<li>        Set \(f = f \cdot g_{T, P}\)</li>
+<li>        Set \(T = T + P\)</li>
+<li>    <strong>End if</strong></li>
+<li><strong>End for</strong></li>
+<li><strong>Return</strong> \(f\)</li>
+</ol>
+<p>===========================</p>
+<hr />
+<p><strong>Proof:</strong> TBD</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>pub fn get_lambda&lt;F: Field&gt;(
+    p: EllipticCurvePoint&lt;F&gt;,
+    q: EllipticCurvePoint&lt;F&gt;,
+    r: EllipticCurvePoint&lt;F&gt;,
+) -&gt; F {
+    let p_x = p.x.clone().unwrap();
+    let p_y = p.y.clone().unwrap();
+    let q_x = q.x.clone().unwrap();
+    // let q_y = q.y.clone().unwrap();
+    let r_x = r.x.clone().unwrap();
+    let r_y = r.y.clone().unwrap();
+
+    if (p == q &amp;&amp; p_y.clone() == F::zero()) || (p != q &amp;&amp; p_x.clone() == q_x.clone()) {
+        return r_x.clone() - p_x.clone();
+    }
+    let slope = p.line_slope(q.clone());
+    let numerator = r_y.clone() - p_y.clone() - slope.clone() * (r_x.clone() - p_x.clone());
+    let denominator = r_x.clone() + p_x.clone() + q_x.clone() - slope.clone() * slope.clone();
+    return numerator / denominator;
+}
+
+pub fn miller&lt;F: Field&gt;(p: EllipticCurvePoint&lt;F&gt;, q: EllipticCurvePoint&lt;F&gt;, m: BigInt) -&gt; F {
+    if p == q {
+        F::one();
+    }
+
+    let mut f = F::one();
+    let mut t = p.clone();
+
+    for i in (1..m.bits()).rev() {
+        f = (f.clone() * f.clone()) * (get_lambda(t.clone(), t.clone(), q.clone()));
+        t = t.clone() + t.clone();
+        if m.bit(i) {
+            f = f * (get_lambda(t.clone(), p.clone(), q.clone()));
+            t = t.clone() + p.clone();
+        }
+    }
+
+    f
+}
+
+pub fn weil_pairing&lt;F: Field&gt;(
+    p: EllipticCurvePoint&lt;F&gt;,
+    q: EllipticCurvePoint&lt;F&gt;,
+    m: BigInt,
+    s: Option&lt;EllipticCurvePoint&lt;F&gt;&gt;,
+) -&gt; F {
+    let s_value = s.unwrap();
+    let fp_qs = miller(p.clone(), q.clone() + s_value.clone(), m.clone());
+    let fp_s = miller(p.clone(), s_value.clone(), m.clone());
+    let fq_ps = miller(q.clone(), p.clone() - s_value.clone(), m.clone());
+    let fq_s = miller(q.clone(), -s_value.clone(), m.clone());
+
+    return (fp_qs / fp_s) / (fq_ps / fq_s);
+}
+<span class="boring">}</span></code></pre></pre>
+
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+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html" class="active"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
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+                        <h1 id="assumptions"><a class="header" href="#assumptions">Assumptions</a></h1>
+<h3 id="assumption-discrete-logarithm-problem"><a class="header" href="#assumption-discrete-logarithm-problem">Assumption: Discrete Logarithm Problem</a></h3>
+<hr />
+<p><em>Let \(G\) be a finite cyclic group of order \(n\), with \(\gamma\) as its generator and \(1\) as the identity element. For any element \(\alpha \in G\), there is currently no known efficient (polynomial-time) algorithm to compute the smallest non-negative integer \(x\) such that \(\alpha = \gamma^{x}\).</em></p>
+<hr />
+<p>The Discrete Logarithm Problem can be thought of as a one-way function. It's easy to compute \(g^{x}\) given \(g\) and \(x\), but it's computationally difficult to find \(x\) given \(g\) and \(g^{x}\).</p>
+<h3 id="assumption-elliptic-curve-discrete-logarithm-problem"><a class="header" href="#assumption-elliptic-curve-discrete-logarithm-problem">Assumption: Elliptic Curve Discrete Logarithm Problem</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve defined over a finite field \(\mathbb{F}_q\), where \(q\) is a prime power. Let \(P\) be a point on \(E\) of point order \(n\), and let \(\langle P \rangle\) be the cyclic subgroup of \(E\) generated by \(P\). For any element \(Q \in \langle P \rangle\), there is currently no known efficient (polynomial-time) algorithm to compute the unique integer \(k\), \(0 \leq k &lt; n\), such that \(Q = kP\).</em></p>
+<hr />
+<p>This assumption is an elliptic curve version of the Discrete Logarithm Problem.</p>
+<h3 id="assumption-knowledge-of-exponent-assumption"><a class="header" href="#assumption-knowledge-of-exponent-assumption">Assumption: Knowledge of Exponent Assumption</a></h3>
+<hr />
+<p>Let \(G\) be a cyclic group of prime order \(q\) with generator \(g \in G\). For any probabilistic polynomial-time algorithm \(\mathcal{A}\) that outputs:</p>
+<p>\begin{equation}
+\mathcal{A}(g, g^x) = (h, h') \quad s.t. \quad h' = h^x
+\end{equation}
+, there exists an efficient extractor \(\mathcal{E}\) such that:
+\begin{equation}
+\mathcal{E}(\mathcal{A}, g, g^x) = y \quad s.t. \quad h = g^y
+\end{equation}</p>
+<hr />
+<p>This assumption states that if \(\mathcal{A}\) can compute the pair \((g^y, g^{xy})\) from \((g, g^x)\), then \(\mathcal{A}\) must "know" the value \(y\), in the sense that \(\mathcal{E}\) can extract \(y\) from \(\mathcal{A}\)'s internal state.
+The Knowledge of Exponent Assumption is useful for constructing verifiable exponential calculation algorithms. Consider a scenario where Alice has a secret value \(a\), and Bob has a secret value \(b\). Bob wants to obtain \(g^{ab}\). This can be achieved through the following protocol:</p>
+<p><strong>Verifiable Exponential Calculation Algorithm</strong></p>
+<ol>
+<li>Bob sends \((g, g'=g^{b})\) to Alice</li>
+<li>Alice sends \((h=g^{a}, h'=g'^{a})\) to Bob</li>
+<li>Bob checks \(h^{b} = h'\).</li>
+</ol>
+<p>Thanks to the Discrete Logarithm Assumption and the Knowledge of Exponent Assumption, the following properties hold:</p>
+<ul>
+<li>Bob cannot derive \(a\) from \((h, h')\).</li>
+<li>Alice cannot derive \(b\) from \((g, g')\).</li>
+<li>Alice cannot generate \((t, t')\) such that \(t \neq h\) and \(t^{b} = t'\).</li>
+<li>If \(h^{b} = h'\), Bob can conclude that \(h\) is the power of \(g\).</li>
+</ul>
+
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@@ -91,7 +92,7 @@
 
         <nav id="sidebar" class="sidebar" aria-label="Table of contents">
             <div class="sidebar-scrollbox">
-                <ol class="chapter"><li class="chapter-item expanded "><a href="number_theory/index.html"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="number_theory/subsec1.html"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="zksnark/index.html"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="zkstark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="zkvm/index.html"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
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@@ -176,22 +177,1252 @@ <h1 class="menu-title">MyZKP: Building Zero Knowledge Proof from Scratch</h1>
 
                 <div id="content" class="content">
                     <main>
-                        <h1 id="basics-of-number-theory"><a class="header" href="#basics-of-number-theory">Basics of Number Theory</a></h1>
-<div style="break-before: page; page-break-before: always;"></div><h2 id="computation-rule-and-properties"><a class="header" href="#computation-rule-and-properties">Computation Rule and Properties</a></h2>
+                        <div style="break-before: page; page-break-before: always;"></div><h1 id="basics-of-number-theory"><a class="header" href="#basics-of-number-theory">Basics of Number Theory</a></h1>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="computation-rule-and-properties"><a class="header" href="#computation-rule-and-properties">Computation Rule and Properties</a></h1>
+<h3 id="definition-binary-operation"><a class="header" href="#definition-binary-operation">Definition: Binary Operation</a></h3>
+<hr />
+<p><em>A mapping \(\circ: X \times X \rightarrow X\) is a binary operation on \(X\) if for any pair of elements \((x_1, x_2)\) in \(X\), \(x_1 \circ x_2\) is also in \(X\).</em></p>
 <hr />
-<p><em><strong>Definition: Binary Operation</strong></em></p>
-<p>A mapping \(\circ: X \times X \rightarrow X\) is a binary operation on \(X\) if for any pair of elements \((x_1, x_2)\) in \(X\), \(x_1 \circ x_2\) is also in \(X\).</p>
 <p><strong>Example</strong>: Addition (+) on the set of integers is a binary operation. For example, \(5 + 3 = 8\), and both \(5, 3, 8\) are integers, staying within the set of integers.</p>
+<h3 id="definition-associative-property"><a class="header" href="#definition-associative-property">Definition: Associative Property</a></h3>
+<hr />
+<p><em>A binary operation \(\circ\) is associative if \((a \circ b) \circ c = a \circ (b \circ c)\) for all \(a, b, c \in X\).</em></p>
 <hr />
-<p><em><strong>Definition: Associative Property</strong></em></p>
-<p>A binary operation \(\circ\) is associative if \((a \circ b) \circ c = a \circ (b \circ c)\) for all \(a, b, c \in X\).</p>
 <p><strong>Example</strong>: Multiplication of real numbers is associative: \((2 \times 3) \times 4 = 2 \times (3 \times 4) = 24\). In a modular context, we also have addition modulo \(n\) being associative. For example, for \(n = 5\), \((2 + 3) \bmod 5 + 4 \bmod 5 = 2 + (3 \bmod 5 + 4) \bmod 5 = 4\).</p>
+<h3 id="definition-commutative-property"><a class="header" href="#definition-commutative-property">Definition: Commutative Property</a></h3>
+<hr />
+<p><em>A binary operation \(\circ\) is commutative if \(a \circ b = b \circ a\) for all \(a, b \in X\).</em></p>
 <hr />
-<p><em><strong>Definition: Commutative Property</strong></em></p>
-<p>A binary operation \(\circ\) is commutative if \(a \circ b = b \circ a\) for all \(a, b \in X\).</p>
 <p><strong>Example</strong>: Addition modulo \(n\) is also commutative. For \(n = 7\), \(5 + 3 \bmod 7 = 3 + 5 \bmod 7 = 1\).</p>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="semigroup-group-ring"><a class="header" href="#semigroup-group-ring">Semigroup, Group, Ring</a></h1>
+<h3 id="definition-semigroup"><a class="header" href="#definition-semigroup">Definition: Semigroup</a></h3>
+<hr />
+<p><em>A pair \((H, \circ)\), where \(H\) is a non-empty set and \(\circ\) is an associative binary operation on \(H\), is called a semigroup.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of positive integers under multiplication modulo \(n\) forms a semigroup. For instance, with \(n = 6\), the elements \(\{1, 2, 3, 4, 5\}\) under multiplication modulo 6 form a semigroup, since multiplication modulo 6 is associative.</p>
+<h3 id="definition-abelian-semigroup"><a class="header" href="#definition-abelian-semigroup">Definition: Abelian Semigroup</a></h3>
+<hr />
+<p><em>A semigroup whose operation is commutative is called an abelian semigroup.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of natural numbers under addition modulo \(n\) forms an abelian semigroup. For \(n = 7\), addition modulo 7 is both associative and commutative, so it is an abelian semigroup.</p>
+<h3 id="definition-identity-element"><a class="header" href="#definition-identity-element">Definition: Identity Element</a></h3>
+<hr />
+<p><em>An element \(e \in H\) is an identity element of \(H\) if it satisfies \(e \circ a = a \circ e = a\) for any \(a \in H\).</em></p>
+<hr />
+<p><strong>Example</strong>: 0 is the identity element for addition modulo \(n\). For example, \(0 + a \bmod 5 = a + 0 \bmod 5 = a\). Similarly, 1 is the identity element for multiplication modulo \(n\). For example, \(1 \times a \bmod 7 = a \times 1 \bmod 7 = a\).</p>
+<h3 id="definition-monoid"><a class="header" href="#definition-monoid">Definition: Monoid</a></h3>
+<hr />
+<p><em>A semigroup with an identity element is called a monoid.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of non-negative integers under addition modulo \(n\) forms a monoid. For \(n = 5\), the set \(\{0, 1, 2, 3, 4\}\) under addition modulo 5 forms a monoid with 0 as the identity element.</p>
+<h3 id="definition-inverse"><a class="header" href="#definition-inverse">Definition: Inverse</a></h3>
+<hr />
+<p><em>For an element \(a \in H\), an element \(b \in H\) is an inverse of \(a\) if \(a \circ b = b \circ a = e\), where \(e\) is the identity element.</em></p>
+<hr />
+<p><strong>Example</strong>: In modulo \(n\) arithmetic (addition), the inverse of an element exists if it can cancel itself out to yield the identity element. In the set of integers modulo 7, the inverse of 3 is 5, because \(3 \times 5 \bmod 7 = 1\), where 1 is the identity element for multiplication.</p>
+<h3 id="definition-group"><a class="header" href="#definition-group">Definition: Group</a></h3>
+<hr />
+<p><em>A monoid in which every element has an inverse is called a group.</em></p>
+<hr />
+<p><strong>Example</strong>: The set of integers modulo a prime \(p\) under multiplication forms a group (Can you prove it?). For instance, in \(\mathbb{Z}/5\mathbb{Z}\), every non-zero element \(\{1 + 5\mathbb{Z}, 2 + 5\mathbb{Z}, 3 + 5\mathbb{Z}, 4 + 5\mathbb{Z}\}\) has an inverse, making it a group.</p>
+<h3 id="definition-order-of-a-group"><a class="header" href="#definition-order-of-a-group">Definition: Order of a Group</a></h3>
+<hr />
+<p><em>The order of a group is the number of elements in the group.</em></p>
+<hr />
+<p><strong>Example</strong>: The group of integers modulo 4 under addition has order 4, because the set of elements is \(\{0, 1, 2, 3\}\).</p>
+<h3 id="definition-ring"><a class="header" href="#definition-ring">Definition: Ring</a></h3>
+<hr />
+<p><em>A triple \((R, +, \cdot)\) is a ring if \((R, +)\) is an abelian group, \((R, \cdot)\) is a semigroup, and the distributive property holds: \(x \cdot (y + z) = (x \cdot y) + (x \cdot z)\) and \((x + y) \cdot z = (x \cdot z) + (y \cdot z)\) for all \(x, y, z \in R\).</em></p>
+<hr />
+<p><strong>Example</strong>: The set of integers with usual addition and multiplication modulo \(n\) forms a ring. For example, in \(\mathbb{Z}/6\mathbb{Z}\), addition and multiplication modulo 6 form a ring.</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>use std::fmt;
+use std::ops::{Add, Mul, Neg, Sub};
+
+pub trait Ring:
+    Sized
+    + Clone
+    + PartialEq
+    + fmt::Display
+    + Add&lt;Output = Self&gt;
+    + Sub&lt;Output = Self&gt;
+    + Mul&lt;Output = Self&gt;
+    + Mul&lt;i64, Output = Self&gt;
+    + Neg&lt;Output = Self&gt;
+{
+    // A ring is an algebraic structure with addition and multiplication
+    fn zero() -&gt; Self;
+    fn one() -&gt; Self;
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-communicative-ring"><a class="header" href="#definition-communicative-ring">Definition: Communicative Ring</a></h3>
+<hr />
+<p><em>A ring is called a commutative ring if its multiplication operation is commutative.</em></p>
+<hr />
+<p><strong>Example</strong> The set of real numbers under usual addition and multiplication forms a commutative ring.</p>
+<h3 id="definition-field"><a class="header" href="#definition-field">Definition: Field</a></h3>
+<hr />
+<p><em>A commutative ring with a multiplicative identity element where every non-zero element has a multiplicative inverse is called a field.</em></p>
+<hr />
+<p><strong>Example</strong> The set of rational numbers under usual addition and multiplication forms a field.</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>use crate::modules::ring::Ring;
+
+pub trait Field: Ring + Div&lt;Output = Self&gt; {
+    fn inverse(&amp;self) -&gt; Self;
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-residue-class"><a class="header" href="#definition-residue-class">Definition: Residue Class</a></h3>
+<hr />
+<p><em>The residue class of \( a \) modulo \( m \), denoted as \( a + m\mathbb{Z} \), is the set \( \{b : b \equiv a \pmod{m}\} \).</em></p>
+<hr />
+<p><strong>Example:</strong> For \( m = 3 \), the residue class of 2 is \( 2 + 3\mathbb{Z} = \{\ldots, -4, -1, 2, 5, 8, \ldots\} \).</p>
+<h3 id="definition-inverse-of-residue-class"><a class="header" href="#definition-inverse-of-residue-class">Definition: Inverse of Residue Class</a></h3>
+<hr />
+<p><em>We denote the set of all residue classes modulo \( m \) as \( \mathbb{Z} / m\mathbb{Z} \). We say that \( a + m\mathbb{Z} \) is invertible in \( \mathbb{Z} / m\mathbb{Z} \) if and only if there exists a solution for \( ax \equiv 1 \pmod{m} \).</em></p>
+<hr />
+<p><strong>Example:</strong> In \( \mathbb{Z}/5\mathbb{Z} \), \( 3 + 5\mathbb{Z} \) is invertible because \( \gcd(3, 5) = 1 \) (since \( 3\cdot2 \equiv 1 \pmod{5} \)). However, in \( \mathbb{Z}/6\mathbb{Z} \), \( 3 + 6\mathbb{Z} \) is not invertible because \( \gcd(3, 6) = 3 \neq 1 \).</p>
+<h3 id="lemma-121"><a class="header" href="#lemma-121">Lemma 1.2.1</a></h3>
+<hr />
+<p><em>If \( a \) and \( b \) are coprime, the residues of \( a \), \( 2a \), \( 3a \), ..., \( (b-1)a \) modulo \( b \) are all distinct.</em></p>
+<hr />
+<p><strong>Proof:</strong> Suppose, for contradiction, that there exist \( x, y \in \{1, 2, \dots, b-1\} \) with \( x \neq y \) such that \( xa \equiv ya \pmod{b} \). Then \( (x-y)a \equiv 0 \pmod{b} \), implying \( b \mid (x-y)a \). Since \( a \) and \( b \) are coprime, we must have \( b \mid (x-y) \). However, \( |x-y| &lt; b \), so this is only possible if \( x = y \), contradicting our assumption. Therefore, all residues must be distinct.</p>
+<h3 id="theorem-122"><a class="header" href="#theorem-122">Theorem 1.2.2</a></h3>
+<hr />
+<p><em>For any integers \( a \) and \( b \), the equation \( ax + by = 1 \) has a solution in integers \( x \) and \( y \) if and only if \( a \) and \( b \) are coprime.</em></p>
+<hr />
+<p><strong>Proof:</strong></p>
+<ul>
+<li><strong>(\( \Rightarrow \))</strong> Suppose \( a \) and \( b \) are not coprime, let \( d = \gcd(a,b) &gt; 1 \). Then \( d \mid a \) and \( d \mid b \), so \( d \mid (ax + by) \) for any integers \( x \) and \( y \). Thus, \( ax + by \neq 1 \) for any \( x \) and \( y \).</li>
+<li><strong>(\( \Leftarrow \))</strong> Suppose \( a \) and \( b \) are coprime. By the previous lemma, the residues of \( a \), \( 2a \), ..., \( (b-1)a \) modulo \( b \) are all distinct. Therefore, there exists an \( m \in \{1, 2, ..., b-1\} \) such that \( ma \equiv 1 \pmod{b} \). This means there exists an integer \( n \) such that \( ma = bn + 1 \), or equivalently, \( ma - bn = 1 \).</li>
+</ul>
+<h3 id="theorem-bézouts-identity"><a class="header" href="#theorem-bézouts-identity">Theorem: Bézout's Identity</a></h3>
+<hr />
+<p><em>For any integers \( a \) and \( b \), we have:
+\(
+a\mathbb{Z} + b \mathbb{Z} = \gcd(a, b)\mathbb{Z}
+\)</em></p>
+<hr />
+<p><strong>Proof:</strong></p>
+<p>This statement is equivalent to proving that \( ax + by = c \) has an integer solution if and only if \( c \) is a multiple of \( \gcd(a,b) \).</p>
+<ul>
+<li><strong>(\( \Rightarrow \))</strong> If \( ax + by = c \) for some integers \( x \) and \( y \), then \( \gcd(a,b) \mid a \) and \( \gcd(a,b) \mid b \), so \( \gcd(a,b) \mid (ax + by) = c \).</li>
+<li><strong>(\( \Leftarrow \))</strong> Let \( c = k\gcd(a,b) \) for some integer \( k \). We can write \( a = p\gcd(a,b) \) and \( b = q\gcd(a,b) \), where \( p \) and \( q \) are coprime. By the previous theorem, there exist integers \( m \) and \( n \) such that \( pm + qn = 1 \). Multiplying both sides by \( k\gcd(a,b) \), we get:
+[
+akm + bkn = c
+]
+Thus, \( x = km \) and \( y = kn \) are integer solutions to \( ax + by = c \).</li>
+</ul>
+<p>This theorem implies that for any integers \( a \), \( b \), and \( n \), the equation \( ax + by = n \) has an integer solution if and only if \( \gcd(a,b) \mid n \).</p>
+<h3 id="theorem-123"><a class="header" href="#theorem-123">Theorem 1.2.3</a></h3>
+<hr />
+<p><em>\( a + m\mathbb{Z} \) is invertible in \( \mathbb{Z}/m\mathbb{Z} \) if and only if \( \gcd(a,m) = 1 \).</em></p>
+<hr />
+<p><strong>Proof:</strong></p>
+<ul>
+<li><strong>(\( \Rightarrow \))</strong> Suppose \( a + m\mathbb{Z} \) is invertible in \( \mathbb{Z}/m\mathbb{Z} \). Then there exists an integer \( x \) such that \( ax \equiv 1 \pmod{m} \). Let \( g = \gcd(a,m) \). Since \( g \mid ax \) and \( g \mid (ax - 1) \), we must have \( g \mid 1 \), so \( g = 1 \).</li>
+<li><strong>(\( \Leftarrow \))</strong> Suppose \( \gcd(a,m) = 1 \). By Bézout's identity, there exist integers \( x \) and \( y \) such that \( ax + my = 1 \). Thus, \( x + m\mathbb{Z} \) is the multiplicative inverse of \( a + m\mathbb{Z} \) in \( \mathbb{Z}/m\mathbb{Z} \).</li>
+</ul>
+<h3 id="definition-residue-class-ring"><a class="header" href="#definition-residue-class-ring">Definition: Residue Class Ring</a></h3>
+<p><em>\( (\mathbb{Z} / m \mathbb{Z}, +, \cdot) \) is a commutative ring where \( 1 + m \mathbb{Z} \) is the multiplicative identity element. This ring is called the residue class ring modulo \( m \).</em></p>
+<p><strong>Example:</strong> \( \mathbb{Z}/4\mathbb{Z} = \{0 + 4\mathbb{Z}, 1 + 4\mathbb{Z}, 2 + 4\mathbb{Z}, 3 + 4\mathbb{Z}\} \).</p>
+<h3 id="definition-primitive-residue-class"><a class="header" href="#definition-primitive-residue-class">Definition: Primitive Residue Class</a></h3>
+<hr />
+<p><em>A residue class \( a + m\mathbb{Z} \) is called primitive if \( \gcd(a, m) = 1 \).</em></p>
+<hr />
+<p><strong>Example:</strong> In \( \mathbb{Z}/6\mathbb{Z} \), the primitive residue classes are \( 1 + 6\mathbb{Z} \) and \( 5 + 6\mathbb{Z} \).</p>
+<h3 id="theorem-124"><a class="header" href="#theorem-124">Theorem 1.2.4</a></h3>
+<hr />
+<p><em>A residue ring \( \mathbb{Z} / m\mathbb{Z} \) is a field if and only if \( m \) is a prime number.</em></p>
+<hr />
+<p><strong>Proof:</strong> TBD</p>
+<p><strong>Example:</strong> For \( m = 5 \), \( \mathbb{Z}/5\mathbb{Z} = \{0 + 5\mathbb{Z}, 1 + 5\mathbb{Z}, 2 + 5\mathbb{Z}, 3 + 5\mathbb{Z}, 4 + 5\mathbb{Z}\} \) forms a field because 5 is a prime number, and every non-zero element has a multiplicative inverse.</p>
+<h3 id="definition-primitive-residue-class-group"><a class="header" href="#definition-primitive-residue-class-group">Definition: Primitive Residue Class Group</a></h3>
+<hr />
+<p><em>The group of all primitive residue classes modulo \( m \) is called the primitive residue class group, denoted by \( (\mathbb{Z}/m\mathbb{Z})^{\times} \).</em></p>
+<hr />
+<p><strong>Example:</strong> For \( m = 8 \), the set of all primitive residue classes is \( (\mathbb{Z}/8\mathbb{Z})^{\times} = \{1 + 8\mathbb{Z}, 3 + 8\mathbb{Z}, 5 + 8\mathbb{Z}, 7 + 8\mathbb{Z}\} \). These are the integers less than 8 that are coprime to 8 (i.e., \(\gcd(a, 8) = 1\)).</p>
+<p>Contrast this with \( m = 9 \). The primitive residue class group is \((\mathbb{Z}/9\mathbb{Z})^{\times} = \{1 + 9\mathbb{Z}, 2 + 9\mathbb{Z}, 4 + 9\mathbb{Z}, 5 + 9\mathbb{Z}, 7 + 9\mathbb{Z}, 8 + 9\mathbb{Z}\}\), as these are the integers less than 9 that are coprime to 9.</p>
+<h3 id="definition-eulers-totient-function"><a class="header" href="#definition-eulers-totient-function">Definition: Euler's Totient Function</a></h3>
+<hr />
+<p><em>Euler's totient function \(\phi(m)\) is equal to the order of the primitive residue class group modulo \(m\), which is the number of integers less than \(m\) and coprime to \(m\).</em></p>
+<hr />
+<p><strong>Example</strong>: For \(m = 12\), \(\phi(12) = 4\) because there are 4 integers less than 12 that are coprime to 12: \({1, 5, 7, 11}\).</p>
+<p>For \(m = 10\), \(\phi(10) = 4\), as there are also 4 integers less than 10 that are coprime to 10: \({1, 3, 7, 9}\).</p>
+<h3 id="definition-order-of-an-element-within-a-group"><a class="header" href="#definition-order-of-an-element-within-a-group">Definition: Order of an Element within a Group</a></h3>
+<hr />
+<p><em>The order of \(g \in G\) is the smallest natural number \(e\) such that \(g^{e} = 1\). We denote it as \(\text{order}_g(g)\)or \(\text{order } g\).</em></p>
+<hr />
+<p><strong>Example</strong>: In \((\mathbb{Z}/7\mathbb{Z})^{\times}\), the element 3 has order 6 because \(3^6 \bmod 7 = 1\). In other words, \(3 \times 3 \times 3 \times 3 \times 3 \times 3 \bmod 7 = 1\), and 6 is the smallest such exponent.</p>
+<h3 id="definition-subgroup"><a class="header" href="#definition-subgroup">Definition: Subgroup</a></h3>
+<hr />
+<p><em>A subset \(U \subseteq G\) is a subgroup of \(G\) if \(U\) itself is a group under the operation of \(G\).</em></p>
+<hr />
+<p><strong>Example</strong>: Consider \((\mathbb{Z}/8\mathbb{Z})^{\times} = \{1 + 8\mathbb{Z}, 3+ 8\mathbb{Z}, 5+ 8\mathbb{Z}, 7+ 8\mathbb{Z}\}\) under multiplication modulo 8. The subset \({1+ 8\mathbb{Z}, 7+ 8\mathbb{Z}}\) forms a subgroup because it satisfies the group properties: closed under multiplication, contains the identity element (1), and every element has an inverse (\(7 \times 7 \equiv 1 \bmod 8\)).</p>
+<h3 id="definition-subgroup-generated-by-g"><a class="header" href="#definition-subgroup-generated-by-g">Definition: Subgroup Generated by \(g\)</a></h3>
+<hr />
+<p><em>The set \(\{g^{k} : k \in \mathbb{Z}\}\), for some element \(g \in G\), forms a subgroup of \(G\) and is called the subgroup generated by \(g\), denoted by \(\langle g \rangle\).</em></p>
+<hr />
+<p><strong>Example</strong>: Consider the group \((\mathbb{Z}/7\mathbb{Z})^{\times} = \{1+ 7\mathbb{Z}, 2+ 7\mathbb{Z}, 3+ 7\mathbb{Z}, 4+ 7\mathbb{Z}, 5+ 7\mathbb{Z}, 6+ 7\mathbb{Z}\}\) under multiplication modulo 7. If we take \(g = 3\), then \(\langle 3 +7\mathbb{Z} \rangle =\) \(\{3^1+7\mathbb{Z}, 3^2+7\mathbb{Z}, 3^3+7\mathbb{Z}, 3^4+7\mathbb{Z}, 3^5+7\mathbb{Z}, 3^6+7\mathbb{Z}\} \bmod 7 =\) \(\{3+7\mathbb{Z}, 2+7\mathbb{Z}, 6+7\mathbb{Z}, 4+7\mathbb{Z}, 5+7\mathbb{Z}, 1+7\mathbb{Z}\}\), which forms a subgroup generated by 3. This subgroup contains all elements of \((\mathbb{Z}/7\mathbb{Z})^{\times}\), making 3 a generator of the entire group.</p>
+<p>If \(g\) has a finite order \(e\), we have that \(\langle g \rangle = \{g^{k}: 0 \leq k \leq e\}\), meaning \(e\) is the order of \(\langle g \rangle\).</p>
+<h3 id="definition-cyclic-group"><a class="header" href="#definition-cyclic-group">Definition: Cyclic Group</a></h3>
+<hr />
+<p><em>A group \(G\) is called a cyclic group if there exists an element \(g \in G\) such that \(G = \langle g \rangle\). In this case, \(g\) is called a generator of \(G\).</em></p>
+<hr />
+<p><strong>Example</strong>: The group \((\mathbb{Z}/6\mathbb{Z})^{\times} = \{1+6\mathbb{Z}, 5+6\mathbb{Z}\}\) under multiplication modulo 6 is a cyclic group. In this case, both 1 and 5 are generators of the group because \(\langle 5 +6\mathbb{Z} \rangle = \{(5^1 \bmod 6)+6\mathbb{Z} = 5 +6\mathbb{Z}, (5^2 \bmod 6)+6\mathbb{Z} = 1+6\mathbb{Z}\}\). Since 5 generates all the elements of the group, \(G\) is cyclic.</p>
+<h3 id="theorem-125"><a class="header" href="#theorem-125">Theorem 1.2.5</a></h3>
+<hr />
+<p><em>If \(G\) is a finite cyclic group, it has \(\phi(|G|)\)generators, and each generator has order \(|G|\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: Consider the group \((\mathbb{Z}/8\mathbb{Z})^{\times} = \{1+8\mathbb{Z}, 3+8\mathbb{Z}, 5+8\mathbb{Z}, 7+8\mathbb{Z}\}\). This group is cyclic, and \(\phi(8) = 4\). The generators of this group are \(\{1+8\mathbb{Z}, 3+8\mathbb{Z}, 5+8\mathbb{Z}, 7+8\mathbb{Z}\}\), each of which generates the entire group when raised to successive powers modulo 8. Each generator has the same order, which is \(|G| = 4\).</p>
+<h3 id="theorem-126"><a class="header" href="#theorem-126">Theorem 1.2.6</a></h3>
+<hr />
+<p><em>If \(G\) is a finite cyclic group, the order of any subgroup of \(G\) divides the order of \(G\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: Consider the cyclic group \((\mathbb{Z}/6\mathbb{Z})^{\times} = \{1+6\mathbb{Z}, 5+6\mathbb{Z}\}\) under multiplication modulo 6. If we take the subgroup \(\langle 5+6\mathbb{Z} \rangle = \{1+6\mathbb{Z}, 5+6\mathbb{Z}\}\), this is a subgroup of order 2, and 2 divides the order of the original group, which is 6. This theorem generalizes this property: for any subgroup of a cyclic group, its order divides the order of the group.</p>
+<h3 id="theorem-fermats-little-theorem"><a class="header" href="#theorem-fermats-little-theorem">Theorem: Fermat's Little Theorem</a></h3>
+<hr />
+<p><em>If \(\gcd(a, m) = 1\), then \(a^{\phi(m)} \equiv 1 \pmod{m}\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: Take \(a = 2\) and \(m = 5\). Since \(\gcd(2, 5) = 1\), Fermat's Little Theorem tells us that \(2^{\phi(5)} = 2^4 \equiv 1 \bmod 5\). Indeed, \(2^4 = 16\) and \(16 \bmod 5 = 1\).</p>
+<p>This theorem suggests that \(a^{\phi(m) - 1} + m \mathbb{Z}\) is the inverse residue class of \(a + m \mathbb{Z}\).</p>
+<h3 id="theorem-127"><a class="header" href="#theorem-127">Theorem 1.2.7</a></h3>
+<hr />
+<p><em>The order of any element in a group divides the order of the group.</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: In the group \((\mathbb{Z}/7\mathbb{Z})^{\times}\), consider the element \(3 + 7\mathbb{Z}\). The order of \(3 + 7\mathbb{Z}\) is 6, as \(3^6 \equiv 1 \bmod 7\). The order of the group itself is also 6, and indeed, the order of the element divides the order of the group.</p>
+<h3 id="theorem-generalization-of-fermats-little-theorem"><a class="header" href="#theorem-generalization-of-fermats-little-theorem">Theorem: Generalization of Fermat's Little Theorem</a></h3>
+<hr />
+<p><em>For any element \(g \in G\), we have \(g^{|G|} = 1\).</em></p>
+<hr />
+<p><strong>Proof</strong>: TBD</p>
+<p><strong>Example</strong>: In the group \((\mathbb{Z}/7\mathbb{Z})^{\times}\), for any element \(g\), such as \(g = 3 + 7\mathbb{Z}\), we have \(3^6 \equiv 1 \bmod 7\). This holds for any \(g \in (\mathbb{Z}/7\mathbb{Z})^{\times}\) because the order of the group is 6. Thus, \(g^{|G|} = 1\) is satisfied.</p>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="polynomials"><a class="header" href="#polynomials">Polynomials</a></h1>
+<p>Let \(K\) be a field.</p>
+<h3 id="definition-polynomial"><a class="header" href="#definition-polynomial">Definition: Polynomial</a></h3>
+<p><em>A univariate polynomial over a commutative ring \(R\) with unity \(1\) is an expression of the form \(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where \(x\) is a variable and coefficients \(a_0, \ldots, a_n\) belong to \(R\). The set of all polynomials over \(R\) in the variable \(x\) is denoted as \(R[x]\).</em></p>
+<p><strong>Example:</strong> In \(\mathbb{Z}[x]\), we have polynomials such as \(2x^3 + x + 1\), \(x\), and \(1\). In \(\mathbb{R}[x]\), we have polynomials like \(\pi x^2 - \sqrt{2}x + e\).</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>/// Polynomial struct representing a polynomial over Field.
+#[derive(Debug, Clone, PartialEq)]
+pub struct Polynomial&lt;F: Field&gt; {
+    pub poly: Vec&lt;F&gt;,
+    pub var: String,
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-degree"><a class="header" href="#definition-degree">Definition: Degree</a></h3>
+<p><em>The degree of a non-zero polynomial \(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), denoted as \(\deg f\), is the largest integer \(n\) such that \(a_n \neq 0\). The zero polynomial is defined to have degree \(-1\).</em></p>
+<p><strong>Example</strong></p>
+<ul>
+<li>\(\deg(2x^3 + x + 1) = 3\)</li>
+<li>\(\deg(x) = 1\)</li>
+<li>\(\deg(1) = 0\)</li>
+<li>\(\deg(0) = -1\)</li>
+</ul>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Polynomial&lt;F&gt; {
+    /// Removes trailing zeroes from a polynomial's coefficients.
+    fn trim_trailing_zeros(poly: Vec&lt;F&gt;) -&gt; Vec&lt;F&gt; {
+        let mut trimmed = poly;
+        while trimmed.last() == Some(&amp;F::zero(None)) {
+            trimmed.pop();
+        }
+        trimmed
+    }
+
+    /// Returns the degree of the polynomial.
+    pub fn degree(&amp;self) -&gt; isize {
+        let trimmed = Self::trim_trailing_zeros(self.poly.clone());
+        if trimmed.is_empty() {
+            -1
+        } else {
+            (trimmed.len() - 1) as isize
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="sum-of-polynomials"><a class="header" href="#sum-of-polynomials">Sum of Polynomials</a></h3>
+<p><em>For polynomials \(f(x) = \sum_{i=0}^n a_i x^i\) and \(g(x) = \sum_{i=0}^m b_i x^i\), their sum is defined as: \((f + g)(x) = \sum_{i=0}^{\max(n,m)} (a_i + b_i) x^i\), where we set \(a_i = 0\) for \(i &gt; n\) and \(b_i = 0\) for \(i &gt; m\).</em></p>
+<p><strong>Example:</strong> Let \(f(x) = 2x^2 + 3x + 1\) and \(g(x) = x^3 - x + 4\). Then, \((f + g)(x) = x^3 + 2x^2 + 2x + 5\)</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Add for Polynomial&lt;F&gt; {
+    type Output = Self;
+
+    fn add(self, other: Self) -&gt; Polynomial&lt;F&gt; {
+        let max_len = std::cmp::max(self.poly.len(), other.poly.len());
+        let mut result = Vec::with_capacity(max_len);
+
+        let zero = F::zero(None);
+
+        for i in 0..max_len {
+            let a = self.poly.get(i).unwrap_or(&amp;zero);
+            let b = other.poly.get(i).unwrap_or(&amp;zero);
+            result.push(a.clone() + b.clone());
+        }
+        Polynomial {
+            poly: Self::trim_trailing_zeros(result),
+            var: self.var.clone(),
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="product-of-polynomials"><a class="header" href="#product-of-polynomials">Product of polynomials</a></h3>
+<p><em>For polynomials \(f(x) = \sum_{i=0}^n a_i x^i\) and \(g(x) = \sum_{j=0}^m b_j x^j\), their product is defined as: \((fg)(x) = \sum_{k=0}^{n+m} c_k x^k\), where \(c_k = \sum_{i+j=k} a_i b_j\)</em></p>
+<p><strong>Example:</strong> Let \(f(x) = x + 1\) and \(g(x) = x^2 - 1\). Then, \((fg)(x) = x^3 + x^2 - x - 1\)</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Mul for Polynomial&lt;F&gt; {
+    type Output = Self;
+
+    /// Multiplication of two polynomials.
+    fn mul(self, other: Polynomial&lt;F&gt;) -&gt; Polynomial&lt;F&gt; {
+        let mut result = vec![F::zero(None); self.degree() as usize + other.degree() as usize + 1];
+
+        for (i, a) in self.poly.iter().enumerate() {
+            for (j, b) in other.poly.iter().enumerate() {
+                result[i + j] = result[i + j].clone() + (a.clone() * b.clone());
+            }
+        }
+        Polynomial {
+            poly: Self::trim_trailing_zeros(result),
+            var: self.var.clone(),
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="lemma"><a class="header" href="#lemma">Lemma</a></h3>
+<p><em>Let \(f, g \in K[x]\) be non-zero polynomials. Then, \(\deg(fg) = \deg f + \deg g\).</em></p>
+<p><strong>Example:</strong> Let \(f(x) = x^2 + 1\) and \(g(x) = x^3 - x\) in \(\mathbb{R}[x]\). Then, \(\deg(fg) = \deg(x^5 - x^3 + x^2 + 1) = 5 = 2 + 3 = \deg f + \deg g\)</p>
+<p>We can also define division in the polynomial ring \(K[x]\).</p>
+<h3 id="theorem"><a class="header" href="#theorem">Theorem</a></h3>
+<p><em>Let \(f, g \in K[x]\), with \(g \neq 0\). There exist unique polynomials \(q, r \in K[x]\) that satisfy \(f = qg + r\) and either \(\deg r &lt; \deg g\) or \(r = 0\).</em></p>
+<p><strong>Proof</strong>
+TBD</p>
+<p>\(q\) is called the quotient of \(f\) divided by \(g\), and \(r\) is called the remainder; we write \(r = f \bmod g\).</p>
+<p><strong>Example:</strong> In \(\mathbb{R}[x]\), let \(f(x) = x^3 + 2x^2 - x + 3\) and \(g(x) = x^2 + 1\).  Then \(f = qg + r\) where \(q(x) = x + 2\) and \(r(x) = -3x + 1\).</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; Div for Polynomial&lt;F&gt; {
+    type Output = Self;
+
+    /// Division of two polynomials, returns quotient.
+    fn div(self, other: Polynomial&lt;F&gt;) -&gt; Polynomial&lt;F&gt; {
+        let mut remainder_coeffs = Self::trim_trailing_zeros(self.poly.clone());
+        let divisor_coeffs = Self::trim_trailing_zeros(other.poly.clone());
+        let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse();
+
+        let mut quotient =
+            vec![F::zero(None); self.degree() as usize - other.degree() as usize + 1];
+
+        let mut i = 0_i32;
+        while remainder_coeffs.len() &gt;= divisor_coeffs.len() {
+            let lead_term = remainder_coeffs.last().unwrap().clone() * divisor_lead_inv.clone();
+            let deg_diff = remainder_coeffs.len() - divisor_coeffs.len();
+            quotient[deg_diff] = lead_term.clone();
+
+            for i in 0..divisor_coeffs.len() {
+                remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i].clone()
+                    - (lead_term.clone() * divisor_coeffs[i].clone());
+            }
+            remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs);
+            i += 1;
+        }
+
+        Polynomial {
+            poly: Self::trim_trailing_zeros(quotient),
+            var: self.var.clone(),
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="corollary"><a class="header" href="#corollary">Corollary</a></h3>
+<p><em>Let \(f \in K[x]\) be a non-zero polynomial, and \(a \in K\) such that \(f(a) = 0\). Then, there exists a polynomial \(q \in K[x]\) such that \(f(x) = (x - a)q(x)\). In other words, \((x - a)\) is a factor of \(f(x)\).</em></p>
+<p><strong>Example:</strong> Let \(f(x) = x^2 + 1 \in (\mathbb{Z}/2\mathbb{Z})[x]\). We have \(f(1) = 1^2 + 1 = 0\) in \(\mathbb{Z}/2\mathbb{Z}\), and indeed: \(x^2 + 1 = (x - 1)^2 = x^2 - 2x + 1 = x^2 + 1\) in \((\mathbb{Z}/2\mathbb{Z})[x]\)</p>
+<h3 id="theorem-lagrange-interpolation"><a class="header" href="#theorem-lagrange-interpolation">Theorem: Lagrange Interpolation</a></h3>
+<p><em>A \(n\)-degre polynomial \(P(x)\) that goes through different \(n + 1\) points \(\{(x_1, y_1), (x_2, y_2), \cdots (x_{n + 1}, y_{n + 1})\}\) is uniquely represented as follows:</em></p>
+<p>\begin{align*}
+P(x) = \sum^{n+1}_{i=1} y_i \frac{f_i(x)}{f_i(x_i)}
+\end{align*}</p>
+<p><em>, where \(f_i(x) = \prod_{k \neq i} (x - x_k)\)</em></p>
+<p><strong>Proof</strong> TBD</p>
+<p>For example, the quadratic polynomial that goes through \(\{(1, 0), (2, 3), (3, 8)\}\) is as follows:</p>
+<p>\begin{align*}
+P(x) = 0 \frac{(x - 2)(x - 3)}{(1 - 2) (1 - 3)} + 3 \frac{(x - 1)(x - 3)}{(2 - 1) (2 - 3)} + 8 \frac{(x - 1)(x - 2)}{(3 - 1) (3 - 2)} = x^{2} - 1
+\end{align*}</p>
+<p>Note that Lagrange interpolation finds the lowest degree of interpolating polynomial for the given vector.</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>pub fn interpolate(x_values: &amp;[F], y_values: &amp;[F]) -&gt; Polynomial&lt;F&gt; {
+    let mut lagrange_polys = vec![];
+    let numerators = Polynomial::from_monomials(x_values);
+
+    for j in 0..x_values.len() {
+        let mut denominator = F::one(None);
+        for i in 0..x_values.len() {
+            if i != j {
+                denominator = denominator * (x_values[j].clone() - x_values[i].clone());
+            }
+        }
+        let cur_poly = numerators
+            .clone()
+            .div(Polynomial::from_monomials(&amp;[x_values[j].clone()]).scalar_mul(&amp;denominator));
+        lagrange_polys.push(cur_poly);
+    }
+
+    let mut result = Polynomial {
+        poly: vec![],
+        var: "x".to_string(),
+    };
+    for (j, lagrange_poly) in lagrange_polys.iter().enumerate() {
+        result = result + lagrange_poly.scalar_mul(&amp;y_values[j]);
+    }
+    result
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="proposition-homomorphisms-of-lagrange-interpolation"><a class="header" href="#proposition-homomorphisms-of-lagrange-interpolation">Proposition: Homomorphisms of Lagrange Interpolation</a></h3>
+<p><em>Let \(L(v)\) and \(L(w)\) be the polynomial resulting from Lagrange Interpolation on the output (\(y\)) vector \(v\) and \(w\) for the same inputs (\(x\)). Then, the following properties hold:</em></p>
+<ul>
+<li><em>Additivity: \(L(v + w) = L(v) + L(w)\) for any vectors \(v\) and \(w\)</em></li>
+<li><em>Scalar multiplication: \(L(\gamma v) = \gamma L(v)\) for any scalar \(\gamma\) and vector \(v\)</em></li>
+</ul>
+<p><strong>Proof</strong></p>
+<p>Let \(v = (v_1, \ldots, v_n)\) and \(w = (w_1, \ldots, w_n)\) be vectors, and \(x_1, \ldots, x_n\) be the interpolation points.</p>
+<p>\begin{align*}
+L(v + w) &amp;= \sum_{i=1}^n (v_i + w_i) \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \sum_{i=1}^n v_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} + \sum_{i=1}^n w_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = L(v) + L(w) \\
+L(\gamma v) &amp;= \sum_{i=1}^n (\gamma v_i) \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \gamma \sum_{i=1}^n v_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \gamma L(v)
+\end{align*}</p>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="galois-field"><a class="header" href="#galois-field">Galois Field</a></h1>
+<p>We will now discuss the construction of finite fields with \(p^n\) elements, where \(p\) is a prime number and \(n\) is a positive integer. These fields are also known as Galois fields, denoted as \(GF(p^n)\). It is evident that \(\mathbb{Z}/p\mathbb{Z}\) is isomorphic to \(GF(p)\).</p>
+<h3 id="definition-irreducible-polynomial"><a class="header" href="#definition-irreducible-polynomial">Definition: Irreducible Polynomial</a></h3>
+<hr />
+<p><em>A polynomial \(f \in (\mathbb{Z}/p\mathbb{Z})[X]\) of degree \(n\) is called irreducible over \(\mathbb{Z}/p\mathbb{Z}\) if it cannot be factored as a product of two polynomials of lower degree in \((\mathbb{Z}/p\mathbb{Z})[X]\).</em></p>
+<hr />
+<p><strong>Example:</strong> In \((\mathbb{Z}/2\mathbb{Z})[X]\):</p>
+<ul>
+<li>\(X^2 + X + 1\) is irreducible</li>
+<li>\(X^2 + 1 = (X + 1)^2\) is reducible</li>
+</ul>
+<p>To construct \(GF(p^n)\), we use an irreducible polynomial of degree \(n\) over \(\mathbb{Z}/p\mathbb{Z}\).</p>
+<h3 id="definition-residue-class-modulo-a-polynomial"><a class="header" href="#definition-residue-class-modulo-a-polynomial">Definition: Residue Class modulo a Polynomial</a></h3>
+<hr />
+<p><em>For \(f, g \in (\mathbb{Z}/p\mathbb{Z})[X]\), the residue class of \(g \bmod f\) is the set of all polynomials \(h \in (\mathbb{Z}/p\mathbb{Z})[X]\) such that \(h \equiv g \pmod{f}\). This class is denoted as:</em></p>
+<p>\[ g + f(\mathbb{Z}/p\mathbb{Z})[X] = \{g + hf : h \in (\mathbb{Z}/p\mathbb{Z})[X]\} \]</p>
+<hr />
+<p><strong>Example:</strong> In \((\mathbb{Z}/2\mathbb{Z})[X]\), with \(f(X) = X^2 + X + 1\), the residue classes \(\bmod f\) are:</p>
+<ul>
+<li>\(0 + f(\mathbb{Z}/2\mathbb{Z})[X] = \{0, X^2 + X + 1, X^2 + X, X^2 + 1, X^2, X + 1, X, 1\}\)</li>
+<li>\(1 + f(\mathbb{Z}/2\mathbb{Z})[X] = \{1, X^2 + X, X^2 + 1, X^2, X + 1, X, 0, X^2 + X + 1\}\)</li>
+<li>\(X + f(\mathbb{Z}/2\mathbb{Z})[X] = \{X, X^2 + 1, X^2, X^2 + X + 1, X + 1, 1, X^2 + X, 0\}\)</li>
+<li>\((X + 1) + f(\mathbb{Z}/2\mathbb{Z})[X] = \{X + 1, X^2, X^2 + X + 1, X^2 + X, 1, 0, X^2 + 1, X\}\)</li>
+</ul>
+<p>These four residue classes form \(GF(4)\).</p>
+<h3 id="theorem-1"><a class="header" href="#theorem-1">Theorem:</a></h3>
+<hr />
+<p><em>If \(f \in (\mathbb{Z}/p\mathbb{Z})[X]\) is an irreducible polynomial of degree \(n\), then the residue ring \((\mathbb{Z}/p\mathbb{Z})[X]/(f)\) is a field with \(p^n\) elements, isomorphic to \(GF(p^n)\).</em></p>
+<hr />
+<p><strong>Proof</strong> (Outline) The irreducibility of \(f\) ensures that \((f)\) is a maximal ideal in \((\mathbb{Z}/p\mathbb{Z})[X]\), making the quotient ring a field. The number of elements is \(p^n\) because there are \(p^n\) polynomials of degree less than \(n\) over \(\mathbb{Z}/p\mathbb{Z}\).</p>
+<p>This construction allows us to represent elements of \(GF(p^n)\) as polynomials of degree less than \(n\) over \(\mathbb{Z}/p\mathbb{Z}\). Addition is performed coefficient-wise modulo \(p\), while multiplication is performed modulo the irreducible polynomial \(f\).</p>
+<p><strong>Example:</strong> To construct \(GF(8)\), we can use the irreducible polynomial \(f(X) = X^3 + X + 1\) over \(\mathbb{Z}/2\mathbb{Z}\). The elements of \(GF(8)\) are represented by:
+[ \{0, 1, X, X+1, X^2, X^2+1, X^2+X, X^2+X+1\} ]
+For instance, multiplication in \(GF(8)\):
+[ (X^2 + 1)(X + 1) = X^3 + X^2 + X + 1 \equiv X^2 \pmod{X^3 + X + 1} ]</p>
+<h3 id="lemma-schwartz---zippel-lemma"><a class="header" href="#lemma-schwartz---zippel-lemma">Lemma: Schwartz - Zippel Lemma</a></h3>
+<hr />
+<p><em>Let \(\mathbb{F}\) be a field and \(P: F^m \rightarrow \mathbb{F}\) and \(Q: \mathbb{F}^m \rightarrow \mathbb{F}\) be two distinct multivariate polynomials of total degree at most \(n\). For any finite subset \(\mathbb{S} \subseteq \mathbb{F}\), we have:</em></p>
+<p>\begin{align}
+Pr_{u \sim \mathbb{S}^{m}}[P(u) = Q(u)] \leq \frac{n}{|\mathbb{S}|}
+\end{align}</p>
+<p><em>, where \(u\) is drawn uniformly at random from \(\mathbb{S}^m\).</em></p>
+<hr />
+<p><strong>Proof</strong> TBD</p>
+<p>This lemma states that if \(\mathbb{S}\) is sufficiently large and \(n\) is relatively small, the probability that the two different polynomials return the same value for a randomly chosen input is negligibly small. In other words, if we observe \(P(u) = Q(u)\) for a random input \(u\), we can conclude with high probability that \(P\) and \(Q\) are identical polynomials.</p>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="elliptic-curve"><a class="header" href="#elliptic-curve">Elliptic curve</a></h1>
+<h3 id="definition-elliptic-curve"><a class="header" href="#definition-elliptic-curve">Definition: Elliptic Curve</a></h3>
+<hr />
+<p><em>An elliptic curve \(E\) over a finite field \(\mathbb{F}_{p}\) is defined by the equation:</em></p>
+<p>\begin{equation}
+y^2 = x^3 + a x + b
+\end{equation}</p>
+<p><em>, where \(a, b \in \mathbb{F}_{p}\) and the discriminant \(\Delta_{E} = 4a^3 + 27b^2 \neq 0\).</em></p>
+<hr />
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>use crate::modules::field::Field;
+
+#[derive(Debug, Clone, PartialEq)]
+pub struct EllipticCurve&lt;F: Field&gt; {
+    pub a: F,
+    pub b: F,
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-mathbbf_p-rational-point"><a class="header" href="#definition-mathbbf_p-rational-point">Definition: \(\mathbb{F}_{p}\)-Rational Point</a></h3>
+<hr />
+<p><em>An \(\mathbb{F}_{p}\)-rational point on an elliptic curve \(E\) is a point \((x, y)\) where both \(x\) and \(y\) are elements of \(\mathbb{F}_{p}\) and satisfy the curve equation, or the point at infinity \(\mathcal{O}\).</em></p>
+<hr />
+<p>Example: For \(E: y^2 = x^3 + 3x + 4\) over \(\mathbb{F}_7\), the \(\mathbb{F}_{7}\)-rational points are:
+\(\{(0, 2), (0, 5), (1, 1), (1, 6), (2, 2), (2, 5), (5, 1), (5, 6), \mathcal{O}\}\)</p>
+<h3 id="definition-the-point-at-infinity"><a class="header" href="#definition-the-point-at-infinity">Definition: The point at infinity</a></h3>
+<hr />
+<p>The point at infinity denoted \(\mathcal{O}\), is a special point on the elliptic curve that serves as the identity element for the group operation. It can be visualized as the point where all vertical lines on the curve meet.</p>
+<hr />
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>#[derive(Debug, Clone, PartialEq)]
+pub struct EllipticCurvePoint&lt;F: Field&gt; {
+    pub x: Option&lt;F&gt;,
+    pub y: Option&lt;F&gt;,
+    pub curve: EllipticCurve&lt;F&gt;,
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-addition-on-elliptic-curve"><a class="header" href="#definition-addition-on-elliptic-curve">Definition: Addition on Elliptic Curve</a></h3>
+<hr />
+<p><em>For an elliptic curve \(E: y^2 = x^3 + ax + b\), the addition of points \(P\) and \(Q\) to get \(R = P + Q\) is defined as follows:</em></p>
+<ul>
+<li>
+<p><em>If \(P = \mathcal{O}\), \(R = Q\).</em></p>
+</li>
+<li>
+<p><em>If \(Q = \mathcal{O}\), \(R = P\).</em></p>
+</li>
+<li>
+<p><em>Otherwise, let \(P = (x_P, y_P), Q = (x_Q, y_Q)\). Then:</em></p>
+<ul>
+<li><em>If \(y_P = -y_Q\), \(R = \mathcal{O}\)</em></li>
+<li><em>If \(y_P \neq -y_Q\), \(R = (x_R = \lambda^2 - x_P - x_Q, y_R = \lambda(x_P - x_R) - y_P)\), where \(\lambda = \)</em> \( \begin{cases}
+\frac{y_P - y_Q}{x_P - x_Q} \quad &amp;\hbox{If } (x_P \neq x_Q) \\
+\frac{3^{2}_{P} + a}{2y_P} \quad &amp;\hbox{Otherwise}
+\end{cases}\)</li>
+</ul>
+</li>
+</ul>
+<hr />
+<p>Example: On \(E: y^2 = x^3 + 2x + 3\) over \(\mathbb{F}_{7}\), let \(P = (5, 1)\) and \(Q = (4, 4)\). Then, \(P + Q = (0, 5)\), where \(\lambda = \frac{1 - 4}{5 - 4} \equiv 4 \bmod 7\).</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>impl&lt;F: Field&gt; EllipticCurvePoint&lt;F&gt; {
+    fn new(x: F, y: F, curve: EllipticCurve&lt;F&gt;) -&gt; Self {
+        EllipticCurvePoint {
+            x: Some(x),
+            y: Some(y),
+            curve: curve,
+        }
+    }
+
+    pub fn point_at_infinity(curve: EllipticCurve&lt;F&gt;) -&gt; Self {
+        EllipticCurvePoint {
+            x: None,
+            y: None,
+            curve: curve,
+        }
+    }
+
+    pub fn is_point_at_infinity(&amp;self) -&gt; bool {
+        self.x.is_none() || self.y.is_none()
+    }
+
+    pub fn line_slope(&amp;self, other: Self) -&gt; F {
+        let x1 = self.x.clone().unwrap();
+        let y1 = self.y.clone().unwrap();
+        let x2 = other.x.clone().unwrap();
+        let y2 = other.y.clone().unwrap();
+
+        if self.x.clone() == other.x.clone() {
+            ((x1.clone() * x1.clone()) * (3_i64) + self.curve.a.clone()) / (y1.clone() * (2_i64))
+        } else {
+            (y2.clone() - y1.clone()) / (x2.clone() - x1.clone())
+        }
+    }
+}
+
+impl&lt;F: Field&gt; Add for EllipticCurvePoint&lt;F&gt; {
+    type Output = Self;
+
+    fn add(self, other: Self) -&gt; Self {
+        if self.is_point_at_infinity() {
+            return other;
+        }
+        if other.is_point_at_infinity() {
+            return self;
+        }
+
+        let m = self.line_slope(other.clone());
+
+        if self.x == other.x {
+            if self.y != other.y {
+                return EllipticCurvePoint::point_at_infinity(self.curve.clone());
+            } else {
+                let x1 = self.x.clone().unwrap();
+                let y1 = self.y.clone().unwrap();
+
+                let x3 = m.clone() * m.clone() - x1.clone() - x1.clone();
+                let y3 = m * (x1 - x3.clone()) - y1;
+
+                return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+            }
+        } else {
+            let x1 = self.x.clone().unwrap();
+            let y1 = self.y.clone().unwrap();
+            let x2 = other.x.clone().unwrap();
+            let x3 = m.clone() * m.clone() - x1.clone() - x2.clone();
+            let y3 = m * (x1 - x3.clone()) - y1;
+
+            return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+        }
+    }
+}
+<span class="boring">}</span></code></pre></pre>
+<h3 id="definition-mordell-weil-group"><a class="header" href="#definition-mordell-weil-group">Definition: Mordell-Weil Group</a></h3>
+<hr />
+<p><em>The Mordell-Weil group of an elliptic curve \(E\) is the group of rational points on \(E\) under the addition operation defined above.</em></p>
+<hr />
+<p>Example: For \(E: y^2 = x^3 + x + 6\) over \(\mathbb{F}_{11}\), the Mordell-Weil group is the set of all \(\mathbb{F}_{11}\)-rational points on \(E\) with the elliptic curve addition operation.</p>
+<h3 id="definition-group-order"><a class="header" href="#definition-group-order">Definition: Group Order</a></h3>
+<hr />
+<p><em>The group order of an elliptic curve \(E\) over \(\mathbb{F}_{p}\), denoted \(\#E(\mathbb{F}_{p})\), is the number of \(\mathbb{F}_{p}\)-rational points on \(E\), including the point at infinity.</em></p>
+<hr />
+<p>Example: For \(E: y^2 = x^3 + x + 6\) over \(\mathbb{F}_{11}\), \(\#E(\mathbb{F}_{11}) = 13\).</p>
+<h3 id="theorem-hasse-weil"><a class="header" href="#theorem-hasse-weil">Theorem: Hasse-Weil</a></h3>
+<hr />
+<p><em>Let \(\#E(\mathbb{F}_{p})\) be the group order of the elliptic curve \(E\) over \(\mathbb{F}_{p}\). Then:</em></p>
+<p>\begin{equation}
+p + 1 - 2 \sqrt{p} \leq \#E \leq p + 1 + 2 \sqrt{p}
+\end{equation}</p>
+<hr />
+<p><strong>Example:</strong> For an elliptic curve over \(\mathbb{F}_{23}\), the Hasse-Weil theorem guarantees that:</p>
+<p>\begin{equation*}
+23 + 1 - 2 \sqrt{23} \simeq 14.42 \#E(\mathbb{F}_{23}) \geq 23 + 1 + 2 \sqrt{23} \simeq 33.58
+\end{equation*}</p>
+<h3 id="definition-point-order"><a class="header" href="#definition-point-order">Definition: Point Order</a></h3>
+<p>The order of a point \(P\) on an elliptic curve is the smallest positive integer \(n\) such that \(nP = \mathcal{O}\) (where \(nP\) denotes \(P\) added to itself \(n\) times). We also denote the set of points of order \(n\), also called \textit{torsion} group, by</p>
+<p>\begin{equation}
+E[n] = \{P \in E: [n]P = \mathcal{O}\}
+\end{equation}</p>
+<p>Example: On \(E: y^2 = x^3 + 2x + 2\) over \(\mathbb{F}_{17}\), the point \(P = (5, 1)\) has order 18 because \(18P = \mathcal{O}\), and no smaller positive multiple of \(P\) equals \(\mathcal{O}\).</p>
+<p>The intuitive view is that if you continue to add points to themselves (doubling, tripling, etc.), the lines drawn between the prior point and the next point will eventually become more vertical. When the line becomes vertical, it does not intersect the elliptic curve at any finite point. In elliptic curve geometry, a vertical line is considered to "intersect" the curve at a special place called the "point at infinity," This point is like a north pole in geographic terms: no matter which direction you go, if the line becomes vertical (reaching infinitely high), it converges to this point.</p>
+<h3 id="definition-field-extension"><a class="header" href="#definition-field-extension">Definition: Field Extension</a></h3>
+<hr />
+<p><em>Let \(F\) and \(L\) be fields. If \(F \subseteq L\) and the operations of \(F\) are the same as those of \(L\), we call \(L\) a field extension of \(F\). This is denoted as \(L/F\).</em></p>
+<hr />
+<p>A field extension \(L/F\) naturally gives \(L\) the structure of a vector space over \(F\). The dimension of this vector space is called the degree of the extension.</p>
+<p><strong>Examples:</strong></p>
+<ul>
+<li>\(\mathbb{C}/\mathbb{R}\) is a field extension with basis \(\{1, i\}\) and degree 2.</li>
+<li>\(\mathbb{R}/\mathbb{Q}\) is an infinite degree extension.</li>
+<li>\(\mathbb{Q}(\sqrt{2})/\mathbb{Q}\) is a degree 2 extension with basis \(\{1, \sqrt{2}\}\).</li>
+</ul>
+<h3 id="definition-algebraic-extension"><a class="header" href="#definition-algebraic-extension">Definition: Algebraic Extension</a></h3>
+<hr />
+<p><em>A field extension \(L/K\) is called algebraic if every element \(\alpha \in L\) is algebraic over \(K\), i.e., \(\alpha\) is the root of some non-zero polynomial with coefficients in \(K\).</em></p>
+<p><em>For an algebraic element \(\alpha \in L\) over \(K\), we denote by \(K(\alpha)\) the smallest field containing both \(K\) and \(\alpha\).</em></p>
+<hr />
+<p><strong>Example:</strong></p>
+<ul>
+<li>\(\mathbb{C}/\mathbb{R}\) is algebraic: any \(z = a + bi \in \mathbb{C}\) is a root of \(x^2 - 2ax + (a^2 + b^2) \in \mathbb{R}[x]\).</li>
+<li>\(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}\) is algebraic: \(\sqrt[3]{2}\) is a root of \(x^3 - 2 \in \mathbb{Q}[x]\).</li>
+<li>\(\mathbb{R}/\mathbb{Q}\) is not algebraic (a field extension that is not algebraic is called \textit{transcendental}).</li>
+</ul>
+<h3 id="definition-field-of-rational-functions"><a class="header" href="#definition-field-of-rational-functions">Definition: Field of Rational Functions</a></h3>
+<hr />
+<p><em>Let \(K\) be a field and \(X\) be indeterminate. The field of rational functions over \(K\), denoted \(K(X)\), is defined as:</em></p>
+<p>\begin{equation*}
+K(X) = \left\{ \frac{f(X)}{g(X)} \middle|\ f(X), g(X) \in K[X], g(X) \neq 0 \right\}
+\end{equation*}</p>
+<p><em>where \(K[X]\) is the ring of polynomials in \(X\) with coefficients in \(K\).</em></p>
+<hr />
+<p>\(K(X)\) can be viewed as the field obtained by adjoining a letter \(X\) to \(K\). This construction generalizes to multiple variables, e.g., \(K(X,Y)\).</p>
+<p>The concept of a function field naturally arises in the context of algebraic curves, particularly elliptic curves. Intuitively, the function field encapsulates the algebraic structure of rational functions on the curve.</p>
+<p>We first construct the coordinate ring for an elliptic curve \(E: y^2 = x^3 + ax + b\). Consider functions \(X: E \to K\) and \(Y: E \to K\) that extract the \(x\) and \(y\) coordinates, respectively, from an arbitrary point \(P \in E\). These functions generate the polynomial ring \(K[X,Y]\), subject to the relation \(Y^2 = X^3 + aX + b\).</p>
+<p>To put it simply, the function field is a field that consists of all functions that determine the value based on the point on the curve.</p>
+<h3 id="definition-coordinate-ring-of-an-elliptic-curve"><a class="header" href="#definition-coordinate-ring-of-an-elliptic-curve">Definition: Coordinate Ring of an Elliptic Curve</a></h3>
+<hr />
+<p><em>The <strong>coordinate ring</strong> of an elliptic curve \(E: y^2 = x^3 + ax + b\) over a field \(K\) is defined as:</em>
+\begin{equation}
+K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)
+\end{equation}</p>
+<hr />
+<p>In other words, we can view \(K[E]\) as a ring representing all polynomial functions on \(E\). Recall that \(K[X, Y]\) is the polynomial ring in two variables \(X\) and \(Y\) over the field K, meaning that it contains all polynomials in \(X\) and \(Y\) with coefficients from \(K\). Then, the notation \(K[X, Y]/(Y^2 - X^3 - aX - b)\) denotes the quotient ring obtained by taking \(K[X, Y]\) and "modding out" by the ideal \((Y^2 - X^3 - aX - b)\).</p>
+<p><strong>Example</strong> For example, for an elliptic curve \(E: y^2 = x^3 - x\) over \(\mathbb{Q}\), some elements of the coordinate ring \(\mathbb{Q}[E]\) include:</p>
+<ul>
+<li>Constants: \(3, -2, \frac{1}{7}, \ldots\)</li>
+<li>Linear functions: \(X, Y, 2X+3Y, \ldots\)</li>
+<li>Quadratic functions: \(X^2, XY, Y^2 (= X^3 - X), \ldots\)</li>
+<li>Higher-degree functions: \(X^3, X^2Y, XY^2 (= X^4 - X^2), \ldots\)</li>
+</ul>
+<p>Then, the function field is defined as follows:</p>
+<h3 id="definition-function-field-of-an-elliptic-curve"><a class="header" href="#definition-function-field-of-an-elliptic-curve">Definition: Function Field of an Elliptic Curve</a></h3>
+<hr />
+<p><em>Let \(E: y^2 = x^3 + ax + b\) be an elliptic curve over a field \(K\). The <strong>function field</strong> of \(E\), denoted \(K(E)\), is defined as:</em>
+\begin{equation}
+K(E) = \left\{\frac{f}{g} ,\middle|, f, g \in K[E], g \neq 0 \right\}
+\end{equation}
+<em>where \(K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)\) is the coordinate ring of \(E\).</em></p>
+<hr />
+<p>\(K(E)\) can be viewed as the field of all rational functions on \(E\).</p>
+<h3 id="definition-zero-of-a-function"><a class="header" href="#definition-zero-of-a-function">Definition: Zero of a Function</a></h3>
+<hr />
+<p><em>Let \(h \in K(E)\) be a non-zero function. A point \(P \in E\) is called a <strong>zero</strong> of \(h\) if \(h(P) = 0\).</em></p>
+<hr />
+<h3 id="definition-pole-of-a-function"><a class="header" href="#definition-pole-of-a-function">Definition: Pole of a Function</a></h3>
+<hr />
+<p><em>Let \(h \in K(E)\) be a non-zero function. A point \(P \in E\) is called a <strong>pole</strong> of \(h\) if \(h\) is not defined at \(P\) or, equivalently if \(1/h\) has a zero at \(P\).</em></p>
+<hr />
+<p><strong>Example</strong> Consider the elliptic curve \(E: Y^2 = X^3 - X\) over a field \(K\) of characteristic \(\neq 2, 3\). Let \(P_{-1} = (-1, 0)\), \(P_0 = (0, 0)\), and \(P_1 = (1, 0)\) be the points where \(Y = 0\).</p>
+<ul>
+<li>The function \(Y \in K(E)\) has three simple zeros: \(P_{-1}\), \(P_0\), and \(P_1\).</li>
+<li>The function \(X \in K(E)\) has a double zero at \(P_0\) (since \(P_0 = -P_0\)).</li>
+<li>The function \(X - 1 \in K(E)\) has a simple zero at \(P_1\).</li>
+<li>The function \(X^2 - 1 \in K(E)\) has two simple zeros at \(P_{-1}\) and \(P_1\).</li>
+<li>The function \(\frac{Y}{X} \in K(E)\) has a simple zero at \(P_{-1}\) and a simple pole at \(P_0\).</li>
+</ul>
+<p>An important property of functions in \(K(E)\) is that they have the same number of zeros and poles when counted with multiplicity. This is a consequence of a more general result known as the Degree-Genus Formula.</p>
+<h3 id="theorem-degree-genus-formula-for-elliptic-curves"><a class="header" href="#theorem-degree-genus-formula-for-elliptic-curves">Theorem: Degree-Genus Formula for Elliptic Curves</a></h3>
+<hr />
+<p><em>Let \(f \in K(E)\) be a non-zero rational function on an elliptic curve \(E\). Then:</em></p>
+<p>\begin{equation}
+\sum_{P \in E} ord_{P(f)} = 0
+\end{equation}</p>
+<p><em>, where \(ord_{P(f)}\) denotes the order of \(f\) at \(P\), which is positive for zeros and negative for poles.</em></p>
+<hr />
+<p>This theorem implies that the total number of zeros (counting multiplicity) equals the total number of poles for any non-zero function in \(K(E)\).</p>
+<p>We now introduce a powerful tool for analyzing functions on elliptic curves: the concept of divisors.</p>
+<h3 id="definition-divisor-of-a-function-on-an-elliptic-curve"><a class="header" href="#definition-divisor-of-a-function-on-an-elliptic-curve">Definition: Divisor of a Function on an Elliptic Curve</a></h3>
+<hr />
+<p><em>Let \(E: Y^2 = X^3 + AX + B\) be an elliptic curve over a field \(K\), and let \(f \in K(E)\) be a non-zero rational function on \(E\). The <strong>divisor</strong> of \(f\), denoted \(div(f)\), is defined as:</em>
+\begin{equation}
+div(f) = \sum_{P \in E} ord_P(f) [P]
+\end{equation}
+<em>, where \(ord_P(f)\) is the order of \(f\) at \(P\) (positive for zeros, negative for poles), and the sum is taken over all points \(P \in E\), including the point at infinity. This sum has only finitely many non-zero terms.</em></p>
+<hr />
+<p>Note that this \(\sum_{P \in E}\) is a symbolic summation, and we do not calculate the concrete value of a divisor.</p>
+<p><strong>Example</strong> Consider the elliptic curve \(E: Y^2 = X^3 - X\) over \(\mathbb{Q}\).</p>
+<ul>
+<li>For \(f = X\), we have \(div(X) = 2[(0,0)] - 2[\mathcal{O}]\).</li>
+<li>For \(g = Y\), we have \(div(Y) = [(1,0)] + [(0,0)] + [(-1,0)] - 3[\mathcal{O}]\).</li>
+<li>For \(h = \frac{X-1}{Y}\), we have \(div(h) = [(1,0)] - [(-1,0)]\).</li>
+</ul>
+<p>The concept of divisors can be extended to the elliptic curve itself:</p>
+<h3 id="definition-divisor-on-an-elliptic-curve"><a class="header" href="#definition-divisor-on-an-elliptic-curve">Definition: Divisor on an Elliptic Curve</a></h3>
+<hr />
+<p><em>A <strong>divisor</strong> \(D\) on an elliptic curve \(E\) is a formal sum</em>
+\begin{equation}
+D = \sum_{P \in E} n_P [P]
+\end{equation}
+<em>where \(n_P \in \mathbb{Z}\) and \(n_P = 0\) for all but finitely many \(P\).</em></p>
+<hr />
+<p><strong>Example</strong> On the curve \(E: Y^2 = X^3 - X\):</p>
+<ul>
+<li>\(D_1 = 3[(0,0)] - 2[(1,1)] - [(2,\sqrt{6})]\) is a divisor.</li>
+<li>\(D_2 = [(1,0)] + [(-1,0)] - 2[\mathcal{O}]\) is a divisor.</li>
+<li>\(D_3 = \sum_{P \in E[2]} [P] - 4[\mathcal{O}]\) is a divisor (where \(E[2]\) are the 2-torsion points).</li>
+</ul>
+<p>To quantify the properties of divisors, we introduce two important metrics:</p>
+<h3 id="definition-degree-of-a-divisor"><a class="header" href="#definition-degree-of-a-divisor">Definition: Degree of a Divisor</a></h3>
+<hr />
+<p><em>The <strong>degree</strong> of a divisor \(D = \sum_{P \in E} n_P [P]\) is defined as:</em>
+\begin{equation}
+deg(D) = \sum_{P \in E} n_P
+\end{equation}</p>
+<hr />
+<h3 id="definition-sum-of-a-divisor"><a class="header" href="#definition-sum-of-a-divisor">Definition: Sum of a Divisor</a></h3>
+<hr />
+<p><em>The <strong>sum</strong> of a divisor \(D = \sum_{P \in E} n_P [P]\) is defined as:</em>
+\begin{equation}
+Sum(D) = \sum_{P \in E} n_P P
+\end{equation}
+<em>where \(n_P P\) denotes the point addition of \(P\) to itself \(n_P\) times in the group law of \(E\).</em></p>
+<hr />
+<p><strong>Example</strong> For the divisors in the previous example:</p>
+<ul>
+<li>\(deg(D_1) = 3 - 2 - 1 = 0\)</li>
+<li>\(deg(D_2) = 1 + 1 - 2 = 0\)</li>
+<li>\(deg(D_3) = 4 - 4 = 0\)</li>
+<li>\(Sum(D_2) = (1,0) + (-1,0) - 2\mathcal{O} = \mathcal{O}\) (since \((1,0)\) and \((-1,0)\) are 2-torsion points)</li>
+</ul>
+<p>The following theorem characterizes divisors of functions and provides a criterion for when a divisor is the divisor of a function:</p>
+<h3 id="theorem-2"><a class="header" href="#theorem-2">Theorem</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve over a field \(K\).</em></p>
+<ul>
+<li><em>If \(f, f' \in K(E)\) are non-zero rational functions on \(E\) with \(div(f) = div(f')\), then there exists a non-zero constant \(c \in K^</em>\) such that \(f = cf'\).*</li>
+<li><em>A divisor \(D\) on \(E\) is the divisor of a rational function on \(E\) if and only if \(deg(D) = 0\) and \(Sum(D) = \mathcal{O}\).</em></li>
+</ul>
+<hr />
+<p><strong>Example</strong> On \(E: Y^2 = X^3 - X\):</p>
+<ul>
+<li>The function \(f = \frac{Y}{X}\) has \(div(f) = [(1,0)] + [(-1,0)] - 2[(0,0)]\). Note that \(deg(div(f)) = 0\) and \(Sum(div(f)) = (1,0) + (-1,0) - 2(0,0) = \mathcal{O}\).</li>
+<li>The divisor \(D = 2[(1,1)] - [(2,\sqrt{6})] - [(0,0)]\) has \(deg(D) = 0\), but \(Sum(D) \neq \mathcal{O}\). Therefore, \(D\) is not the divisor of any rational function on \(E\).</li>
+</ul>
+<h3 id="definition-pairing"><a class="header" href="#definition-pairing">Definition: Pairing</a></h3>
+<hr />
+<p><em>Let \(G_1\) and \(G_2\) be cyclic groups under addition, both of prime order \(p\), with generators \(P\) and \(Q\) respectively:</em></p>
+<p>\begin{align}
+G_1 &amp;= \{0, P, 2P, ..., (p-1)P\} \
+G_2 &amp;= \{0, Q, 2Q, ..., (p-1)Q\}
+\end{align}</p>
+<p><em>Let \(G_T\) be a cyclic group under multiplication, also of order \(p\). A pairing is a map \(e: G_1 \times G_2 \rightarrow G_T\) that satisfies the following bilinear property:</em></p>
+<p>\begin{equation}
+e(aP, bQ) = e(P, Q)^{ab}
+\end{equation} <em>for all \(a, b \in \mathbb{Z}_p\).</em></p>
+<hr />
+<p>Imagine \(G_1\) represents length, \(G_2\) represents width, and \(G_T\) represents area. The pairing function \(e\) is like calculating the area: If you double the length and triple the width, the area becomes six times larger: \(e(2P, 3Q) = e(P, Q)^{6}\)</p>
+<p>The Weil pairing is one of the bilinear pairings for elliptic curves. We begin with its formal definition.</p>
+<h3 id="definition-the-weil-pairing"><a class="header" href="#definition-the-weil-pairing">Definition: The Weil Pairing</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve and \(n\) be a positive integer. For points \(P, Q \in E[n]\), where \(E[n]\) denotes the \(n\)-torsion subgroup of \(E\), we define the Weil pairing \(e_n(P, Q)\) as follows:</em></p>
+<p>Let \(f_P\) and \(f_Q\) be rational functions on \(E\) satisfying:
+\begin{align}
+div(f_P) &amp;= n[P] - n[\mathcal{O}] \
+div(f_Q) &amp;= n[Q] - n[\mathcal{O}]
+\end{align}</p>
+<p><em>Then, for an arbitrary point \(S \in E\) such that \(S \notin \{\mathcal{O}, P, -Q, P-Q\}\), the Weil pairing is given by:</em></p>
+<p>\begin{equation}
+e_n(P, Q) = \frac{f_P(Q + S)}{f_P(S)} /\ \frac{f_Q(P - S)}{f_Q(-S)}
+\end{equation}</p>
+<hr />
+<p>We introduce a crucial theorem about a specific rational function on elliptic curves to construct the functions required for the Weil pairing.</p>
+<h3 id="theorem-3"><a class="header" href="#theorem-3">Theorem</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve over a field \(K\), and let \(P = (x_P, y_P)\) and \(Q = (x_Q, y_Q)\) be non-zero points on \(E\). Define \(\lambda\) as:</em></p>
+<p>\begin{equation}
+\lambda = \begin{cases}
+\hbox{slope of the line through \(P\) and \(Q\)} &amp;\quad \hbox{if \(P \neq Q\)} \\
+\hbox{slope of the tangent line to \(E\) at \(P\)} &amp;\quad \hbox{if \(P = Q\)} \\
+\infty &amp;\quad \hbox{if the line is vertical}
+\end{cases}
+\end{equation}</p>
+<p><em>Then, the function \(g_{P,Q}: E \to K\) defined by:</em>
+\begin{equation}
+g_{P,Q} = \begin{cases}
+\frac{y - y_P - \lambda(x - x_P)}{x + x_P + x_Q - \lambda^2} &amp;\quad \hbox{if } \lambda \neq \infty \\
+x - x_P &amp;\quad \hbox{if } \lambda = \infty
+\end{cases}
+\end{equation} <em>has the following divisor:</em></p>
+<p>\begin{equation}
+div(g_{P,Q}) = [P] + [Q] - [P + Q] - [\mathcal{O}]
+\end{equation}</p>
+<hr />
+<p><strong>Proof:</strong> We consider two cases based on the value of \(\lambda\).</p>
+<p>Case 1: \(\lambda \neq \infty\)</p>
+<p>Let \(y = \lambda x + v\) be the line through \(P\) and \(Q\) (or the tangent line at \(P\) if \(P = Q\)). This line intersects \(E\) at three points: \(P\), \(Q\), and \(-P-Q\). Thus,
+\begin{equation}
+div(y - \lambda x - v) = [P] + [Q] + [-P - Q] - 3[\mathcal{O}]
+\end{equation}
+Vertical lines intersect \(E\) at points and their negatives, so:
+\begin{equation}
+div(x - x_{P+Q}) = [P + Q] + [-P - Q] - 2[\mathcal{O}]
+\end{equation}
+It follows that \(g_{P,Q} = \frac{y - \lambda x - v}{x - x_{P+Q}}\) has the desired divisor.</p>
+<p>Case 2: \(\lambda = \infty\)</p>
+<p>In this case, \(P + Q = \mathcal{O}\), so we want \(g_{P,Q}\) to have divisor \([P] + [-P] - 2[\mathcal{O}]\). The function \(x - x_P\) has this divisor.
+\end{proof}</p>
+<h3 id="theorem-millers-algorithm"><a class="header" href="#theorem-millers-algorithm">Theorem: Miller's Algorithm</a></h3>
+<hr />
+<p><em>Let \(m \geq 1\) and write its binary expansion as:</em>
+\begin{equation}
+m = m_0 + m_1 \cdot 2 + m_2 \cdot 2^2 + \cdots + m_{n-1} \cdot 2^{n-1}
+\end{equation}
+<em>where \(m_i \in \{0, 1\}\) and \(m_{n-1} \neq 0\).</em></p>
+<p><em>The following algorithm, using the function \(g_{P,Q}\) defined in the previous theorem, returns a function \(f_P\) whose divisor satisfies:</em></p>
+<p>\begin{equation}
+div(f_P) = m[P] - m[P] - (m - 1)[\mathcal{O}]
+\end{equation}</p>
+<p>===========================</p>
+<p><strong>Miller's Algorithm</strong></p>
+<ol>
+<li>Set \(T = P\) and \(f = 1\)</li>
+<li><strong>For</strong> \(i \gets n-2 \cdots 0\) <strong>do</strong></li>
+<li>    Set \(f = f^2 \cdot g_{T, T}\)</li>
+<li>    Set \(T = 2T\)</li>
+<li>    <strong>If</strong> \(m_i = 1\) <strong>then</strong></li>
+<li>        Set \(f = f \cdot g_{T, P}\)</li>
+<li>        Set \(T = T + P\)</li>
+<li>    <strong>End if</strong></li>
+<li><strong>End for</strong></li>
+<li><strong>Return</strong> \(f\)</li>
+</ol>
+<p>===========================</p>
+<hr />
+<p><strong>Proof:</strong> TBD</p>
+<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
+</span><span class="boring">fn main() {
+</span>pub fn get_lambda&lt;F: Field&gt;(
+    p: EllipticCurvePoint&lt;F&gt;,
+    q: EllipticCurvePoint&lt;F&gt;,
+    r: EllipticCurvePoint&lt;F&gt;,
+) -&gt; F {
+    let p_x = p.x.clone().unwrap();
+    let p_y = p.y.clone().unwrap();
+    let q_x = q.x.clone().unwrap();
+    // let q_y = q.y.clone().unwrap();
+    let r_x = r.x.clone().unwrap();
+    let r_y = r.y.clone().unwrap();
+
+    if (p == q &amp;&amp; p_y.clone() == F::zero()) || (p != q &amp;&amp; p_x.clone() == q_x.clone()) {
+        return r_x.clone() - p_x.clone();
+    }
+    let slope = p.line_slope(q.clone());
+    let numerator = r_y.clone() - p_y.clone() - slope.clone() * (r_x.clone() - p_x.clone());
+    let denominator = r_x.clone() + p_x.clone() + q_x.clone() - slope.clone() * slope.clone();
+    return numerator / denominator;
+}
+
+pub fn miller&lt;F: Field&gt;(p: EllipticCurvePoint&lt;F&gt;, q: EllipticCurvePoint&lt;F&gt;, m: BigInt) -&gt; F {
+    if p == q {
+        F::one();
+    }
+
+    let mut f = F::one();
+    let mut t = p.clone();
+
+    for i in (1..m.bits()).rev() {
+        f = (f.clone() * f.clone()) * (get_lambda(t.clone(), t.clone(), q.clone()));
+        t = t.clone() + t.clone();
+        if m.bit(i) {
+            f = f * (get_lambda(t.clone(), p.clone(), q.clone()));
+            t = t.clone() + p.clone();
+        }
+    }
+
+    f
+}
+
+pub fn weil_pairing&lt;F: Field&gt;(
+    p: EllipticCurvePoint&lt;F&gt;,
+    q: EllipticCurvePoint&lt;F&gt;,
+    m: BigInt,
+    s: Option&lt;EllipticCurvePoint&lt;F&gt;&gt;,
+) -&gt; F {
+    let s_value = s.unwrap();
+    let fp_qs = miller(p.clone(), q.clone() + s_value.clone(), m.clone());
+    let fp_s = miller(p.clone(), s_value.clone(), m.clone());
+    let fq_ps = miller(q.clone(), p.clone() - s_value.clone(), m.clone());
+    let fq_s = miller(q.clone(), -s_value.clone(), m.clone());
+
+    return (fp_qs / fp_s) / (fq_ps / fq_s);
+}
+<span class="boring">}</span></code></pre></pre>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="assumptions"><a class="header" href="#assumptions">Assumptions</a></h1>
+<h3 id="assumption-discrete-logarithm-problem"><a class="header" href="#assumption-discrete-logarithm-problem">Assumption: Discrete Logarithm Problem</a></h3>
+<hr />
+<p><em>Let \(G\) be a finite cyclic group of order \(n\), with \(\gamma\) as its generator and \(1\) as the identity element. For any element \(\alpha \in G\), there is currently no known efficient (polynomial-time) algorithm to compute the smallest non-negative integer \(x\) such that \(\alpha = \gamma^{x}\).</em></p>
+<hr />
+<p>The Discrete Logarithm Problem can be thought of as a one-way function. It's easy to compute \(g^{x}\) given \(g\) and \(x\), but it's computationally difficult to find \(x\) given \(g\) and \(g^{x}\).</p>
+<h3 id="assumption-elliptic-curve-discrete-logarithm-problem"><a class="header" href="#assumption-elliptic-curve-discrete-logarithm-problem">Assumption: Elliptic Curve Discrete Logarithm Problem</a></h3>
+<hr />
+<p><em>Let \(E\) be an elliptic curve defined over a finite field \(\mathbb{F}_q\), where \(q\) is a prime power. Let \(P\) be a point on \(E\) of point order \(n\), and let \(\langle P \rangle\) be the cyclic subgroup of \(E\) generated by \(P\). For any element \(Q \in \langle P \rangle\), there is currently no known efficient (polynomial-time) algorithm to compute the unique integer \(k\), \(0 \leq k &lt; n\), such that \(Q = kP\).</em></p>
+<hr />
+<p>This assumption is an elliptic curve version of the Discrete Logarithm Problem.</p>
+<h3 id="assumption-knowledge-of-exponent-assumption"><a class="header" href="#assumption-knowledge-of-exponent-assumption">Assumption: Knowledge of Exponent Assumption</a></h3>
+<hr />
+<p>Let \(G\) be a cyclic group of prime order \(q\) with generator \(g \in G\). For any probabilistic polynomial-time algorithm \(\mathcal{A}\) that outputs:</p>
+<p>\begin{equation}
+\mathcal{A}(g, g^x) = (h, h') \quad s.t. \quad h' = h^x
+\end{equation}
+, there exists an efficient extractor \(\mathcal{E}\) such that:
+\begin{equation}
+\mathcal{E}(\mathcal{A}, g, g^x) = y \quad s.t. \quad h = g^y
+\end{equation}</p>
 <hr />
-<div style="break-before: page; page-break-before: always;"></div><div style="break-before: page; page-break-before: always;"></div><div style="break-before: page; page-break-before: always;"></div>
+<p>This assumption states that if \(\mathcal{A}\) can compute the pair \((g^y, g^{xy})\) from \((g, g^x)\), then \(\mathcal{A}\) must "know" the value \(y\), in the sense that \(\mathcal{E}\) can extract \(y\) from \(\mathcal{A}\)'s internal state.
+The Knowledge of Exponent Assumption is useful for constructing verifiable exponential calculation algorithms. Consider a scenario where Alice has a secret value \(a\), and Bob has a secret value \(b\). Bob wants to obtain \(g^{ab}\). This can be achieved through the following protocol:</p>
+<p><strong>Verifiable Exponential Calculation Algorithm</strong></p>
+<ol>
+<li>Bob sends \((g, g'=g^{b})\) to Alice</li>
+<li>Alice sends \((h=g^{a}, h'=g'^{a})\) to Bob</li>
+<li>Bob checks \(h^{b} = h'\).</li>
+</ol>
+<p>Thanks to the Discrete Logarithm Assumption and the Knowledge of Exponent Assumption, the following properties hold:</p>
+<ul>
+<li>Bob cannot derive \(a\) from \((h, h')\).</li>
+<li>Alice cannot derive \(b\) from \((g, g')\).</li>
+<li>Alice cannot generate \((t, t')\) such that \(t \neq h\) and \(t^{b} = t'\).</li>
+<li>If \(h^{b} = h'\), Bob can conclude that \(h\) is the power of \(g\).</li>
+</ul>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="basics-of-zk-snark"><a class="header" href="#basics-of-zk-snark">Basics of zk-SNARK</a></h1>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="arithmetization"><a class="header" href="#arithmetization">Arithmetization</a></h1>
+<p>The ultimate goal of Zero-Knowledge Proofs (ZKP) is to allow the prover to demonstrate their knowledge to the verifier without revealing any additional information. This knowledge typically involves solving complex problems, such as finding a secret input value that corresponds to a given public hash. ZKP protocols usually convert these statements into polynomial constraints. This process is often called \textit{arithmetization}.</p>
+<p>To make the protocol flexible, we need to encode this knowledge in a specific format, and one common approach is using Boolean circuits. It's well-known that problems in P (those solvable in polynomial time) and NP (those where the solution can be verified in polynomial time) can be represented as Boolean circuits. This means adopting Boolean circuits allows us to handle both P and NP problems.</p>
+<p>However, Boolean circuits are often large and inefficient. Even a simple operation, like adding two 256-bit integers, can require hundreds of Boolean operators. In contrast, arithmetic circuits—essentially systems of equations involving addition, multiplication, and equality—offer a much more compact representation. Additionally, any Boolean circuit can be converted into an arithmetic circuit. For instance, the Boolean expression \(z = x \land y\) can be represented as \(x(x-1) = 0\), \(y(y-1) = 0\), and \(z = xy\) in an arithmetic circuit. Furthermore, as we'll see in this section, converting arithmetic circuits into polynomial constraints allows for much faster evaluation.</p>
+<h2 id="rank-1-constraint-system-r1cs"><a class="header" href="#rank-1-constraint-system-r1cs">Rank-1 Constraint System (R1CS)</a></h2>
+<p>There are many formats to represent arithmetic circuits, and one of the most popular ones is R1CS (Rank-1 Constraint System), which represents arithmetic circuits as \underline{a set of equality constraints, each involving only one multiplication}. In an arithmetic circuit, we call the concrete values assigned to the variables within the constraints witness. We first provide the formal definition of R1CS as follows:</p>
+<h3 id="definition-r1cs"><a class="header" href="#definition-r1cs">Definition: R1CS</a></h3>
+<p>An R1CS structure \(\mathcal{S}\) consists of:</p>
+<ul>
+<li>Size bounds \(m, d, \ell \in \mathbb{N}\) where \(d &gt; \ell\)</li>
+<li>Three matrices \(O, L, R \in \mathbb{F}^{m\ times d}\) with at most \(\Omega(\max(m, d))\) non-zero entries in total</li>
+</ul>
+<p>An R1CS instance includes a public input \(p \in \mathbb{F}^\ell\), while an R1CS witness is a vector \(w \in \mathbb{F}^{d - \ell - 1}\).
+A structure-instance tuple \((S, p)\) is satisfied by a witness \(w\) if:
+\begin{equation}
+(L \cdot a) \circ (R \cdot a) - O \cdot a = \mathbf{0}
+\end{equation}
+where \(a = (1, w, p) \in \mathbb{F}^d\), \(\cdot\) denotes matrix-vector multiplication, and \(\circ\) is the Hadamard product.</p>
+<p>The intuitive interpretation of each matrix is as follows:</p>
+<ul>
+<li>\(L\): Encodes the left input of each gate</li>
+<li>\(R\): Encodes the right input of each gate</li>
+<li>\(O\): Encodes the output of each gate</li>
+<li>The leading 1 in the witness vector allows for constant terms</li>
+</ul>
+<p><strong>Single Multiplication</strong></p>
+<p>Let's consider a simple example where we want to prove \(z = x \cdot y\), with \(z = 3690\), \(x = 82\), and \(y = 45\).</p>
+<ul>
+<li>\textbf{Witness vector}: \((1, z, x, y) = (1, 3690, 82, 45)\)</li>
+<li>\textbf{Number of witnesses}: \(m = 4\)</li>
+<li>\textbf{Number of constraints}: \(d = 1\)</li>
+</ul>
+<p>The R1CS constraint for \(z = x \cdot y\) is satisfied when:</p>
+<p>\begin{align*}
+&amp;(\begin{bmatrix} 0 &amp; 0 &amp; 1 &amp; 0 \end{bmatrix} \cdot a) \circ (\begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix} \cdot a) - \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; 0 \end{bmatrix} \cdot a \\
+=&amp;(\begin{bmatrix} 0 &amp; 0 &amp; 1 &amp; 0 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\ 3690 \\ 82 \\ 45
+\end{bmatrix}) \circ (\begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\ 3690 \\ 82 \\ 45
+\end{bmatrix}) - \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; 0 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\ 3690 \\ 82 \\ 45
+\end{bmatrix} \\
+=&amp; 82 \cdot 45 - 3690 \\
+=&amp; 3690 - 3690 \\
+=&amp; 0
+\end{align*}</p>
+<p>This example demonstrates how R1CS encodes a simple multiplication constraint:</p>
+<ul>
+<li>\(L = \begin{bmatrix} 0 &amp; 0 &amp; 1 &amp; 0 \end{bmatrix}\) selects \(x\) (left input)</li>
+<li>\(R = \begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix}\) selects \(y\) (right input)</li>
+<li>\(O = \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; 0 \end{bmatrix}\) selects \(z\) (output)</li>
+</ul>
+<p><strong>Multiple Constraints</strong></p>
+<p>Let's examine a more complex example: \(r = a \cdot b \cdot c \cdot d\). Since R1CS requires that each constraint contain only one multiplication, we need to break this down into multiple constraints:</p>
+<p>\begin{align*}
+v_1 &amp;= a \cdot b \\
+v_2 &amp;= c \cdot d \\
+r &amp;= v_1 \cdot v_2
+\end{align*}</p>
+<p>Note that alternative representations are possible, such as \(v_1 = ab, v_2 = v_1c, r = v_2d\). In this example, we use 7 variables \((r, a, b, c, d, v_1, v_2)\), so the dimension of the witness vector will be \(m = 8\) (including the constant 1). We have three constraints, so \(n = 3\).
+To construct the matrices \(L\), \(R\), and \(O\), we can interpret the constraints as linear combinations:</p>
+<p>\begin{align*}
+v_1 &amp;= (0 \cdot 1 + 0 \cdot r + 1 \cdot a + 0 \cdot b + 0 \cdot c + 0 \cdot d + 0 \cdot v_1 + 0 \cdot v_2) \cdot b \\
+v_2 &amp;= (0 \cdot 1 + 0 \cdot r + 0 \cdot a + 0 \cdot b + 1 \cdot c + 0 \cdot d + 0 \cdot v_1 + 0 \cdot v_2) \cdot d \\
+r &amp;= (0 \cdot 1 + 0 \cdot r + 0 \cdot a + 0 \cdot b + 0 \cdot c + 0 \cdot d + 1 \cdot v_1 + 0 \cdot v_2) \cdot v_2
+\end{align*}</p>
+<p>Thus, we can construct \(L\), \(R\), and \(O\) as follows:</p>
+<p>\begin{equation*}
+L = \begin{bmatrix}
+0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0
+\end{bmatrix}
+\end{equation*}
+\begin{equation*}
+R = \begin{bmatrix}
+0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1
+\end{bmatrix}
+\end{equation*}
+\begin{equation*}
+O = \begin{bmatrix}
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 \\
+0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0
+\end{bmatrix}
+\end{equation*}</p>
+<p>Where the columns in each matrix correspond to \((1, r, a, b, c, d, v_1, v_2)\).</p>
+<p><strong>Addition with a Constant</strong></p>
+<p>Let's examine the case \(z = x \cdot y + 3\). We can represent this as \(-3 + z = x \cdot y\). For the witness vector \((1, z, x, y)\), we have:</p>
+<p>\begin{align*}
+L &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 1 &amp; 0
+\end{bmatrix} \\
+R &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 0 &amp; 1
+\end{bmatrix} \\
+O &amp;= \begin{bmatrix}
+-3 &amp; 1 &amp; 0 &amp; 0
+\end{bmatrix}
+\end{align*}</p>
+<p>Note that the constant 3 appears in the \(O\) matrix with a negative sign, effectively moving it to the left side of the equation</p>
+<p><strong>Multiplication with a Constant</strong></p>
+<p>Now, let's consider \(z = 3x^2 + y\). The requirement of "one multiplication per constraint" doesn't apply to multiplication with a constant, as we can treat it as repeated addition.</p>
+<p>\begin{align*}
+L &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 3 &amp; 0
+\end{bmatrix} \\
+R &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 1 &amp; 0
+\end{bmatrix} \\
+O &amp;= \begin{bmatrix}
+0 &amp; 1 &amp; 0 &amp; -1
+\end{bmatrix}
+\end{align*}</p>
+<h2 id="uadratic-arithmetic-program-qap"><a class="header" href="#uadratic-arithmetic-program-qap">uadratic Arithmetic Program (QAP)</a></h2>
+<p>Recall that the prover aims to demonstrate knowledge of a witness \(w\) without revealing it. This is equivalent to knowing a vector \(a\) that satisfies \((L \cdot a) \circ (R \cdot a) = O \cdot a\), where \(\circ\) denotes the Hadamard (element-wise) product. However, evaluating this equivalence directly requires \(\Omega(d)\) operations, where \(d\) is the number of rows. To improve efficiency, we can convert this matrix comparison to a polynomial comparison, leveraging the Schwartz-Zippel Lemma, which allows us to check polynomial equality with \(\Omega(1)\) evaluations.</p>
+<p>Let's consider a simpler example to illustrate this concept. Suppose we want to test the equivalence \(Av = Bu\), where:</p>
+<p>\begin{align*}
+A = \begin{bmatrix}
+2 &amp; 5 \\
+3 &amp; 1 \\
+\end{bmatrix},
+B = \begin{bmatrix}
+4 &amp; 1 \\
+2 &amp; 3 \\
+\end{bmatrix},
+v = \begin{bmatrix}
+3 \\ 1
+\end{bmatrix},
+u = \begin{bmatrix}
+2 \\ 2
+\end{bmatrix}
+\end{align*}</p>
+<p>The equivalence check can be represented as:</p>
+<p>\begin{equation*}
+\begin{bmatrix}
+2 \\ 3
+\end{bmatrix} \cdot 3 + \begin{bmatrix}
+5 \\ 1
+\end{bmatrix} \cdot 1 = \begin{bmatrix}
+4 \\ 2
+\end{bmatrix} \cdot 2 + \begin{bmatrix}
+1 \\ 3
+\end{bmatrix} \cdot 2
+\end{equation*}</p>
+<p>This matrix-vector equality check is equivalent to the following polynomial equality check:</p>
+<p>\begin{equation*}
+3 \cdot L([(1, 2), (2, 3)]) + 1 \cdot L([(1, 5), (2, 1)]) = 2 \cdot L([(1, 4), (2, 2)]) + 2 \cdot L([(1, 1), (2, 3)])
+\end{equation*}</p>
+<p>where \(L\) denotes Lagrange Interpolation. In \(\mathbb{F}_{11}\) (field with 11 elements), we have:</p>
+<p>\begin{align*}
+L([(1, 2), (2, 3)]) &amp;= x + 1 \\
+L([(1, 5), (2, 1)]) &amp;= 7x + 9 \\
+L([(1, 4), (2, 2)]) &amp;= 9x + 6 \\
+L([(1, 1), (2, 3)]) &amp;= 2x + 10
+\end{align*}</p>
+<p>The Schwartz-Zippel Lemma states that we need only one evaluation at a random point to check the equivalence of polynomials with high probability.</p>
+<p>However, the homomorphic property for multiplication doesn't hold for Lagrange Interpolation. Thus, we don't have \(\ell(x) \cdot r(x) = o(x)\), where \(\ell(x)\), \(r(x)\), and \(o(x)\) are the polynomials resulting from the Lagrange interpolation of \(L \cdot a\), \(R \cdot a\), and \(O \cdot a\) respectively.</p>
+<p>While \(\ell(x)\), \(r(x)\), and \(o(x)\) are of degree at most \(d-1\), \(\ell(x) \cdot r(x)\) is of degree at most \(2d-2\).</p>
+<p>To address this discrepancy, we introduce a degree \(d\) polynomial \(t(x) = \prod_{i=1}^{d} (x - i)\). We can then rewrite the equation as:</p>
+<p>\begin{equation}
+\ell(x) \cdot r(x) = o(x) + h(x) \cdot t(x)
+\end{equation}</p>
+<p>where \(h(x) = \frac{\ell(x) \cdot r(x) - o(x)}{t(x)}\). This formulation allows us to maintain the desired polynomial relationships while accounting for the degree differences.</p>
+<div style="break-before: page; page-break-before: always;"></div><h1 id="proving-single-polynomial"><a class="header" href="#proving-single-polynomial">Proving Single Polynomial</a></h1>
+<p>Before dealing with all of \(\ell(x)\), \(r(x)\), and \(o(x)\) at once, we design a protocol that allows the Prover \(\mathcal{A}\) to convince the Verifier \(\mathcal{B}\) that \(\mathcal{A}\) knows a specific polynomial. Let's denote this polynomial of degree \(n\) with coefficients in a finite field as:</p>
+<p>\begin{equation}
+P(x) = c_0 + c_1 x + c_2 x^{2} + \cdots c_n x^{n}
+\end{equation}</p>
+<p>Assume \(P(x)\) has \(n\) roots, \(a_1, a_2, \ldots, a_n \in \mathbb{F}\), such that \(P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)\). The Verifier \(\mathcal{B}\) knows \(d &lt; n\) roots of \(P(x)\), namely \(a_1, a_2, \ldots, a_d\). Let \(T(x) = (x - a_1)(x - a_2)\cdots(x - a_d)\). Note that the Prover also knows \(T(x)\).</p>
+<p>The Prover's objective is to convince the Verifier that \(\mathcal{A}\) knows a polynomial \(H(x) = \frac{P(x)}{T(x)}\).</p>
+<h3 id="naive-approach"><a class="header" href="#naive-approach">Naive Approach</a></h3>
+<p>The simplest approach to prove that \(\mathcal{A}\) knows \(H(x)\) is as follows:</p>
+<p><em>Naive Approach</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) sends all possible values in \(\mathbb{F}\) to \(\mathcal{A}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends all possible outputs of \(H(x)\) and \(P(x)\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(H(a)T(a) = P(a)\) holds for any \(a\) in \(\mathbb{F}\).</em></li>
+</ul>
+<p>This protocol is highly inefficient, requiring \(\mathcal{O}(|\mathbb{F}|)\) evaluations and communications.</p>
+<h3 id="-schwartz-zippel-lemma"><a class="header" href="#-schwartz-zippel-lemma">\(+\) Schwartz-Zippel Lemma</a></h3>
+<p>Instead of evaluating polynomials at all values in \(\mathbb{F}\), we can leverage the Schwartz-Zippel Lemma: if \(H(s) = \frac{P(s)}{T(s)}\) or equivalently \(H(s)T(s) = P(s)\) for a random element \(s\), we can conclude that \(H(x) = \frac{P(x)}{T(x)}\) with high probability. Thus, the Prover \(\mathcal{A}\) only needs to send evaluations of \(P(s)\) and \(H(s)\) for a random input \(s\) received from \(\mathcal{B}\).</p>
+<p><em>\(+\) Schwartz-Zippel Lemma</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) draws random \(s\) from \(\mathbb{F}\) and sends it to \(\mathcal{A}\).</em></li>
+<li><em>\(\mathcal{A}\) computes \(h = H(s)\) and \(p = P(s)\) and send them to \(\mathcal{B}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(p = t h\), where \(t\) denotes \(T(s)\).</em></li>
+</ul>
+<p>This protocol is efficient, requiring only a constant number of evaluations and communications. However, it is vulnerable to a malicious prover who could send an arbitrary value \(h'\) and the corresponding \(p'\) such that \(p' = h't\).</p>
+<h3 id="-discrete-logarithm-assumption"><a class="header" href="#-discrete-logarithm-assumption">\(+\) Discrete Logarithm Assumption</a></h3>
+<p>To address this vulnerability, the Verifier must hide the randomly chosen input \(s\) from the Prover. This can be achieved using the discrete logarithm assumption: it is computationally hard to determine \(s\) from \(\alpha\), where \(\alpha = g^s \bmod p\). Thus, it's safe for the Verifier to send \(\alpha\), as the Prover cannot easily derive \(s\) from it.</p>
+<p>An interesting property of polynomial exponentiation is:</p>
+<p>\begin{align}
+g^{P(x)} &amp;= g^{c_0 + c_1 x + c_2 x^{2} + \cdots c_n x^{n}} = g^{c_0} (g^{x})^{c_1}  (g^{(x^2)})^{c_2} \cdots (g^{(x^n)})^{c_n}
+\end{align}</p>
+<p>Instead of sending \(s\), the Verifier can send \(g\) and \(\alpha_{i} = g^{(s^i)}\) for \(i = 1, \cdots n\). BE CAREFUL THAT <strong>\(g^{(s^i)} \neq (g^s)^i\)</strong>. The Prover can still evaluate \(g^p = g^{P(s)}\) using these powers of \(g\):</p>
+<p>\begin{equation}
+g^{p} = g^{P(s)} = g^{c_0} \alpha_{1}^{c_1} (\alpha_{2})^{c_2} \cdots (\alpha_{n})^{c_n}
+\end{equation}</p>
+<p>Similarly, the Prover can evaluate \(g^h = g^{H(s)}\). The Verifier can then check \(p = ht \iff g^p = (g^h)^t\).</p>
+<p><em>\(+\) Discrete Logarithm Assumption</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) randomly draw \(s\) from \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{B}\) computes and sends \(\{\alpha, \alpha_2, ..., \alpha_{n}\}\), where \(\alpha_i= g^{(s^{i})}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends \(u = g^{p}\) and \(v = g^{h}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u = v^{t}\).</em></li>
+</ul>
+<p>This approach prevents the Prover from obtaining \(s\) or \(t = T(s)\), making it impossible to send fake \((h', p')\) such that \(p' = h't\).</p>
+<p>However, this protocol still has a flaw. Since the Prover can compute \(g^t = \alpha_{c_1}(\alpha_2)^{c_2}\cdots(\alpha_d)^{c_d}\), they could send fake values \(((g^{t})^{z}, g^{z})\) instead of \((g^p, g^h)\) for an arbitrary value \(z\). The verifier's check would still pass, and they could not detect this deception.</p>
+<h3 id="-knowledge-of-exponent-assumption"><a class="header" href="#-knowledge-of-exponent-assumption">\(+\) Knowledge of Exponent Assumption</a></h3>
+<p>To address the vulnerability where the verifier \(\mathcal{B}\) cannot distinguish if \(v (= g^h)\) from the prover is a power of \(\alpha_i = g^{(s^i)}\), we can employ the Knowledge of Exponent Assumption. This approach involves the following steps:</p>
+<ul>
+<li>\(\mathcal{B}\) sends both \(\alpha_i\) and \(\alpha'_i = \alpha_i^r\) for a new random value \(r\).</li>
+<li>The prover returns \(a = (\alpha_i)^{c_i}\) and \(a' = (\alpha'_i)^{c_i}\) for \(i = 1, ..., n\).</li>
+<li>\(\mathcal{B}\) can conclude that \(a\) is a power of \(\alpha_i\) if \(a^r = a'\).</li>
+</ul>
+<p>Based on this assumption, we can design an improved protocol:</p>
+<p><em>\(+\) Knowledge of Exponent Assumption</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) randomly selects \(s\) and \(r\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{B}\) computes and sends \({\alpha_1, \alpha_2, ..., \alpha_{n}}\) and \({\alpha'_1, \alpha'_2, ..., \alpha'_{n}}\), where \(\alpha_i = g^{(s^i)}\) and \(\alpha' = \alpha_{r} = g^{(s^{i})r}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends \(u = g^{p}\), \(v = g^{h}\), and \(w = g^{p'}\), where \(g^{p'} = g^{P(sr)}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u^{r} = w\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u = v^{t}\).</em></li>
+</ul>
+<p>The prover can compute \(g^{p'} = g^{P(sr)} = \alpha'^{c_1} (\alpha'^{2})^{c_2} \cdots (\alpha'^{n})^{c_n}\) using powers of \(\alpha'\). This protocol now satisfies the properties of a SNARK: completeness, soundness, and efficiency.</p>
+<h3 id="-zero-knowledge"><a class="header" href="#-zero-knowledge">\(+\) Zero Knowledge</a></h3>
+<p>To transform the above protocol into a zk-SNARK, we need to ensure that the verifier cannot learn anything about \(P(x)\) from the prover's information. This is achieved by having the prover obfuscate all information with a random secret value \(\delta\):</p>
+<p><em>\(+\) Zero Knowledge</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) randomly selects \(s\) and \(r\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{B}\) computes and sends \(\{\alpha_1, \alpha_2, ..., \alpha_{n}\}\) and \(\{\alpha_1', \alpha'_2, ..., \alpha'_{n}\}\), where \(\alpha_i = g^{(s^{i})}\) and \(\alpha_i' = \alpha_i^{r} = g^{(s^{i})r}\).</em></li>
+<li><em>\(\mathcal{A}\) randomly selects \(\delta\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends \(u' = (g^{p})^{\delta}\), \(v' = (g^{h})^{\delta}\), and \(w' = (g^{p'})^{\delta}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u'^{r} = w'\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u' = v'^{t}\).</em></li>
+</ul>
+<p>By introducing the random value \(\delta\), the verifier can no longer learn anything about \(p\), \(h\), or \(w\), thus achieving zero knowledge.</p>
+<h3 id="-non-interactivity"><a class="header" href="#-non-interactivity">\(+\) Non-interactivity</a></h3>
+<p>The previously described protocol requires each verifier to generate unique random values, which becomes inefficient when a prover needs to demonstrate knowledge to multiple verifiers. To address this, we aim to eliminate the interaction between the prover and verifier. One effective solution is the use of a trusted setup.</p>
+<p><em>\(+\) Non-Interactive: Trusted Setup</em></p>
+<ul>
+<li><em><strong>Secret Seed:</strong> A trusted third party generates the random values \(s\) and \(r\)</em></li>
+<li><em><strong>Proof Key:</strong> Provided to the prover</em>
+<ul>
+<li><em>\(\{\alpha_1, \alpha_2, ..., \alpha_{n}\}\), where \(\alpha_{i} = g^{(s^i)}\)</em></li>
+<li><em>\(\{\alpha'_1, \alpha'_2, ..., \alpha'_{n}\}\), where \(\alpha_i' = g^{(s^{i})r}\)</em></li>
+</ul>
+</li>
+<li><em><strong>Verification Key:</strong> Distributed to verifiers</em>
+<ul>
+<li><em>\(g^{r}\)</em></li>
+<li><em>\(g^{T(s)}\)</em></li>
+</ul>
+</li>
+<li><em>After distribution, the original \(s\) and \(r\) values are securely destroyed.</em></li>
+</ul>
+<p>Then, the non-interactive protocol consists of two main parts: proof generation and verification.</p>
+<p><em>\(+\) Non-Interactive: Proof</em></p>
+<ul>
+<li><em>\(\mathcal{A}\) receives the proof key</em></li>
+<li><em>\(\mathcal{A}\) randomly selects \(\delta\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{A}\) broadcast the proof \(\pi = (u' = (g^{p})^{\delta}, v' = (g^{h})^{\delta}, w' = (g^{p'})^{\delta})\)</em></li>
+</ul>
+<p>Since \(r\) is not shared and already destroyed, the verifier \(\mathcal{B}\) cannot calculate \(u'^{r}\) to check \(u'^{r} = w'\). Instead, the verifier can use a paring with bilinear mapping; \(u'^{r} = w'\) is equivalent to \(e(u' = (g^{p})^{\delta}, g^{r}) = e(w'=(g^{p'})^{\delta}, g)\).</p>
+<p><em>\(+\) Non-Interactive: Verification</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) receives the verification key.</em></li>
+<li><em>\(\mathcal{B}\) receives the proof \(\pi\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(e(u', g^{r}) = e(w', g)\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u' = v'^{t}\).</em></li>
+</ul>
+<div style="break-before: page; page-break-before: always;"></div><div style="break-before: page; page-break-before: always;"></div>
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of Number Theory » Basics of Number Theory","id":"0","title":"Basics of Number Theory"},"1":{"body":"","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Computation Rule and Properties","id":"1","title":"Computation Rule and Properties"},"10":{"body":"For an element \\(a \\in H\\), an element \\(b \\in H\\) is an inverse of \\(a\\) if \\(a \\circ b = b \\circ a = e\\), where \\(e\\) is the identity element. Example : In modulo \\(n\\) arithmetic (addition), the inverse of an element exists if it can cancel itself out to yield the identity element. In the set of integers modulo 7, the inverse of 3 is 5, because \\(3 \\times 5 \\bmod 7 = 1\\), where 1 is the identity element for multiplication.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Inverse","id":"10","title":"Definition: Inverse"},"11":{"body":"A monoid in which every element has an inverse is called a group. Example : The set of integers modulo a prime \\(p\\) under multiplication forms a group (Can you prove it?). For instance, in \\(\\mathbb{Z}/5\\mathbb{Z}\\), every non-zero element \\(\\{1 + 5\\mathbb{Z}, 2 + 5\\mathbb{Z}, 3 + 5\\mathbb{Z}, 4 + 5\\mathbb{Z}\\}\\) has an inverse, making it a group.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Group","id":"11","title":"Definition: Group"},"12":{"body":"The order of a group is the number of elements in the group. Example : The group of integers modulo 4 under addition has order 4, because the set of elements is \\(\\{0, 1, 2, 3\\}\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Order of a Group","id":"12","title":"Definition: Order of a Group"},"13":{"body":"A triple \\((R, +, \\cdot)\\) is a ring if \\((R, +)\\) is an abelian group, \\((R, \\cdot)\\) is a semigroup, and the distributive property holds: \\(x \\cdot (y + z) = (x \\cdot y) + (x \\cdot z)\\) and \\((x + y) \\cdot z = (x \\cdot z) + (y \\cdot z)\\) for all \\(x, y, z \\in R\\). Example : The set of integers with usual addition and multiplication modulo \\(n\\) forms a ring. For example, in \\(\\mathbb{Z}/6\\mathbb{Z}\\), addition and multiplication modulo 6 form a ring. use std::fmt;\nuse std::ops::{Add, Mul, Neg, Sub}; pub trait Ring: Sized + Clone + PartialEq + fmt::Display + Add<Output = Self> + Sub<Output = Self> + Mul<Output = Self> + Mul<i64, Output = Self> + Neg<Output = Self>\n{ // A ring is an algebraic structure with addition and multiplication fn zero() -> Self; fn one() -> Self;\n}","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Ring","id":"13","title":"Definition: Ring"},"14":{"body":"A ring is called a commutative ring if its multiplication operation is commutative. Example The set of real numbers under usual addition and multiplication forms a commutative ring.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Communicative Ring","id":"14","title":"Definition: Communicative Ring"},"15":{"body":"A commutative ring with a multiplicative identity element where every non-zero element has a multiplicative inverse is called a field. Example The set of rational numbers under usual addition and multiplication forms a field. use crate::modules::ring::Ring; pub trait Field: Ring + Div<Output = Self> { fn inverse(&self) -> Self;\n}","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Field","id":"15","title":"Definition: Field"},"16":{"body":"The residue class of \\( a \\) modulo \\( m \\), denoted as \\( a + m\\mathbb{Z} \\), is the set \\( \\{b : b \\equiv a \\pmod{m}\\} \\). Example: For \\( m = 3 \\), the residue class of 2 is \\( 2 + 3\\mathbb{Z} = \\{\\ldots, -4, -1, 2, 5, 8, \\ldots\\} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Residue Class","id":"16","title":"Definition: Residue Class"},"17":{"body":"We denote the set of all residue classes modulo \\( m \\) as \\( \\mathbb{Z} / m\\mathbb{Z} \\). We say that \\( a + m\\mathbb{Z} \\) is invertible in \\( \\mathbb{Z} / m\\mathbb{Z} \\) if and only if there exists a solution for \\( ax \\equiv 1 \\pmod{m} \\). Example: In \\( \\mathbb{Z}/5\\mathbb{Z} \\), \\( 3 + 5\\mathbb{Z} \\) is invertible because \\( \\gcd(3, 5) = 1 \\) (since \\( 3\\cdot2 \\equiv 1 \\pmod{5} \\)). However, in \\( \\mathbb{Z}/6\\mathbb{Z} \\), \\( 3 + 6\\mathbb{Z} \\) is not invertible because \\( \\gcd(3, 6) = 3 \\neq 1 \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Inverse of Residue Class","id":"17","title":"Definition: Inverse of Residue Class"},"18":{"body":"If \\( a \\) and \\( b \\) are coprime, the residues of \\( a \\), \\( 2a \\), \\( 3a \\), ..., \\( (b-1)a \\) modulo \\( b \\) are all distinct. Proof: Suppose, for contradiction, that there exist \\( x, y \\in \\{1, 2, \\dots, b-1\\} \\) with \\( x \\neq y \\) such that \\( xa \\equiv ya \\pmod{b} \\). Then \\( (x-y)a \\equiv 0 \\pmod{b} \\), implying \\( b \\mid (x-y)a \\). Since \\( a \\) and \\( b \\) are coprime, we must have \\( b \\mid (x-y) \\). However, \\( |x-y| < b \\), so this is only possible if \\( x = y \\), contradicting our assumption. Therefore, all residues must be distinct.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Lemma 1.2.1","id":"18","title":"Lemma 1.2.1"},"19":{"body":"For any integers \\( a \\) and \\( b \\), the equation \\( ax + by = 1 \\) has a solution in integers \\( x \\) and \\( y \\) if and only if \\( a \\) and \\( b \\) are coprime. Proof: (\\( \\Rightarrow \\)) Suppose \\( a \\) and \\( b \\) are not coprime, let \\( d = \\gcd(a,b) > 1 \\). Then \\( d \\mid a \\) and \\( d \\mid b \\), so \\( d \\mid (ax + by) \\) for any integers \\( x \\) and \\( y \\). Thus, \\( ax + by \\neq 1 \\) for any \\( x \\) and \\( y \\). (\\( \\Leftarrow \\)) Suppose \\( a \\) and \\( b \\) are coprime. By the previous lemma, the residues of \\( a \\), \\( 2a \\), ..., \\( (b-1)a \\) modulo \\( b \\) are all distinct. Therefore, there exists an \\( m \\in \\{1, 2, ..., b-1\\} \\) such that \\( ma \\equiv 1 \\pmod{b} \\). This means there exists an integer \\( n \\) such that \\( ma = bn + 1 \\), or equivalently, \\( ma - bn = 1 \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.2","id":"19","title":"Theorem 1.2.2"},"2":{"body":"A mapping \\(\\circ: X \\times X \\rightarrow X\\) is a binary operation on \\(X\\) if for any pair of elements \\((x_1, x_2)\\) in \\(X\\), \\(x_1 \\circ x_2\\) is also in \\(X\\). Example : Addition (+) on the set of integers is a binary operation. For example, \\(5 + 3 = 8\\), and both \\(5, 3, 8\\) are integers, staying within the set of integers.","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Definition: Binary Operation","id":"2","title":"Definition: Binary Operation"},"20":{"body":"For any integers \\( a \\) and \\( b \\), we have: \\( a\\mathbb{Z} + b \\mathbb{Z} = \\gcd(a, b)\\mathbb{Z} \\) Proof: This statement is equivalent to proving that \\( ax + by = c \\) has an integer solution if and only if \\( c \\) is a multiple of \\( \\gcd(a,b) \\). (\\( \\Rightarrow \\)) If \\( ax + by = c \\) for some integers \\( x \\) and \\( y \\), then \\( \\gcd(a,b) \\mid a \\) and \\( \\gcd(a,b) \\mid b \\), so \\( \\gcd(a,b) \\mid (ax + by) = c \\). (\\( \\Leftarrow \\)) Let \\( c = k\\gcd(a,b) \\) for some integer \\( k \\). We can write \\( a = p\\gcd(a,b) \\) and \\( b = q\\gcd(a,b) \\), where \\( p \\) and \\( q \\) are coprime. By the previous theorem, there exist integers \\( m \\) and \\( n \\) such that \\( pm + qn = 1 \\). Multiplying both sides by \\( k\\gcd(a,b) \\), we get: [ akm + bkn = c ] Thus, \\( x = km \\) and \\( y = kn \\) are integer solutions to \\( ax + by = c \\). This theorem implies that for any integers \\( a \\), \\( b \\), and \\( n \\), the equation \\( ax + by = n \\) has an integer solution if and only if \\( \\gcd(a,b) \\mid n \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem: Bézout's Identity","id":"20","title":"Theorem: Bézout's Identity"},"21":{"body":"\\( a + m\\mathbb{Z} \\) is invertible in \\( \\mathbb{Z}/m\\mathbb{Z} \\) if and only if \\( \\gcd(a,m) = 1 \\). Proof: (\\( \\Rightarrow \\)) Suppose \\( a + m\\mathbb{Z} \\) is invertible in \\( \\mathbb{Z}/m\\mathbb{Z} \\). Then there exists an integer \\( x \\) such that \\( ax \\equiv 1 \\pmod{m} \\). Let \\( g = \\gcd(a,m) \\). Since \\( g \\mid ax \\) and \\( g \\mid (ax - 1) \\), we must have \\( g \\mid 1 \\), so \\( g = 1 \\). (\\( \\Leftarrow \\)) Suppose \\( \\gcd(a,m) = 1 \\). By Bézout's identity, there exist integers \\( x \\) and \\( y \\) such that \\( ax + my = 1 \\). Thus, \\( x + m\\mathbb{Z} \\) is the multiplicative inverse of \\( a + m\\mathbb{Z} \\) in \\( \\mathbb{Z}/m\\mathbb{Z} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.3","id":"21","title":"Theorem 1.2.3"},"22":{"body":"\\( (\\mathbb{Z} / m \\mathbb{Z}, +, \\cdot) \\) is a commutative ring where \\( 1 + m \\mathbb{Z} \\) is the multiplicative identity element. This ring is called the residue class ring modulo \\( m \\). Example: \\( \\mathbb{Z}/4\\mathbb{Z} = \\{0 + 4\\mathbb{Z}, 1 + 4\\mathbb{Z}, 2 + 4\\mathbb{Z}, 3 + 4\\mathbb{Z}\\} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Residue Class Ring","id":"22","title":"Definition: Residue Class Ring"},"23":{"body":"A residue class \\( a + m\\mathbb{Z} \\) is called primitive if \\( \\gcd(a, m) = 1 \\). Example: In \\( \\mathbb{Z}/6\\mathbb{Z} \\), the primitive residue classes are \\( 1 + 6\\mathbb{Z} \\) and \\( 5 + 6\\mathbb{Z} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Primitive Residue Class","id":"23","title":"Definition: Primitive Residue Class"},"24":{"body":"A residue ring \\( \\mathbb{Z} / m\\mathbb{Z} \\) is a field if and only if \\( m \\) is a prime number. Proof: TBD Example: For \\( m = 5 \\), \\( \\mathbb{Z}/5\\mathbb{Z} = \\{0 + 5\\mathbb{Z}, 1 + 5\\mathbb{Z}, 2 + 5\\mathbb{Z}, 3 + 5\\mathbb{Z}, 4 + 5\\mathbb{Z}\\} \\) forms a field because 5 is a prime number, and every non-zero element has a multiplicative inverse.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.4","id":"24","title":"Theorem 1.2.4"},"25":{"body":"The group of all primitive residue classes modulo \\( m \\) is called the primitive residue class group, denoted by \\( (\\mathbb{Z}/m\\mathbb{Z})^{\\times} \\). Example: For \\( m = 8 \\), the set of all primitive residue classes is \\( (\\mathbb{Z}/8\\mathbb{Z})^{\\times} = \\{1 + 8\\mathbb{Z}, 3 + 8\\mathbb{Z}, 5 + 8\\mathbb{Z}, 7 + 8\\mathbb{Z}\\} \\). These are the integers less than 8 that are coprime to 8 (i.e., \\(\\gcd(a, 8) = 1\\)). Contrast this with \\( m = 9 \\). The primitive residue class group is \\((\\mathbb{Z}/9\\mathbb{Z})^{\\times} = \\{1 + 9\\mathbb{Z}, 2 + 9\\mathbb{Z}, 4 + 9\\mathbb{Z}, 5 + 9\\mathbb{Z}, 7 + 9\\mathbb{Z}, 8 + 9\\mathbb{Z}\\}\\), as these are the integers less than 9 that are coprime to 9.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Primitive Residue Class Group","id":"25","title":"Definition: Primitive Residue Class Group"},"26":{"body":"Euler's totient function \\(\\phi(m)\\) is equal to the order of the primitive residue class group modulo \\(m\\), which is the number of integers less than \\(m\\) and coprime to \\(m\\). Example : For \\(m = 12\\), \\(\\phi(12) = 4\\) because there are 4 integers less than 12 that are coprime to 12: \\({1, 5, 7, 11}\\). For \\(m = 10\\), \\(\\phi(10) = 4\\), as there are also 4 integers less than 10 that are coprime to 10: \\({1, 3, 7, 9}\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Euler's Totient Function","id":"26","title":"Definition: Euler's Totient Function"},"27":{"body":"The order of \\(g \\in G\\) is the smallest natural number \\(e\\) such that \\(g^{e} = 1\\). We denote it as \\(\\text{order}_g(g)\\)or \\(\\text{order } g\\). Example : In \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), the element 3 has order 6 because \\(3^6 \\bmod 7 = 1\\). In other words, \\(3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\bmod 7 = 1\\), and 6 is the smallest such exponent.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Order of an Element within a Group","id":"27","title":"Definition: Order of an Element within a Group"},"28":{"body":"A subset \\(U \\subseteq G\\) is a subgroup of \\(G\\) if \\(U\\) itself is a group under the operation of \\(G\\). Example : Consider \\((\\mathbb{Z}/8\\mathbb{Z})^{\\times} = \\{1 + 8\\mathbb{Z}, 3+ 8\\mathbb{Z}, 5+ 8\\mathbb{Z}, 7+ 8\\mathbb{Z}\\}\\) under multiplication modulo 8. The subset \\({1+ 8\\mathbb{Z}, 7+ 8\\mathbb{Z}}\\) forms a subgroup because it satisfies the group properties: closed under multiplication, contains the identity element (1), and every element has an inverse (\\(7 \\times 7 \\equiv 1 \\bmod 8\\)).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Subgroup","id":"28","title":"Definition: Subgroup"},"29":{"body":"The set \\(\\{g^{k} : k \\in \\mathbb{Z}\\}\\), for some element \\(g \\in G\\), forms a subgroup of \\(G\\) and is called the subgroup generated by \\(g\\), denoted by \\(\\langle g \\rangle\\). Example : Consider the group \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times} = \\{1+ 7\\mathbb{Z}, 2+ 7\\mathbb{Z}, 3+ 7\\mathbb{Z}, 4+ 7\\mathbb{Z}, 5+ 7\\mathbb{Z}, 6+ 7\\mathbb{Z}\\}\\) under multiplication modulo 7. If we take \\(g = 3\\), then \\(\\langle 3 +7\\mathbb{Z} \\rangle =\\) \\(\\{3^1+7\\mathbb{Z}, 3^2+7\\mathbb{Z}, 3^3+7\\mathbb{Z}, 3^4+7\\mathbb{Z}, 3^5+7\\mathbb{Z}, 3^6+7\\mathbb{Z}\\} \\bmod 7 =\\) \\(\\{3+7\\mathbb{Z}, 2+7\\mathbb{Z}, 6+7\\mathbb{Z}, 4+7\\mathbb{Z}, 5+7\\mathbb{Z}, 1+7\\mathbb{Z}\\}\\), which forms a subgroup generated by 3. This subgroup contains all elements of \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), making 3 a generator of the entire group. If \\(g\\) has a finite order \\(e\\), we have that \\(\\langle g \\rangle = \\{g^{k}: 0 \\leq k \\leq e\\}\\), meaning \\(e\\) is the order of \\(\\langle g \\rangle\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Subgroup Generated by \\(g\\)","id":"29","title":"Definition: Subgroup Generated by \\(g\\)"},"3":{"body":"A binary operation \\(\\circ\\) is associative if \\((a \\circ b) \\circ c = a \\circ (b \\circ c)\\) for all \\(a, b, c \\in X\\). Example : Multiplication of real numbers is associative: \\((2 \\times 3) \\times 4 = 2 \\times (3 \\times 4) = 24\\). In a modular context, we also have addition modulo \\(n\\) being associative. For example, for \\(n = 5\\), \\((2 + 3) \\bmod 5 + 4 \\bmod 5 = 2 + (3 \\bmod 5 + 4) \\bmod 5 = 4\\).","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Definition: Associative Property","id":"3","title":"Definition: Associative Property"},"30":{"body":"A group \\(G\\) is called a cyclic group if there exists an element \\(g \\in G\\) such that \\(G = \\langle g \\rangle\\). In this case, \\(g\\) is called a generator of \\(G\\). Example : The group \\((\\mathbb{Z}/6\\mathbb{Z})^{\\times} = \\{1+6\\mathbb{Z}, 5+6\\mathbb{Z}\\}\\) under multiplication modulo 6 is a cyclic group. In this case, both 1 and 5 are generators of the group because \\(\\langle 5 +6\\mathbb{Z} \\rangle = \\{(5^1 \\bmod 6)+6\\mathbb{Z} = 5 +6\\mathbb{Z}, (5^2 \\bmod 6)+6\\mathbb{Z} = 1+6\\mathbb{Z}\\}\\). Since 5 generates all the elements of the group, \\(G\\) is cyclic.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Cyclic Group","id":"30","title":"Definition: Cyclic Group"},"31":{"body":"If \\(G\\) is a finite cyclic group, it has \\(\\phi(|G|)\\)generators, and each generator has order \\(|G|\\). Proof : TBD Example : Consider the group \\((\\mathbb{Z}/8\\mathbb{Z})^{\\times} = \\{1+8\\mathbb{Z}, 3+8\\mathbb{Z}, 5+8\\mathbb{Z}, 7+8\\mathbb{Z}\\}\\). This group is cyclic, and \\(\\phi(8) = 4\\). The generators of this group are \\(\\{1+8\\mathbb{Z}, 3+8\\mathbb{Z}, 5+8\\mathbb{Z}, 7+8\\mathbb{Z}\\}\\), each of which generates the entire group when raised to successive powers modulo 8. Each generator has the same order, which is \\(|G| = 4\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.5","id":"31","title":"Theorem 1.2.5"},"32":{"body":"If \\(G\\) is a finite cyclic group, the order of any subgroup of \\(G\\) divides the order of \\(G\\). Proof : TBD Example : Consider the cyclic group \\((\\mathbb{Z}/6\\mathbb{Z})^{\\times} = \\{1+6\\mathbb{Z}, 5+6\\mathbb{Z}\\}\\) under multiplication modulo 6. If we take the subgroup \\(\\langle 5+6\\mathbb{Z} \\rangle = \\{1+6\\mathbb{Z}, 5+6\\mathbb{Z}\\}\\), this is a subgroup of order 2, and 2 divides the order of the original group, which is 6. This theorem generalizes this property: for any subgroup of a cyclic group, its order divides the order of the group.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.6","id":"32","title":"Theorem 1.2.6"},"33":{"body":"If \\(\\gcd(a, m) = 1\\), then \\(a^{\\phi(m)} \\equiv 1 \\pmod{m}\\). Proof : TBD Example : Take \\(a = 2\\) and \\(m = 5\\). Since \\(\\gcd(2, 5) = 1\\), Fermat's Little Theorem tells us that \\(2^{\\phi(5)} = 2^4 \\equiv 1 \\bmod 5\\). Indeed, \\(2^4 = 16\\) and \\(16 \\bmod 5 = 1\\). This theorem suggests that \\(a^{\\phi(m) - 1} + m \\mathbb{Z}\\) is the inverse residue class of \\(a + m \\mathbb{Z}\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem: Fermat's Little Theorem","id":"33","title":"Theorem: Fermat's Little Theorem"},"34":{"body":"The order of any element in a group divides the order of the group. Proof : TBD Example : In the group \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), consider the element \\(3 + 7\\mathbb{Z}\\). The order of \\(3 + 7\\mathbb{Z}\\) is 6, as \\(3^6 \\equiv 1 \\bmod 7\\). The order of the group itself is also 6, and indeed, the order of the element divides the order of the group.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.7","id":"34","title":"Theorem 1.2.7"},"35":{"body":"For any element \\(g \\in G\\), we have \\(g^{|G|} = 1\\). Proof : TBD Example : In the group \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), for any element \\(g\\), such as \\(g = 3 + 7\\mathbb{Z}\\), we have \\(3^6 \\equiv 1 \\bmod 7\\). This holds for any \\(g \\in (\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\) because the order of the group is 6. Thus, \\(g^{|G|} = 1\\) is satisfied.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem: Generalization of Fermat's Little Theorem","id":"35","title":"Theorem: Generalization of Fermat's Little Theorem"},"36":{"body":"Let \\(K\\) be a field.","breadcrumbs":"Basics of Number Theory » Polynomials » Polynomials","id":"36","title":"Polynomials"},"37":{"body":"A univariate polynomial over a commutative ring \\(R\\) with unity \\(1\\) is an expression of the form \\(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0\\), where \\(x\\) is a variable and coefficients \\(a_0, \\ldots, a_n\\) belong to \\(R\\). The set of all polynomials over \\(R\\) in the variable \\(x\\) is denoted as \\(R[x]\\). Example: In \\(\\mathbb{Z}[x]\\), we have polynomials such as \\(2x^3 + x + 1\\), \\(x\\), and \\(1\\). In \\(\\mathbb{R}[x]\\), we have polynomials like \\(\\pi x^2 - \\sqrt{2}x + e\\). /// Polynomial struct representing a polynomial over Field.\n#[derive(Debug, Clone, PartialEq)]\npub struct Polynomial<F: Field> { pub poly: Vec<F>, pub var: String,\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Definition: Polynomial","id":"37","title":"Definition: Polynomial"},"38":{"body":"The degree of a non-zero polynomial \\(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0\\), denoted as \\(\\deg f\\), is the largest integer \\(n\\) such that \\(a_n \\neq 0\\). The zero polynomial is defined to have degree \\(-1\\). Example \\(\\deg(2x^3 + x + 1) = 3\\) \\(\\deg(x) = 1\\) \\(\\deg(1) = 0\\) \\(\\deg(0) = -1\\) impl<F: Field> Polynomial<F> { /// Removes trailing zeroes from a polynomial's coefficients. fn trim_trailing_zeros(poly: Vec<F>) -> Vec<F> { let mut trimmed = poly; while trimmed.last() == Some(&F::zero(None)) { trimmed.pop(); } trimmed } /// Returns the degree of the polynomial. pub fn degree(&self) -> isize { let trimmed = Self::trim_trailing_zeros(self.poly.clone()); if trimmed.is_empty() { -1 } else { (trimmed.len() - 1) as isize } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Definition: Degree","id":"38","title":"Definition: Degree"},"39":{"body":"For polynomials \\(f(x) = \\sum_{i=0}^n a_i x^i\\) and \\(g(x) = \\sum_{i=0}^m b_i x^i\\), their sum is defined as: \\((f + g)(x) = \\sum_{i=0}^{\\max(n,m)} (a_i + b_i) x^i\\), where we set \\(a_i = 0\\) for \\(i > n\\) and \\(b_i = 0\\) for \\(i > m\\). Example: Let \\(f(x) = 2x^2 + 3x + 1\\) and \\(g(x) = x^3 - x + 4\\). Then, \\((f + g)(x) = x^3 + 2x^2 + 2x + 5\\) impl<F: Field> Add for Polynomial<F> { type Output = Self; fn add(self, other: Self) -> Polynomial<F> { let max_len = std::cmp::max(self.poly.len(), other.poly.len()); let mut result = Vec::with_capacity(max_len); let zero = F::zero(None); for i in 0..max_len { let a = self.poly.get(i).unwrap_or(&zero); let b = other.poly.get(i).unwrap_or(&zero); result.push(a.clone() + b.clone()); } Polynomial { poly: Self::trim_trailing_zeros(result), var: self.var.clone(), } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Sum of Polynomials","id":"39","title":"Sum of Polynomials"},"4":{"body":"A binary operation \\(\\circ\\) is commutative if \\(a \\circ b = b \\circ a\\) for all \\(a, b \\in X\\). Example : Addition modulo \\(n\\) is also commutative. For \\(n = 7\\), \\(5 + 3 \\bmod 7 = 3 + 5 \\bmod 7 = 1\\).","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Definition: Commutative Property","id":"4","title":"Definition: Commutative Property"},"40":{"body":"For polynomials \\(f(x) = \\sum_{i=0}^n a_i x^i\\) and \\(g(x) = \\sum_{j=0}^m b_j x^j\\), their product is defined as: \\((fg)(x) = \\sum_{k=0}^{n+m} c_k x^k\\), where \\(c_k = \\sum_{i+j=k} a_i b_j\\) Example: Let \\(f(x) = x + 1\\) and \\(g(x) = x^2 - 1\\). Then, \\((fg)(x) = x^3 + x^2 - x - 1\\) impl<F: Field> Mul for Polynomial<F> { type Output = Self; /// Multiplication of two polynomials. fn mul(self, other: Polynomial<F>) -> Polynomial<F> { let mut result = vec![F::zero(None); self.degree() as usize + other.degree() as usize + 1]; for (i, a) in self.poly.iter().enumerate() { for (j, b) in other.poly.iter().enumerate() { result[i + j] = result[i + j].clone() + (a.clone() * b.clone()); } } Polynomial { poly: Self::trim_trailing_zeros(result), var: self.var.clone(), } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Product of polynomials","id":"40","title":"Product of polynomials"},"41":{"body":"Let \\(f, g \\in K[x]\\) be non-zero polynomials. Then, \\(\\deg(fg) = \\deg f + \\deg g\\). Example: Let \\(f(x) = x^2 + 1\\) and \\(g(x) = x^3 - x\\) in \\(\\mathbb{R}[x]\\). Then, \\(\\deg(fg) = \\deg(x^5 - x^3 + x^2 + 1) = 5 = 2 + 3 = \\deg f + \\deg g\\) We can also define division in the polynomial ring \\(K[x]\\).","breadcrumbs":"Basics of Number Theory » Polynomials » Lemma","id":"41","title":"Lemma"},"42":{"body":"Let \\(f, g \\in K[x]\\), with \\(g \\neq 0\\). There exist unique polynomials \\(q, r \\in K[x]\\) that satisfy \\(f = qg + r\\) and either \\(\\deg r < \\deg g\\) or \\(r = 0\\). Proof TBD \\(q\\) is called the quotient of \\(f\\) divided by \\(g\\), and \\(r\\) is called the remainder; we write \\(r = f \\bmod g\\). Example: In \\(\\mathbb{R}[x]\\), let \\(f(x) = x^3 + 2x^2 - x + 3\\) and \\(g(x) = x^2 + 1\\). Then \\(f = qg + r\\) where \\(q(x) = x + 2\\) and \\(r(x) = -3x + 1\\). impl<F: Field> Div for Polynomial<F> { type Output = Self; /// Division of two polynomials, returns quotient. fn div(self, other: Polynomial<F>) -> Polynomial<F> { let mut remainder_coeffs = Self::trim_trailing_zeros(self.poly.clone()); let divisor_coeffs = Self::trim_trailing_zeros(other.poly.clone()); let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse(); let mut quotient = vec![F::zero(None); self.degree() as usize - other.degree() as usize + 1]; let mut i = 0_i32; while remainder_coeffs.len() >= divisor_coeffs.len() { let lead_term = remainder_coeffs.last().unwrap().clone() * divisor_lead_inv.clone(); let deg_diff = remainder_coeffs.len() - divisor_coeffs.len(); quotient[deg_diff] = lead_term.clone(); for i in 0..divisor_coeffs.len() { remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i].clone() - (lead_term.clone() * divisor_coeffs[i].clone()); } remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs); i += 1; } Polynomial { poly: Self::trim_trailing_zeros(quotient), var: self.var.clone(), } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Theorem","id":"42","title":"Theorem"},"43":{"body":"Let \\(f \\in K[x]\\) be a non-zero polynomial, and \\(a \\in K\\) such that \\(f(a) = 0\\). Then, there exists a polynomial \\(q \\in K[x]\\) such that \\(f(x) = (x - a)q(x)\\). In other words, \\((x - a)\\) is a factor of \\(f(x)\\). Example: Let \\(f(x) = x^2 + 1 \\in (\\mathbb{Z}/2\\mathbb{Z})[x]\\). We have \\(f(1) = 1^2 + 1 = 0\\) in \\(\\mathbb{Z}/2\\mathbb{Z}\\), and indeed: \\(x^2 + 1 = (x - 1)^2 = x^2 - 2x + 1 = x^2 + 1\\) in \\((\\mathbb{Z}/2\\mathbb{Z})[x]\\)","breadcrumbs":"Basics of Number Theory » Polynomials » Corollary","id":"43","title":"Corollary"},"44":{"body":"A \\(n\\)-degre polynomial \\(P(x)\\) that goes through different \\(n + 1\\) points \\(\\{(x_1, y_1), (x_2, y_2), \\cdots (x_{n + 1}, y_{n + 1})\\}\\) is uniquely represented as follows: \\begin{align*} P(x) = \\sum^{n+1}_{i=1} y_i \\frac{f_i(x)}{f_i(x_i)} \\end{align*} , where \\(f_i(x) = \\prod_{k \\neq i} (x - x_k)\\) Proof TBD For example, the quadratic polynomial that goes through \\(\\{(1, 0), (2, 3), (3, 8)\\}\\) is as follows: \\begin{align*} P(x) = 0 \\frac{(x - 2)(x - 3)}{(1 - 2) (1 - 3)} + 3 \\frac{(x - 1)(x - 3)}{(2 - 1) (2 - 3)} + 8 \\frac{(x - 1)(x - 2)}{(3 - 1) (3 - 2)} = x^{2} - 1 \\end{align*} Note that Lagrange interpolation finds the lowest degree of interpolating polynomial for the given vector. pub fn interpolate(x_values: &[F], y_values: &[F]) -> Polynomial<F> { let mut lagrange_polys = vec![]; let numerators = Polynomial::from_monomials(x_values); for j in 0..x_values.len() { let mut denominator = F::one(None); for i in 0..x_values.len() { if i != j { denominator = denominator * (x_values[j].clone() - x_values[i].clone()); } } let cur_poly = numerators .clone() .div(Polynomial::from_monomials(&[x_values[j].clone()]).scalar_mul(&denominator)); lagrange_polys.push(cur_poly); } let mut result = Polynomial { poly: vec![], var: \"x\".to_string(), }; for (j, lagrange_poly) in lagrange_polys.iter().enumerate() { result = result + lagrange_poly.scalar_mul(&y_values[j]); } result\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Theorem: Lagrange Interpolation","id":"44","title":"Theorem: Lagrange Interpolation"},"45":{"body":"Let \\(L(v)\\) and \\(L(w)\\) be the polynomial resulting from Lagrange Interpolation on the output (\\(y\\)) vector \\(v\\) and \\(w\\) for the same inputs (\\(x\\)). Then, the following properties hold: Additivity: \\(L(v + w) = L(v) + L(w)\\) for any vectors \\(v\\) and \\(w\\) Scalar multiplication: \\(L(\\gamma v) = \\gamma L(v)\\) for any scalar \\(\\gamma\\) and vector \\(v\\) Proof Let \\(v = (v_1, \\ldots, v_n)\\) and \\(w = (w_1, \\ldots, w_n)\\) be vectors, and \\(x_1, \\ldots, x_n\\) be the interpolation points. \\begin{align*} L(v + w) &= \\sum_{i=1}^n (v_i + w_i) \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = \\sum_{i=1}^n v_i \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} + \\sum_{i=1}^n w_i \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = L(v) + L(w) \\\\ L(\\gamma v) &= \\sum_{i=1}^n (\\gamma v_i) \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = \\gamma \\sum_{i=1}^n v_i \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = \\gamma L(v) \\end{align*}","breadcrumbs":"Basics of Number Theory » Polynomials » Proposition: Homomorphisms of Lagrange Interpolation","id":"45","title":"Proposition: Homomorphisms of Lagrange Interpolation"},"46":{"body":"We will now discuss the construction of finite fields with \\(p^n\\) elements, where \\(p\\) is a prime number and \\(n\\) is a positive integer. These fields are also known as Galois fields, denoted as \\(GF(p^n)\\). It is evident that \\(\\mathbb{Z}/p\\mathbb{Z}\\) is isomorphic to \\(GF(p)\\).","breadcrumbs":"Basics of Number Theory » Galoid Field » Galois Field","id":"46","title":"Galois Field"},"47":{"body":"A polynomial \\(f \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\) of degree \\(n\\) is called irreducible over \\(\\mathbb{Z}/p\\mathbb{Z}\\) if it cannot be factored as a product of two polynomials of lower degree in \\((\\mathbb{Z}/p\\mathbb{Z})[X]\\). Example: In \\((\\mathbb{Z}/2\\mathbb{Z})[X]\\): \\(X^2 + X + 1\\) is irreducible \\(X^2 + 1 = (X + 1)^2\\) is reducible To construct \\(GF(p^n)\\), we use an irreducible polynomial of degree \\(n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\).","breadcrumbs":"Basics of Number Theory » Galoid Field » Definition: Irreducible Polynomial","id":"47","title":"Definition: Irreducible Polynomial"},"48":{"body":"For \\(f, g \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\), the residue class of \\(g \\bmod f\\) is the set of all polynomials \\(h \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\) such that \\(h \\equiv g \\pmod{f}\\). This class is denoted as: \\[ g + f(\\mathbb{Z}/p\\mathbb{Z})[X] = \\{g + hf : h \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\} \\] Example: In \\((\\mathbb{Z}/2\\mathbb{Z})[X]\\), with \\(f(X) = X^2 + X + 1\\), the residue classes \\(\\bmod f\\) are: \\(0 + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{0, X^2 + X + 1, X^2 + X, X^2 + 1, X^2, X + 1, X, 1\\}\\) \\(1 + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{1, X^2 + X, X^2 + 1, X^2, X + 1, X, 0, X^2 + X + 1\\}\\) \\(X + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{X, X^2 + 1, X^2, X^2 + X + 1, X + 1, 1, X^2 + X, 0\\}\\) \\((X + 1) + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{X + 1, X^2, X^2 + X + 1, X^2 + X, 1, 0, X^2 + 1, X\\}\\) These four residue classes form \\(GF(4)\\).","breadcrumbs":"Basics of Number Theory » Galoid Field » Definition: Residue Class modulo a Polynomial","id":"48","title":"Definition: Residue Class modulo a Polynomial"},"49":{"body":"If \\(f \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\) is an irreducible polynomial of degree \\(n\\), then the residue ring \\((\\mathbb{Z}/p\\mathbb{Z})[X]/(f)\\) is a field with \\(p^n\\) elements, isomorphic to \\(GF(p^n)\\). Proof (Outline) The irreducibility of \\(f\\) ensures that \\((f)\\) is a maximal ideal in \\((\\mathbb{Z}/p\\mathbb{Z})[X]\\), making the quotient ring a field. The number of elements is \\(p^n\\) because there are \\(p^n\\) polynomials of degree less than \\(n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\). This construction allows us to represent elements of \\(GF(p^n)\\) as polynomials of degree less than \\(n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\). Addition is performed coefficient-wise modulo \\(p\\), while multiplication is performed modulo the irreducible polynomial \\(f\\). Example: To construct \\(GF(8)\\), we can use the irreducible polynomial \\(f(X) = X^3 + X + 1\\) over \\(\\mathbb{Z}/2\\mathbb{Z}\\). The elements of \\(GF(8)\\) are represented by: [ \\{0, 1, X, X+1, X^2, X^2+1, X^2+X, X^2+X+1\\} ] For instance, multiplication in \\(GF(8)\\): [ (X^2 + 1)(X + 1) = X^3 + X^2 + X + 1 \\equiv X^2 \\pmod{X^3 + X + 1} ]","breadcrumbs":"Basics of Number Theory » Galoid Field » Theorem:","id":"49","title":"Theorem:"},"5":{"body":"","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Semigroup, Group, Ring","id":"5","title":"Semigroup, Group, Ring"},"50":{"body":"Let \\(\\mathbb{F}\\) be a field and \\(P: F^m \\rightarrow \\mathbb{F}\\) and \\(Q: \\mathbb{F}^m \\rightarrow \\mathbb{F}\\) be two distinct multivariate polynomials of total degree at most \\(n\\). For any finite subset \\(\\mathbb{S} \\subseteq \\mathbb{F}\\), we have: \\begin{align} Pr_{u \\sim \\mathbb{S}^{m}}[P(u) = Q(u)] \\leq \\frac{n}{|\\mathbb{S}|} \\end{align} , where \\(u\\) is drawn uniformly at random from \\(\\mathbb{S}^m\\). Proof TBD This lemma states that if \\(\\mathbb{S}\\) is sufficiently large and \\(n\\) is relatively small, the probability that the two different polynomials return the same value for a randomly chosen input is negligibly small. In other words, if we observe \\(P(u) = Q(u)\\) for a random input \\(u\\), we can conclude with high probability that \\(P\\) and \\(Q\\) are identical polynomials.","breadcrumbs":"Basics of Number Theory » Galoid Field » Lemma: Schwartz - Zippel Lemma","id":"50","title":"Lemma: Schwartz - Zippel Lemma"},"51":{"body":"","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Elliptic curve","id":"51","title":"Elliptic curve"},"52":{"body":"An elliptic curve \\(E\\) over a finite field \\(\\mathbb{F}_{p}\\) is defined by the equation: \\begin{equation} y^2 = x^3 + a x + b \\end{equation} , where \\(a, b \\in \\mathbb{F}_{p}\\) and the discriminant \\(\\Delta_{E} = 4a^3 + 27b^2 \\neq 0\\). use crate::modules::field::Field; #[derive(Debug, Clone, PartialEq)]\npub struct EllipticCurve<F: Field> { pub a: F, pub b: F,\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Elliptic Curve","id":"52","title":"Definition: Elliptic Curve"},"53":{"body":"An \\(\\mathbb{F}_{p}\\)-rational point on an elliptic curve \\(E\\) is a point \\((x, y)\\) where both \\(x\\) and \\(y\\) are elements of \\(\\mathbb{F}_{p}\\) and satisfy the curve equation, or the point at infinity \\(\\mathcal{O}\\). Example: For \\(E: y^2 = x^3 + 3x + 4\\) over \\(\\mathbb{F}_7\\), the \\(\\mathbb{F}_{7}\\)-rational points are: \\(\\{(0, 2), (0, 5), (1, 1), (1, 6), (2, 2), (2, 5), (5, 1), (5, 6), \\mathcal{O}\\}\\)","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: \\(\\mathbb{F}_{p}\\)-Rational Point","id":"53","title":"Definition: \\(\\mathbb{F}_{p}\\)-Rational Point"},"54":{"body":"The point at infinity denoted \\(\\mathcal{O}\\), is a special point on the elliptic curve that serves as the identity element for the group operation. It can be visualized as the point where all vertical lines on the curve meet. #[derive(Debug, Clone, PartialEq)]\npub struct EllipticCurvePoint<F: Field> { pub x: Option<F>, pub y: Option<F>, pub curve: EllipticCurve<F>,\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: The point at infinity","id":"54","title":"Definition: The point at infinity"},"55":{"body":"For an elliptic curve \\(E: y^2 = x^3 + ax + b\\), the addition of points \\(P\\) and \\(Q\\) to get \\(R = P + Q\\) is defined as follows: If \\(P = \\mathcal{O}\\), \\(R = Q\\). If \\(Q = \\mathcal{O}\\), \\(R = P\\). Otherwise, let \\(P = (x_P, y_P), Q = (x_Q, y_Q)\\). Then: If \\(y_P = -y_Q\\), \\(R = \\mathcal{O}\\) If \\(y_P \\neq -y_Q\\), \\(R = (x_R = \\lambda^2 - x_P - x_Q, y_R = \\lambda(x_P - x_R) - y_P)\\), where \\(\\lambda = \\) \\( \\begin{cases} \\frac{y_P - y_Q}{x_P - x_Q} \\quad &\\hbox{If } (x_P \\neq x_Q) \\\\ \\frac{3^{2}_{P} + a}{2y_P} \\quad &\\hbox{Otherwise} \\end{cases}\\) Example: On \\(E: y^2 = x^3 + 2x + 3\\) over \\(\\mathbb{F}_{7}\\), let \\(P = (5, 1)\\) and \\(Q = (4, 4)\\). Then, \\(P + Q = (0, 5)\\), where \\(\\lambda = \\frac{1 - 4}{5 - 4} \\equiv 4 \\bmod 7\\). impl<F: Field> EllipticCurvePoint<F> { fn new(x: F, y: F, curve: EllipticCurve<F>) -> Self { EllipticCurvePoint { x: Some(x), y: Some(y), curve: curve, } } pub fn point_at_infinity(curve: EllipticCurve<F>) -> Self { EllipticCurvePoint { x: None, y: None, curve: curve, } } pub fn is_point_at_infinity(&self) -> bool { self.x.is_none() || self.y.is_none() } pub fn line_slope(&self, other: Self) -> F { let x1 = self.x.clone().unwrap(); let y1 = self.y.clone().unwrap(); let x2 = other.x.clone().unwrap(); let y2 = other.y.clone().unwrap(); if self.x.clone() == other.x.clone() { ((x1.clone() * x1.clone()) * (3_i64) + self.curve.a.clone()) / (y1.clone() * (2_i64)) } else { (y2.clone() - y1.clone()) / (x2.clone() - x1.clone()) } }\n} impl<F: Field> Add for EllipticCurvePoint<F> { type Output = Self; fn add(self, other: Self) -> Self { if self.is_point_at_infinity() { return other; } if other.is_point_at_infinity() { return self; } let m = self.line_slope(other.clone()); if self.x == other.x { if self.y != other.y { return EllipticCurvePoint::point_at_infinity(self.curve.clone()); } else { let x1 = self.x.clone().unwrap(); let y1 = self.y.clone().unwrap(); let x3 = m.clone() * m.clone() - x1.clone() - x1.clone(); let y3 = m * (x1 - x3.clone()) - y1; return EllipticCurvePoint::new(x3, y3, self.curve.clone()); } } else { let x1 = self.x.clone().unwrap(); let y1 = self.y.clone().unwrap(); let x2 = other.x.clone().unwrap(); let x3 = m.clone() * m.clone() - x1.clone() - x2.clone(); let y3 = m * (x1 - x3.clone()) - y1; return EllipticCurvePoint::new(x3, y3, self.curve.clone()); } }\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Addition on Elliptic Curve","id":"55","title":"Definition: Addition on Elliptic Curve"},"56":{"body":"The Mordell-Weil group of an elliptic curve \\(E\\) is the group of rational points on \\(E\\) under the addition operation defined above. Example: For \\(E: y^2 = x^3 + x + 6\\) over \\(\\mathbb{F}_{11}\\), the Mordell-Weil group is the set of all \\(\\mathbb{F}_{11}\\)-rational points on \\(E\\) with the elliptic curve addition operation.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Mordell-Weil Group","id":"56","title":"Definition: Mordell-Weil Group"},"57":{"body":"The group order of an elliptic curve \\(E\\) over \\(\\mathbb{F}_{p}\\), denoted \\(\\#E(\\mathbb{F}_{p})\\), is the number of \\(\\mathbb{F}_{p}\\)-rational points on \\(E\\), including the point at infinity. Example: For \\(E: y^2 = x^3 + x + 6\\) over \\(\\mathbb{F}_{11}\\), \\(\\#E(\\mathbb{F}_{11}) = 13\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Group Order","id":"57","title":"Definition: Group Order"},"58":{"body":"Let \\(\\#E(\\mathbb{F}_{p})\\) be the group order of the elliptic curve \\(E\\) over \\(\\mathbb{F}_{p}\\). Then: \\begin{equation} p + 1 - 2 \\sqrt{p} \\leq \\#E \\leq p + 1 + 2 \\sqrt{p} \\end{equation} Example: For an elliptic curve over \\(\\mathbb{F}_{23}\\), the Hasse-Weil theorem guarantees that: \\begin{equation*} 23 + 1 - 2 \\sqrt{23} \\simeq 14.42 \\#E(\\mathbb{F}_{23}) \\geq 23 + 1 + 2 \\sqrt{23} \\simeq 33.58 \\end{equation*}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem: Hasse-Weil","id":"58","title":"Theorem: Hasse-Weil"},"59":{"body":"The order of a point \\(P\\) on an elliptic curve is the smallest positive integer \\(n\\) such that \\(nP = \\mathcal{O}\\) (where \\(nP\\) denotes \\(P\\) added to itself \\(n\\) times). We also denote the set of points of order \\(n\\), also called \\textit{torsion} group, by \\begin{equation} E[n] = \\{P \\in E: [n]P = \\mathcal{O}\\} \\end{equation} Example: On \\(E: y^2 = x^3 + 2x + 2\\) over \\(\\mathbb{F}_{17}\\), the point \\(P = (5, 1)\\) has order 18 because \\(18P = \\mathcal{O}\\), and no smaller positive multiple of \\(P\\) equals \\(\\mathcal{O}\\). The intuitive view is that if you continue to add points to themselves (doubling, tripling, etc.), the lines drawn between the prior point and the next point will eventually become more vertical. When the line becomes vertical, it does not intersect the elliptic curve at any finite point. In elliptic curve geometry, a vertical line is considered to \"intersect\" the curve at a special place called the \"point at infinity,\" This point is like a north pole in geographic terms: no matter which direction you go, if the line becomes vertical (reaching infinitely high), it converges to this point.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Point Order","id":"59","title":"Definition: Point Order"},"6":{"body":"A pair \\((H, \\circ)\\), where \\(H\\) is a non-empty set and \\(\\circ\\) is an associative binary operation on \\(H\\), is called a semigroup. Example : The set of positive integers under multiplication modulo \\(n\\) forms a semigroup. For instance, with \\(n = 6\\), the elements \\(\\{1, 2, 3, 4, 5\\}\\) under multiplication modulo 6 form a semigroup, since multiplication modulo 6 is associative.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Semigroup","id":"6","title":"Definition: Semigroup"},"60":{"body":"Let \\(F\\) and \\(L\\) be fields. If \\(F \\subseteq L\\) and the operations of \\(F\\) are the same as those of \\(L\\), we call \\(L\\) a field extension of \\(F\\). This is denoted as \\(L/F\\). A field extension \\(L/F\\) naturally gives \\(L\\) the structure of a vector space over \\(F\\). The dimension of this vector space is called the degree of the extension. Examples: \\(\\mathbb{C}/\\mathbb{R}\\) is a field extension with basis \\(\\{1, i\\}\\) and degree 2. \\(\\mathbb{R}/\\mathbb{Q}\\) is an infinite degree extension. \\(\\mathbb{Q}(\\sqrt{2})/\\mathbb{Q}\\) is a degree 2 extension with basis \\(\\{1, \\sqrt{2}\\}\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Field Extension","id":"60","title":"Definition: Field Extension"},"61":{"body":"A field extension \\(L/K\\) is called algebraic if every element \\(\\alpha \\in L\\) is algebraic over \\(K\\), i.e., \\(\\alpha\\) is the root of some non-zero polynomial with coefficients in \\(K\\). For an algebraic element \\(\\alpha \\in L\\) over \\(K\\), we denote by \\(K(\\alpha)\\) the smallest field containing both \\(K\\) and \\(\\alpha\\). Example: \\(\\mathbb{C}/\\mathbb{R}\\) is algebraic: any \\(z = a + bi \\in \\mathbb{C}\\) is a root of \\(x^2 - 2ax + (a^2 + b^2) \\in \\mathbb{R}[x]\\). \\(\\mathbb{Q}(\\sqrt[3]{2})/\\mathbb{Q}\\) is algebraic: \\(\\sqrt[3]{2}\\) is a root of \\(x^3 - 2 \\in \\mathbb{Q}[x]\\). \\(\\mathbb{R}/\\mathbb{Q}\\) is not algebraic (a field extension that is not algebraic is called \\textit{transcendental}).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Algebraic Extension","id":"61","title":"Definition: Algebraic Extension"},"62":{"body":"Let \\(K\\) be a field and \\(X\\) be indeterminate. The field of rational functions over \\(K\\), denoted \\(K(X)\\), is defined as: \\begin{equation*} K(X) = \\left\\{ \\frac{f(X)}{g(X)} \\middle|\\ f(X), g(X) \\in K[X], g(X) \\neq 0 \\right\\} \\end{equation*} where \\(K[X]\\) is the ring of polynomials in \\(X\\) with coefficients in \\(K\\). \\(K(X)\\) can be viewed as the field obtained by adjoining a letter \\(X\\) to \\(K\\). This construction generalizes to multiple variables, e.g., \\(K(X,Y)\\). The concept of a function field naturally arises in the context of algebraic curves, particularly elliptic curves. Intuitively, the function field encapsulates the algebraic structure of rational functions on the curve. We first construct the coordinate ring for an elliptic curve \\(E: y^2 = x^3 + ax + b\\). Consider functions \\(X: E \\to K\\) and \\(Y: E \\to K\\) that extract the \\(x\\) and \\(y\\) coordinates, respectively, from an arbitrary point \\(P \\in E\\). These functions generate the polynomial ring \\(K[X,Y]\\), subject to the relation \\(Y^2 = X^3 + aX + b\\). To put it simply, the function field is a field that consists of all functions that determine the value based on the point on the curve.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Field of Rational Functions","id":"62","title":"Definition: Field of Rational Functions"},"63":{"body":"The coordinate ring of an elliptic curve \\(E: y^2 = x^3 + ax + b\\) over a field \\(K\\) is defined as: \\begin{equation} K[E] = K[X, Y]/(Y^2 - X^3 - aX - b) \\end{equation} In other words, we can view \\(K[E]\\) as a ring representing all polynomial functions on \\(E\\). Recall that \\(K[X, Y]\\) is the polynomial ring in two variables \\(X\\) and \\(Y\\) over the field K, meaning that it contains all polynomials in \\(X\\) and \\(Y\\) with coefficients from \\(K\\). Then, the notation \\(K[X, Y]/(Y^2 - X^3 - aX - b)\\) denotes the quotient ring obtained by taking \\(K[X, Y]\\) and \"modding out\" by the ideal \\((Y^2 - X^3 - aX - b)\\). Example For example, for an elliptic curve \\(E: y^2 = x^3 - x\\) over \\(\\mathbb{Q}\\), some elements of the coordinate ring \\(\\mathbb{Q}[E]\\) include: Constants: \\(3, -2, \\frac{1}{7}, \\ldots\\) Linear functions: \\(X, Y, 2X+3Y, \\ldots\\) Quadratic functions: \\(X^2, XY, Y^2 (= X^3 - X), \\ldots\\) Higher-degree functions: \\(X^3, X^2Y, XY^2 (= X^4 - X^2), \\ldots\\) Then, the function field is defined as follows:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Coordinate Ring of an Elliptic Curve","id":"63","title":"Definition: Coordinate Ring of an Elliptic Curve"},"64":{"body":"Let \\(E: y^2 = x^3 + ax + b\\) be an elliptic curve over a field \\(K\\). The function field of \\(E\\), denoted \\(K(E)\\), is defined as: \\begin{equation} K(E) = \\left\\{\\frac{f}{g} ,\\middle|, f, g \\in K[E], g \\neq 0 \\right\\} \\end{equation} where \\(K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)\\) is the coordinate ring of \\(E\\). \\(K(E)\\) can be viewed as the field of all rational functions on \\(E\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Function Field of an Elliptic Curve","id":"64","title":"Definition: Function Field of an Elliptic Curve"},"65":{"body":"Let \\(h \\in K(E)\\) be a non-zero function. A point \\(P \\in E\\) is called a zero of \\(h\\) if \\(h(P) = 0\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Zero of a Function","id":"65","title":"Definition: Zero of a Function"},"66":{"body":"Let \\(h \\in K(E)\\) be a non-zero function. A point \\(P \\in E\\) is called a pole of \\(h\\) if \\(h\\) is not defined at \\(P\\) or, equivalently if \\(1/h\\) has a zero at \\(P\\). Example Consider the elliptic curve \\(E: Y^2 = X^3 - X\\) over a field \\(K\\) of characteristic \\(\\neq 2, 3\\). Let \\(P_{-1} = (-1, 0)\\), \\(P_0 = (0, 0)\\), and \\(P_1 = (1, 0)\\) be the points where \\(Y = 0\\). The function \\(Y \\in K(E)\\) has three simple zeros: \\(P_{-1}\\), \\(P_0\\), and \\(P_1\\). The function \\(X \\in K(E)\\) has a double zero at \\(P_0\\) (since \\(P_0 = -P_0\\)). The function \\(X - 1 \\in K(E)\\) has a simple zero at \\(P_1\\). The function \\(X^2 - 1 \\in K(E)\\) has two simple zeros at \\(P_{-1}\\) and \\(P_1\\). The function \\(\\frac{Y}{X} \\in K(E)\\) has a simple zero at \\(P_{-1}\\) and a simple pole at \\(P_0\\). An important property of functions in \\(K(E)\\) is that they have the same number of zeros and poles when counted with multiplicity. This is a consequence of a more general result known as the Degree-Genus Formula.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Pole of a Function","id":"66","title":"Definition: Pole of a Function"},"67":{"body":"Let \\(f \\in K(E)\\) be a non-zero rational function on an elliptic curve \\(E\\). Then: \\begin{equation} \\sum_{P \\in E} ord_{P(f)} = 0 \\end{equation} , where \\(ord_{P(f)}\\) denotes the order of \\(f\\) at \\(P\\), which is positive for zeros and negative for poles. This theorem implies that the total number of zeros (counting multiplicity) equals the total number of poles for any non-zero function in \\(K(E)\\). We now introduce a powerful tool for analyzing functions on elliptic curves: the concept of divisors.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem: Degree-Genus Formula for Elliptic Curves","id":"67","title":"Theorem: Degree-Genus Formula for Elliptic Curves"},"68":{"body":"Let \\(E: Y^2 = X^3 + AX + B\\) be an elliptic curve over a field \\(K\\), and let \\(f \\in K(E)\\) be a non-zero rational function on \\(E\\). The divisor of \\(f\\), denoted \\(div(f)\\), is defined as: \\begin{equation} div(f) = \\sum_{P \\in E} ord_P(f) [P] \\end{equation} , where \\(ord_P(f)\\) is the order of \\(f\\) at \\(P\\) (positive for zeros, negative for poles), and the sum is taken over all points \\(P \\in E\\), including the point at infinity. This sum has only finitely many non-zero terms. Note that this \\(\\sum_{P \\in E}\\) is a symbolic summation, and we do not calculate the concrete value of a divisor. Example Consider the elliptic curve \\(E: Y^2 = X^3 - X\\) over \\(\\mathbb{Q}\\). For \\(f = X\\), we have \\(div(X) = 2[(0,0)] - 2[\\mathcal{O}]\\). For \\(g = Y\\), we have \\(div(Y) = [(1,0)] + [(0,0)] + [(-1,0)] - 3[\\mathcal{O}]\\). For \\(h = \\frac{X-1}{Y}\\), we have \\(div(h) = [(1,0)] - [(-1,0)]\\). The concept of divisors can be extended to the elliptic curve itself:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Divisor of a Function on an Elliptic Curve","id":"68","title":"Definition: Divisor of a Function on an Elliptic Curve"},"69":{"body":"A divisor \\(D\\) on an elliptic curve \\(E\\) is a formal sum \\begin{equation} D = \\sum_{P \\in E} n_P [P] \\end{equation} where \\(n_P \\in \\mathbb{Z}\\) and \\(n_P = 0\\) for all but finitely many \\(P\\). Example On the curve \\(E: Y^2 = X^3 - X\\): \\(D_1 = 3[(0,0)] - 2[(1,1)] - [(2,\\sqrt{6})]\\) is a divisor. \\(D_2 = [(1,0)] + [(-1,0)] - 2[\\mathcal{O}]\\) is a divisor. \\(D_3 = \\sum_{P \\in E[2]} [P] - 4[\\mathcal{O}]\\) is a divisor (where \\(E[2]\\) are the 2-torsion points). To quantify the properties of divisors, we introduce two important metrics:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Divisor on an Elliptic Curve","id":"69","title":"Definition: Divisor on an Elliptic Curve"},"7":{"body":"A semigroup whose operation is commutative is called an abelian semigroup. Example : The set of natural numbers under addition modulo \\(n\\) forms an abelian semigroup. For \\(n = 7\\), addition modulo 7 is both associative and commutative, so it is an abelian semigroup.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Abelian Semigroup","id":"7","title":"Definition: Abelian Semigroup"},"70":{"body":"The degree of a divisor \\(D = \\sum_{P \\in E} n_P [P]\\) is defined as: \\begin{equation} deg(D) = \\sum_{P \\in E} n_P \\end{equation}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Degree of a Divisor","id":"70","title":"Definition: Degree of a Divisor"},"71":{"body":"The sum of a divisor \\(D = \\sum_{P \\in E} n_P [P]\\) is defined as: \\begin{equation} Sum(D) = \\sum_{P \\in E} n_P P \\end{equation} where \\(n_P P\\) denotes the point addition of \\(P\\) to itself \\(n_P\\) times in the group law of \\(E\\). Example For the divisors in the previous example: \\(deg(D_1) = 3 - 2 - 1 = 0\\) \\(deg(D_2) = 1 + 1 - 2 = 0\\) \\(deg(D_3) = 4 - 4 = 0\\) \\(Sum(D_2) = (1,0) + (-1,0) - 2\\mathcal{O} = \\mathcal{O}\\) (since \\((1,0)\\) and \\((-1,0)\\) are 2-torsion points) The following theorem characterizes divisors of functions and provides a criterion for when a divisor is the divisor of a function:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Sum of a Divisor","id":"71","title":"Definition: Sum of a Divisor"},"72":{"body":"Let \\(E\\) be an elliptic curve over a field \\(K\\). If \\(f, f' \\in K(E)\\) are non-zero rational functions on \\(E\\) with \\(div(f) = div(f')\\), then there exists a non-zero constant \\(c \\in K^ \\) such that \\(f = cf'\\).* A divisor \\(D\\) on \\(E\\) is the divisor of a rational function on \\(E\\) if and only if \\(deg(D) = 0\\) and \\(Sum(D) = \\mathcal{O}\\). Example On \\(E: Y^2 = X^3 - X\\): The function \\(f = \\frac{Y}{X}\\) has \\(div(f) = [(1,0)] + [(-1,0)] - 2[(0,0)]\\). Note that \\(deg(div(f)) = 0\\) and \\(Sum(div(f)) = (1,0) + (-1,0) - 2(0,0) = \\mathcal{O}\\). The divisor \\(D = 2[(1,1)] - [(2,\\sqrt{6})] - [(0,0)]\\) has \\(deg(D) = 0\\), but \\(Sum(D) \\neq \\mathcal{O}\\). Therefore, \\(D\\) is not the divisor of any rational function on \\(E\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem","id":"72","title":"Theorem"},"73":{"body":"Let \\(G_1\\) and \\(G_2\\) be cyclic groups under addition, both of prime order \\(p\\), with generators \\(P\\) and \\(Q\\) respectively: \\begin{align} G_1 &= \\{0, P, 2P, ..., (p-1)P\\} \\ G_2 &= \\{0, Q, 2Q, ..., (p-1)Q\\} \\end{align} Let \\(G_T\\) be a cyclic group under multiplication, also of order \\(p\\). A pairing is a map \\(e: G_1 \\times G_2 \\rightarrow G_T\\) that satisfies the following bilinear property: \\begin{equation} e(aP, bQ) = e(P, Q)^{ab} \\end{equation} for all \\(a, b \\in \\mathbb{Z}_p\\). Imagine \\(G_1\\) represents length, \\(G_2\\) represents width, and \\(G_T\\) represents area. The pairing function \\(e\\) is like calculating the area: If you double the length and triple the width, the area becomes six times larger: \\(e(2P, 3Q) = e(P, Q)^{6}\\) The Weil pairing is one of the bilinear pairings for elliptic curves. We begin with its formal definition.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Pairing","id":"73","title":"Definition: Pairing"},"74":{"body":"Let \\(E\\) be an elliptic curve and \\(n\\) be a positive integer. For points \\(P, Q \\in E[n]\\), where \\(E[n]\\) denotes the \\(n\\)-torsion subgroup of \\(E\\), we define the Weil pairing \\(e_n(P, Q)\\) as follows: Let \\(f_P\\) and \\(f_Q\\) be rational functions on \\(E\\) satisfying: \\begin{align} div(f_P) &= n[P] - n[\\mathcal{O}] \\ div(f_Q) &= n[Q] - n[\\mathcal{O}] \\end{align} Then, for an arbitrary point \\(S \\in E\\) such that \\(S \\notin \\{\\mathcal{O}, P, -Q, P-Q\\}\\), the Weil pairing is given by: \\begin{equation} e_n(P, Q) = \\frac{f_P(Q + S)}{f_P(S)} /\\ \\frac{f_Q(P - S)}{f_Q(-S)} \\end{equation} We introduce a crucial theorem about a specific rational function on elliptic curves to construct the functions required for the Weil pairing.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: The Weil Pairing","id":"74","title":"Definition: The Weil Pairing"},"75":{"body":"Let \\(E\\) be an elliptic curve over a field \\(K\\), and let \\(P = (x_P, y_P)\\) and \\(Q = (x_Q, y_Q)\\) be non-zero points on \\(E\\). Define \\(\\lambda\\) as: \\begin{equation} \\lambda = \\begin{cases} \\hbox{slope of the line through \\(P\\) and \\(Q\\)} &\\quad \\hbox{if \\(P \\neq Q\\)} \\\\ \\hbox{slope of the tangent line to \\(E\\) at \\(P\\)} &\\quad \\hbox{if \\(P = Q\\)} \\\\ \\infty &\\quad \\hbox{if the line is vertical} \\end{cases} \\end{equation} Then, the function \\(g_{P,Q}: E \\to K\\) defined by: \\begin{equation} g_{P,Q} = \\begin{cases} \\frac{y - y_P - \\lambda(x - x_P)}{x + x_P + x_Q - \\lambda^2} &\\quad \\hbox{if } \\lambda \\neq \\infty \\\\ x - x_P &\\quad \\hbox{if } \\lambda = \\infty \\end{cases} \\end{equation} has the following divisor: \\begin{equation} div(g_{P,Q}) = [P] + [Q] - [P + Q] - [\\mathcal{O}] \\end{equation} Proof: We consider two cases based on the value of \\(\\lambda\\). Case 1: \\(\\lambda \\neq \\infty\\) Let \\(y = \\lambda x + v\\) be the line through \\(P\\) and \\(Q\\) (or the tangent line at \\(P\\) if \\(P = Q\\)). This line intersects \\(E\\) at three points: \\(P\\), \\(Q\\), and \\(-P-Q\\). Thus, \\begin{equation} div(y - \\lambda x - v) = [P] + [Q] + [-P - Q] - 3[\\mathcal{O}] \\end{equation} Vertical lines intersect \\(E\\) at points and their negatives, so: \\begin{equation} div(x - x_{P+Q}) = [P + Q] + [-P - Q] - 2[\\mathcal{O}] \\end{equation} It follows that \\(g_{P,Q} = \\frac{y - \\lambda x - v}{x - x_{P+Q}}\\) has the desired divisor. Case 2: \\(\\lambda = \\infty\\) In this case, \\(P + Q = \\mathcal{O}\\), so we want \\(g_{P,Q}\\) to have divisor \\([P] + [-P] - 2[\\mathcal{O}]\\). The function \\(x - x_P\\) has this divisor. \\end{proof}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem","id":"75","title":"Theorem"},"76":{"body":"Let \\(m \\geq 1\\) and write its binary expansion as: \\begin{equation} m = m_0 + m_1 \\cdot 2 + m_2 \\cdot 2^2 + \\cdots + m_{n-1} \\cdot 2^{n-1} \\end{equation} where \\(m_i \\in \\{0, 1\\}\\) and \\(m_{n-1} \\neq 0\\). The following algorithm, using the function \\(g_{P,Q}\\) defined in the previous theorem, returns a function \\(f_P\\) whose divisor satisfies: \\begin{equation} div(f_P) = m[P] - m[P] - (m - 1)[\\mathcal{O}] \\end{equation} =========================== Miller's Algorithm Set \\(T = P\\) and \\(f = 1\\) For \\(i \\gets n-2 \\cdots 0\\) do Set \\(f = f^2 \\cdot g_{T, T}\\) Set \\(T = 2T\\) If \\(m_i = 1\\) then Set \\(f = f \\cdot g_{T, P}\\) Set \\(T = T + P\\) End if End for Return \\(f\\) =========================== Proof: TBD pub fn get_lambda<F: Field>( p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, r: EllipticCurvePoint<F>,\n) -> F { let p_x = p.x.clone().unwrap(); let p_y = p.y.clone().unwrap(); let q_x = q.x.clone().unwrap(); // let q_y = q.y.clone().unwrap(); let r_x = r.x.clone().unwrap(); let r_y = r.y.clone().unwrap(); if (p == q && p_y.clone() == F::zero()) || (p != q && p_x.clone() == q_x.clone()) { return r_x.clone() - p_x.clone(); } let slope = p.line_slope(q.clone()); let numerator = r_y.clone() - p_y.clone() - slope.clone() * (r_x.clone() - p_x.clone()); let denominator = r_x.clone() + p_x.clone() + q_x.clone() - slope.clone() * slope.clone(); return numerator / denominator;\n} pub fn miller<F: Field>(p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, m: BigInt) -> F { if p == q { F::one(); } let mut f = F::one(); let mut t = p.clone(); for i in (1..m.bits()).rev() { f = (f.clone() * f.clone()) * (get_lambda(t.clone(), t.clone(), q.clone())); t = t.clone() + t.clone(); if m.bit(i) { f = f * (get_lambda(t.clone(), p.clone(), q.clone())); t = t.clone() + p.clone(); } } f\n} pub fn weil_pairing<F: Field>( p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, m: BigInt, s: Option<EllipticCurvePoint<F>>,\n) -> F { let s_value = s.unwrap(); let fp_qs = miller(p.clone(), q.clone() + s_value.clone(), m.clone()); let fp_s = miller(p.clone(), s_value.clone(), m.clone()); let fq_ps = miller(q.clone(), p.clone() - s_value.clone(), m.clone()); let fq_s = miller(q.clone(), -s_value.clone(), m.clone()); return (fp_qs / fp_s) / (fq_ps / fq_s);\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem: Miller's Algorithm","id":"76","title":"Theorem: Miller's Algorithm"},"77":{"body":"","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumptions","id":"77","title":"Assumptions"},"78":{"body":"Let \\(G\\) be a finite cyclic group of order \\(n\\), with \\(\\gamma\\) as its generator and \\(1\\) as the identity element. For any element \\(\\alpha \\in G\\), there is currently no known efficient (polynomial-time) algorithm to compute the smallest non-negative integer \\(x\\) such that \\(\\alpha = \\gamma^{x}\\). The Discrete Logarithm Problem can be thought of as a one-way function. It's easy to compute \\(g^{x}\\) given \\(g\\) and \\(x\\), but it's computationally difficult to find \\(x\\) given \\(g\\) and \\(g^{x}\\).","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumption: Discrete Logarithm Problem","id":"78","title":"Assumption: Discrete Logarithm Problem"},"79":{"body":"Let \\(E\\) be an elliptic curve defined over a finite field \\(\\mathbb{F}_q\\), where \\(q\\) is a prime power. Let \\(P\\) be a point on \\(E\\) of point order \\(n\\), and let \\(\\langle P \\rangle\\) be the cyclic subgroup of \\(E\\) generated by \\(P\\). For any element \\(Q \\in \\langle P \\rangle\\), there is currently no known efficient (polynomial-time) algorithm to compute the unique integer \\(k\\), \\(0 \\leq k < n\\), such that \\(Q = kP\\). This assumption is an elliptic curve version of the Discrete Logarithm Problem.","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumption: Elliptic Curve Discrete Logarithm Problem","id":"79","title":"Assumption: Elliptic Curve Discrete Logarithm Problem"},"8":{"body":"An element \\(e \\in H\\) is an identity element of \\(H\\) if it satisfies \\(e \\circ a = a \\circ e = a\\) for any \\(a \\in H\\). Example : 0 is the identity element for addition modulo \\(n\\). For example, \\(0 + a \\bmod 5 = a + 0 \\bmod 5 = a\\). Similarly, 1 is the identity element for multiplication modulo \\(n\\). For example, \\(1 \\times a \\bmod 7 = a \\times 1 \\bmod 7 = a\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Identity Element","id":"8","title":"Definition: Identity Element"},"80":{"body":"Let \\(G\\) be a cyclic group of prime order \\(q\\) with generator \\(g \\in G\\). For any probabilistic polynomial-time algorithm \\(\\mathcal{A}\\) that outputs: \\begin{equation} \\mathcal{A}(g, g^x) = (h, h') \\quad s.t. \\quad h' = h^x \\end{equation} , there exists an efficient extractor \\(\\mathcal{E}\\) such that: \\begin{equation} \\mathcal{E}(\\mathcal{A}, g, g^x) = y \\quad s.t. \\quad h = g^y \\end{equation} This assumption states that if \\(\\mathcal{A}\\) can compute the pair \\((g^y, g^{xy})\\) from \\((g, g^x)\\), then \\(\\mathcal{A}\\) must \"know\" the value \\(y\\), in the sense that \\(\\mathcal{E}\\) can extract \\(y\\) from \\(\\mathcal{A}\\)'s internal state. The Knowledge of Exponent Assumption is useful for constructing verifiable exponential calculation algorithms. Consider a scenario where Alice has a secret value \\(a\\), and Bob has a secret value \\(b\\). Bob wants to obtain \\(g^{ab}\\). This can be achieved through the following protocol: Verifiable Exponential Calculation Algorithm Bob sends \\((g, g'=g^{b})\\) to Alice Alice sends \\((h=g^{a}, h'=g'^{a})\\) to Bob Bob checks \\(h^{b} = h'\\). Thanks to the Discrete Logarithm Assumption and the Knowledge of Exponent Assumption, the following properties hold: Bob cannot derive \\(a\\) from \\((h, h')\\). Alice cannot derive \\(b\\) from \\((g, g')\\). Alice cannot generate \\((t, t')\\) such that \\(t \\neq h\\) and \\(t^{b} = t'\\). If \\(h^{b} = h'\\), Bob can conclude that \\(h\\) is the power of \\(g\\).","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumption: Knowledge of Exponent Assumption","id":"80","title":"Assumption: Knowledge of Exponent Assumption"},"81":{"body":"","breadcrumbs":"Basics of zk-SNARKs » Basics of zk-SNARK","id":"81","title":"Basics of zk-SNARK"},"82":{"body":"The ultimate goal of Zero-Knowledge Proofs (ZKP) is to allow the prover to demonstrate their knowledge to the verifier without revealing any additional information. This knowledge typically involves solving complex problems, such as finding a secret input value that corresponds to a given public hash. ZKP protocols usually convert these statements into polynomial constraints. This process is often called \\textit{arithmetization}. To make the protocol flexible, we need to encode this knowledge in a specific format, and one common approach is using Boolean circuits. It's well-known that problems in P (those solvable in polynomial time) and NP (those where the solution can be verified in polynomial time) can be represented as Boolean circuits. This means adopting Boolean circuits allows us to handle both P and NP problems. However, Boolean circuits are often large and inefficient. Even a simple operation, like adding two 256-bit integers, can require hundreds of Boolean operators. In contrast, arithmetic circuits—essentially systems of equations involving addition, multiplication, and equality—offer a much more compact representation. Additionally, any Boolean circuit can be converted into an arithmetic circuit. For instance, the Boolean expression \\(z = x \\land y\\) can be represented as \\(x(x-1) = 0\\), \\(y(y-1) = 0\\), and \\(z = xy\\) in an arithmetic circuit. Furthermore, as we'll see in this section, converting arithmetic circuits into polynomial constraints allows for much faster evaluation.","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » Arithmetization","id":"82","title":"Arithmetization"},"83":{"body":"There are many formats to represent arithmetic circuits, and one of the most popular ones is R1CS (Rank-1 Constraint System), which represents arithmetic circuits as \\underline{a set of equality constraints, each involving only one multiplication}. In an arithmetic circuit, we call the concrete values assigned to the variables within the constraints witness. We first provide the formal definition of R1CS as follows:","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » Rank-1 Constraint System (R1CS)","id":"83","title":"Rank-1 Constraint System (R1CS)"},"84":{"body":"An R1CS structure \\(\\mathcal{S}\\) consists of: Size bounds \\(m, d, \\ell \\in \\mathbb{N}\\) where \\(d > \\ell\\) Three matrices \\(O, L, R \\in \\mathbb{F}^{m\\ times d}\\) with at most \\(\\Omega(\\max(m, d))\\) non-zero entries in total An R1CS instance includes a public input \\(p \\in \\mathbb{F}^\\ell\\), while an R1CS witness is a vector \\(w \\in \\mathbb{F}^{d - \\ell - 1}\\). A structure-instance tuple \\((S, p)\\) is satisfied by a witness \\(w\\) if: \\begin{equation} (L \\cdot a) \\circ (R \\cdot a) - O \\cdot a = \\mathbf{0} \\end{equation} where \\(a = (1, w, p) \\in \\mathbb{F}^d\\), \\(\\cdot\\) denotes matrix-vector multiplication, and \\(\\circ\\) is the Hadamard product. The intuitive interpretation of each matrix is as follows: \\(L\\): Encodes the left input of each gate \\(R\\): Encodes the right input of each gate \\(O\\): Encodes the output of each gate The leading 1 in the witness vector allows for constant terms Single Multiplication Let's consider a simple example where we want to prove \\(z = x \\cdot y\\), with \\(z = 3690\\), \\(x = 82\\), and \\(y = 45\\). \\textbf{Witness vector}: \\((1, z, x, y) = (1, 3690, 82, 45)\\) \\textbf{Number of witnesses}: \\(m = 4\\) \\textbf{Number of constraints}: \\(d = 1\\) The R1CS constraint for \\(z = x \\cdot y\\) is satisfied when: \\begin{align*} &(\\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\cdot a) \\circ (\\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix} \\cdot a) - \\begin{bmatrix} 0 & 1 & 0 & 0 \\end{bmatrix} \\cdot a \\\\ =&(\\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\cdot \\begin{bmatrix} 1 \\\\ 3690 \\\\ 82 \\\\ 45 \\end{bmatrix}) \\circ (\\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix} \\cdot \\begin{bmatrix} 1 \\\\ 3690 \\\\ 82 \\\\ 45 \\end{bmatrix}) - \\begin{bmatrix} 0 & 1 & 0 & 0 \\end{bmatrix} \\cdot \\begin{bmatrix} 1 \\\\ 3690 \\\\ 82 \\\\ 45 \\end{bmatrix} \\\\ =& 82 \\cdot 45 - 3690 \\\\ =& 3690 - 3690 \\\\ =& 0 \\end{align*} This example demonstrates how R1CS encodes a simple multiplication constraint: \\(L = \\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix}\\) selects \\(x\\) (left input) \\(R = \\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix}\\) selects \\(y\\) (right input) \\(O = \\begin{bmatrix} 0 & 1 & 0 & 0 \\end{bmatrix}\\) selects \\(z\\) (output) Multiple Constraints Let's examine a more complex example: \\(r = a \\cdot b \\cdot c \\cdot d\\). Since R1CS requires that each constraint contain only one multiplication, we need to break this down into multiple constraints: \\begin{align*} v_1 &= a \\cdot b \\\\ v_2 &= c \\cdot d \\\\ r &= v_1 \\cdot v_2 \\end{align*} Note that alternative representations are possible, such as \\(v_1 = ab, v_2 = v_1c, r = v_2d\\). In this example, we use 7 variables \\((r, a, b, c, d, v_1, v_2)\\), so the dimension of the witness vector will be \\(m = 8\\) (including the constant 1). We have three constraints, so \\(n = 3\\). To construct the matrices \\(L\\), \\(R\\), and \\(O\\), we can interpret the constraints as linear combinations: \\begin{align*} v_1 &= (0 \\cdot 1 + 0 \\cdot r + 1 \\cdot a + 0 \\cdot b + 0 \\cdot c + 0 \\cdot d + 0 \\cdot v_1 + 0 \\cdot v_2) \\cdot b \\\\ v_2 &= (0 \\cdot 1 + 0 \\cdot r + 0 \\cdot a + 0 \\cdot b + 1 \\cdot c + 0 \\cdot d + 0 \\cdot v_1 + 0 \\cdot v_2) \\cdot d \\\\ r &= (0 \\cdot 1 + 0 \\cdot r + 0 \\cdot a + 0 \\cdot b + 0 \\cdot c + 0 \\cdot d + 1 \\cdot v_1 + 0 \\cdot v_2) \\cdot v_2 \\end{align*} Thus, we can construct \\(L\\), \\(R\\), and \\(O\\) as follows: \\begin{equation*} L = \\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\end{bmatrix} \\end{equation*} \\begin{equation*} R = \\begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\end{bmatrix} \\end{equation*} \\begin{equation*} O = \\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\end{bmatrix} \\end{equation*} Where the columns in each matrix correspond to \\((1, r, a, b, c, d, v_1, v_2)\\). Addition with a Constant Let's examine the case \\(z = x \\cdot y + 3\\). We can represent this as \\(-3 + z = x \\cdot y\\). For the witness vector \\((1, z, x, y)\\), we have: \\begin{align*} L &= \\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\\\ R &= \\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix} \\\\ O &= \\begin{bmatrix} -3 & 1 & 0 & 0 \\end{bmatrix} \\end{align*} Note that the constant 3 appears in the \\(O\\) matrix with a negative sign, effectively moving it to the left side of the equation Multiplication with a Constant Now, let's consider \\(z = 3x^2 + y\\). The requirement of \"one multiplication per constraint\" doesn't apply to multiplication with a constant, as we can treat it as repeated addition. \\begin{align*} L &= \\begin{bmatrix} 0 & 0 & 3 & 0 \\end{bmatrix} \\\\ R &= \\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\\\ O &= \\begin{bmatrix} 0 & 1 & 0 & -1 \\end{bmatrix} \\end{align*}","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » Definition: R1CS","id":"84","title":"Definition: R1CS"},"85":{"body":"Recall that the prover aims to demonstrate knowledge of a witness \\(w\\) without revealing it. This is equivalent to knowing a vector \\(a\\) that satisfies \\((L \\cdot a) \\circ (R \\cdot a) = O \\cdot a\\), where \\(\\circ\\) denotes the Hadamard (element-wise) product. However, evaluating this equivalence directly requires \\(\\Omega(d)\\) operations, where \\(d\\) is the number of rows. To improve efficiency, we can convert this matrix comparison to a polynomial comparison, leveraging the Schwartz-Zippel Lemma, which allows us to check polynomial equality with \\(\\Omega(1)\\) evaluations. Let's consider a simpler example to illustrate this concept. Suppose we want to test the equivalence \\(Av = Bu\\), where: \\begin{align*} A = \\begin{bmatrix} 2 & 5 \\\\ 3 & 1 \\\\ \\end{bmatrix}, B = \\begin{bmatrix} 4 & 1 \\\\ 2 & 3 \\\\ \\end{bmatrix}, v = \\begin{bmatrix} 3 \\\\ 1 \\end{bmatrix}, u = \\begin{bmatrix} 2 \\\\ 2 \\end{bmatrix} \\end{align*} The equivalence check can be represented as: \\begin{equation*} \\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix} \\cdot 3 + \\begin{bmatrix} 5 \\\\ 1 \\end{bmatrix} \\cdot 1 = \\begin{bmatrix} 4 \\\\ 2 \\end{bmatrix} \\cdot 2 + \\begin{bmatrix} 1 \\\\ 3 \\end{bmatrix} \\cdot 2 \\end{equation*} This matrix-vector equality check is equivalent to the following polynomial equality check: \\begin{equation*} 3 \\cdot L([(1, 2), (2, 3)]) + 1 \\cdot L([(1, 5), (2, 1)]) = 2 \\cdot L([(1, 4), (2, 2)]) + 2 \\cdot L([(1, 1), (2, 3)]) \\end{equation*} where \\(L\\) denotes Lagrange Interpolation. In \\(\\mathbb{F}_{11}\\) (field with 11 elements), we have: \\begin{align*} L([(1, 2), (2, 3)]) &= x + 1 \\\\ L([(1, 5), (2, 1)]) &= 7x + 9 \\\\ L([(1, 4), (2, 2)]) &= 9x + 6 \\\\ L([(1, 1), (2, 3)]) &= 2x + 10 \\end{align*} The Schwartz-Zippel Lemma states that we need only one evaluation at a random point to check the equivalence of polynomials with high probability. However, the homomorphic property for multiplication doesn't hold for Lagrange Interpolation. Thus, we don't have \\(\\ell(x) \\cdot r(x) = o(x)\\), where \\(\\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) are the polynomials resulting from the Lagrange interpolation of \\(L \\cdot a\\), \\(R \\cdot a\\), and \\(O \\cdot a\\) respectively. While \\(\\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) are of degree at most \\(d-1\\), \\(\\ell(x) \\cdot r(x)\\) is of degree at most \\(2d-2\\). To address this discrepancy, we introduce a degree \\(d\\) polynomial \\(t(x) = \\prod_{i=1}^{d} (x - i)\\). We can then rewrite the equation as: \\begin{equation} \\ell(x) \\cdot r(x) = o(x) + h(x) \\cdot t(x) \\end{equation} where \\(h(x) = \\frac{\\ell(x) \\cdot r(x) - o(x)}{t(x)}\\). This formulation allows us to maintain the desired polynomial relationships while accounting for the degree differences.","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » uadratic Arithmetic Program (QAP)","id":"85","title":"uadratic Arithmetic Program (QAP)"},"86":{"body":"Before dealing with all of \\(\\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) at once, we design a protocol that allows the Prover \\(\\mathcal{A}\\) to convince the Verifier \\(\\mathcal{B}\\) that \\(\\mathcal{A}\\) knows a specific polynomial. Let's denote this polynomial of degree \\(n\\) with coefficients in a finite field as: \\begin{equation} P(x) = c_0 + c_1 x + c_2 x^{2} + \\cdots c_n x^{n} \\end{equation} Assume \\(P(x)\\) has \\(n\\) roots, \\(a_1, a_2, \\ldots, a_n \\in \\mathbb{F}\\), such that \\(P(x) = (x - a_1)(x - a_2)\\cdots(x - a_n)\\). The Verifier \\(\\mathcal{B}\\) knows \\(d < n\\) roots of \\(P(x)\\), namely \\(a_1, a_2, \\ldots, a_d\\). Let \\(T(x) = (x - a_1)(x - a_2)\\cdots(x - a_d)\\). Note that the Prover also knows \\(T(x)\\). The Prover's objective is to convince the Verifier that \\(\\mathcal{A}\\) knows a polynomial \\(H(x) = \\frac{P(x)}{T(x)}\\).","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » Proving Single Polynomial","id":"86","title":"Proving Single Polynomial"},"87":{"body":"The simplest approach to prove that \\(\\mathcal{A}\\) knows \\(H(x)\\) is as follows: Naive Approach \\(\\mathcal{B}\\) sends all possible values in \\(\\mathbb{F}\\) to \\(\\mathcal{A}\\). \\(\\mathcal{A}\\) computes and sends all possible outputs of \\(H(x)\\) and \\(P(x)\\). \\(\\mathcal{B}\\) checks whether \\(H(a)T(a) = P(a)\\) holds for any \\(a\\) in \\(\\mathbb{F}\\). This protocol is highly inefficient, requiring \\(\\mathcal{O}(|\\mathbb{F}|)\\) evaluations and communications.","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » Naive Approach","id":"87","title":"Naive Approach"},"88":{"body":"Instead of evaluating polynomials at all values in \\(\\mathbb{F}\\), we can leverage the Schwartz-Zippel Lemma: if \\(H(s) = \\frac{P(s)}{T(s)}\\) or equivalently \\(H(s)T(s) = P(s)\\) for a random element \\(s\\), we can conclude that \\(H(x) = \\frac{P(x)}{T(x)}\\) with high probability. Thus, the Prover \\(\\mathcal{A}\\) only needs to send evaluations of \\(P(s)\\) and \\(H(s)\\) for a random input \\(s\\) received from \\(\\mathcal{B}\\). \\(+\\) Schwartz-Zippel Lemma \\(\\mathcal{B}\\) draws random \\(s\\) from \\(\\mathbb{F}\\) and sends it to \\(\\mathcal{A}\\). \\(\\mathcal{A}\\) computes \\(h = H(s)\\) and \\(p = P(s)\\) and send them to \\(\\mathcal{B}\\). \\(\\mathcal{B}\\) checks whether \\(p = t h\\), where \\(t\\) denotes \\(T(s)\\). This protocol is efficient, requiring only a constant number of evaluations and communications. However, it is vulnerable to a malicious prover who could send an arbitrary value \\(h'\\) and the corresponding \\(p'\\) such that \\(p' = h't\\).","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » \\(+\\) Schwartz-Zippel Lemma","id":"88","title":"\\(+\\) Schwartz-Zippel Lemma"},"89":{"body":"To address this vulnerability, the Verifier must hide the randomly chosen input \\(s\\) from the Prover. This can be achieved using the discrete logarithm assumption: it is computationally hard to determine \\(s\\) from \\(\\alpha\\), where \\(\\alpha = g^s \\bmod p\\). Thus, it's safe for the Verifier to send \\(\\alpha\\), as the Prover cannot easily derive \\(s\\) from it. An interesting property of polynomial exponentiation is: \\begin{align} g^{P(x)} &= g^{c_0 + c_1 x + c_2 x^{2} + \\cdots c_n x^{n}} = g^{c_0} (g^{x})^{c_1} (g^{(x^2)})^{c_2} \\cdots (g^{(x^n)})^{c_n} \\end{align} Instead of sending \\(s\\), the Verifier can send \\(g\\) and \\(\\alpha_{i} = g^{(s^i)}\\) for \\(i = 1, \\cdots n\\). BE CAREFUL THAT \\(g^{(s^i)} \\neq (g^s)^i\\) . The Prover can still evaluate \\(g^p = g^{P(s)}\\) using these powers of \\(g\\): \\begin{equation} g^{p} = g^{P(s)} = g^{c_0} \\alpha_{1}^{c_1} (\\alpha_{2})^{c_2} \\cdots (\\alpha_{n})^{c_n} \\end{equation} Similarly, the Prover can evaluate \\(g^h = g^{H(s)}\\). The Verifier can then check \\(p = ht \\iff g^p = (g^h)^t\\). \\(+\\) Discrete Logarithm Assumption \\(\\mathcal{B}\\) randomly draw \\(s\\) from \\(\\mathbb{F}\\). \\(\\mathcal{B}\\) computes and sends \\(\\{\\alpha, \\alpha_2, ..., \\alpha_{n}\\}\\), where \\(\\alpha_i= g^{(s^{i})}\\). \\(\\mathcal{A}\\) computes and sends \\(u = g^{p}\\) and \\(v = g^{h}\\). \\(\\mathcal{B}\\) checks whether \\(u = v^{t}\\). This approach prevents the Prover from obtaining \\(s\\) or \\(t = T(s)\\), making it impossible to send fake \\((h', p')\\) such that \\(p' = h't\\). However, this protocol still has a flaw. Since the Prover can compute \\(g^t = \\alpha_{c_1}(\\alpha_2)^{c_2}\\cdots(\\alpha_d)^{c_d}\\), they could send fake values \\(((g^{t})^{z}, g^{z})\\) instead of \\((g^p, g^h)\\) for an arbitrary value \\(z\\). The verifier's check would still pass, and they could not detect this deception.","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » \\(+\\) Discrete Logarithm Assumption","id":"89","title":"\\(+\\) Discrete Logarithm Assumption"},"9":{"body":"A semigroup with an identity element is called a monoid. Example : The set of non-negative integers under addition modulo \\(n\\) forms a monoid. For \\(n = 5\\), the set \\(\\{0, 1, 2, 3, 4\\}\\) under addition modulo 5 forms a monoid with 0 as the identity element.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Monoid","id":"9","title":"Definition: Monoid"},"90":{"body":"To address the vulnerability where the verifier \\(\\mathcal{B}\\) cannot distinguish if \\(v (= g^h)\\) from the prover is a power of \\(\\alpha_i = g^{(s^i)}\\), we can employ the Knowledge of Exponent Assumption. This approach involves the following steps: \\(\\mathcal{B}\\) sends both \\(\\alpha_i\\) and \\(\\alpha'_i = \\alpha_i^r\\) for a new random value \\(r\\). The prover returns \\(a = (\\alpha_i)^{c_i}\\) and \\(a' = (\\alpha'_i)^{c_i}\\) for \\(i = 1, ..., n\\). \\(\\mathcal{B}\\) can conclude that \\(a\\) is a power of \\(\\alpha_i\\) if \\(a^r = a'\\). Based on this assumption, we can design an improved protocol: \\(+\\) Knowledge of Exponent Assumption \\(\\mathcal{B}\\) randomly selects \\(s\\) and \\(r\\) from field \\(\\mathbb{F}\\). \\(\\mathcal{B}\\) computes and sends \\({\\alpha_1, \\alpha_2, ..., \\alpha_{n}}\\) and \\({\\alpha'_1, \\alpha'_2, ..., \\alpha'_{n}}\\), where \\(\\alpha_i = g^{(s^i)}\\) and \\(\\alpha' = \\alpha_{r} = g^{(s^{i})r}\\). \\(\\mathcal{A}\\) computes and sends \\(u = g^{p}\\), \\(v = g^{h}\\), and \\(w = g^{p'}\\), where \\(g^{p'} = g^{P(sr)}\\). \\(\\mathcal{B}\\) checks whether \\(u^{r} = w\\). \\(\\mathcal{B}\\) checks whether \\(u = v^{t}\\). The prover can compute \\(g^{p'} = g^{P(sr)} = \\alpha'^{c_1} (\\alpha'^{2})^{c_2} \\cdots (\\alpha'^{n})^{c_n}\\) using powers of \\(\\alpha'\\). This protocol now satisfies the properties of a SNARK: completeness, soundness, and efficiency.","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » \\(+\\) Knowledge of Exponent Assumption","id":"90","title":"\\(+\\) Knowledge of Exponent Assumption"},"91":{"body":"To transform the above protocol into a zk-SNARK, we need to ensure that the verifier cannot learn anything about \\(P(x)\\) from the prover's information. 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of Number Theory » Basics of Number Theory","id":"0","title":"Basics of Number Theory"},"1":{"body":"","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Computation Rule and Properties","id":"1","title":"Computation Rule and Properties"},"10":{"body":"For an element \\(a \\in H\\), an element \\(b \\in H\\) is an inverse of \\(a\\) if \\(a \\circ b = b \\circ a = e\\), where \\(e\\) is the identity element. Example : In modulo \\(n\\) arithmetic (addition), the inverse of an element exists if it can cancel itself out to yield the identity element. In the set of integers modulo 7, the inverse of 3 is 5, because \\(3 \\times 5 \\bmod 7 = 1\\), where 1 is the identity element for multiplication.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Inverse","id":"10","title":"Definition: Inverse"},"11":{"body":"A monoid in which every element has an inverse is called a group. Example : The set of integers modulo a prime \\(p\\) under multiplication forms a group (Can you prove it?). For instance, in \\(\\mathbb{Z}/5\\mathbb{Z}\\), every non-zero element \\(\\{1 + 5\\mathbb{Z}, 2 + 5\\mathbb{Z}, 3 + 5\\mathbb{Z}, 4 + 5\\mathbb{Z}\\}\\) has an inverse, making it a group.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Group","id":"11","title":"Definition: Group"},"12":{"body":"The order of a group is the number of elements in the group. Example : The group of integers modulo 4 under addition has order 4, because the set of elements is \\(\\{0, 1, 2, 3\\}\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Order of a Group","id":"12","title":"Definition: Order of a Group"},"13":{"body":"A triple \\((R, +, \\cdot)\\) is a ring if \\((R, +)\\) is an abelian group, \\((R, \\cdot)\\) is a semigroup, and the distributive property holds: \\(x \\cdot (y + z) = (x \\cdot y) + (x \\cdot z)\\) and \\((x + y) \\cdot z = (x \\cdot z) + (y \\cdot z)\\) for all \\(x, y, z \\in R\\). Example : The set of integers with usual addition and multiplication modulo \\(n\\) forms a ring. For example, in \\(\\mathbb{Z}/6\\mathbb{Z}\\), addition and multiplication modulo 6 form a ring. use std::fmt;\nuse std::ops::{Add, Mul, Neg, Sub}; pub trait Ring: Sized + Clone + PartialEq + fmt::Display + Add<Output = Self> + Sub<Output = Self> + Mul<Output = Self> + Mul<i64, Output = Self> + Neg<Output = Self>\n{ // A ring is an algebraic structure with addition and multiplication fn zero() -> Self; fn one() -> Self;\n}","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Ring","id":"13","title":"Definition: Ring"},"14":{"body":"A ring is called a commutative ring if its multiplication operation is commutative. Example The set of real numbers under usual addition and multiplication forms a commutative ring.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Communicative Ring","id":"14","title":"Definition: Communicative Ring"},"15":{"body":"A commutative ring with a multiplicative identity element where every non-zero element has a multiplicative inverse is called a field. Example The set of rational numbers under usual addition and multiplication forms a field. use crate::modules::ring::Ring; pub trait Field: Ring + Div<Output = Self> { fn inverse(&self) -> Self;\n}","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Field","id":"15","title":"Definition: Field"},"16":{"body":"The residue class of \\( a \\) modulo \\( m \\), denoted as \\( a + m\\mathbb{Z} \\), is the set \\( \\{b : b \\equiv a \\pmod{m}\\} \\). Example: For \\( m = 3 \\), the residue class of 2 is \\( 2 + 3\\mathbb{Z} = \\{\\ldots, -4, -1, 2, 5, 8, \\ldots\\} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Residue Class","id":"16","title":"Definition: Residue Class"},"17":{"body":"We denote the set of all residue classes modulo \\( m \\) as \\( \\mathbb{Z} / m\\mathbb{Z} \\). We say that \\( a + m\\mathbb{Z} \\) is invertible in \\( \\mathbb{Z} / m\\mathbb{Z} \\) if and only if there exists a solution for \\( ax \\equiv 1 \\pmod{m} \\). Example: In \\( \\mathbb{Z}/5\\mathbb{Z} \\), \\( 3 + 5\\mathbb{Z} \\) is invertible because \\( \\gcd(3, 5) = 1 \\) (since \\( 3\\cdot2 \\equiv 1 \\pmod{5} \\)). However, in \\( \\mathbb{Z}/6\\mathbb{Z} \\), \\( 3 + 6\\mathbb{Z} \\) is not invertible because \\( \\gcd(3, 6) = 3 \\neq 1 \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Inverse of Residue Class","id":"17","title":"Definition: Inverse of Residue Class"},"18":{"body":"If \\( a \\) and \\( b \\) are coprime, the residues of \\( a \\), \\( 2a \\), \\( 3a \\), ..., \\( (b-1)a \\) modulo \\( b \\) are all distinct. Proof: Suppose, for contradiction, that there exist \\( x, y \\in \\{1, 2, \\dots, b-1\\} \\) with \\( x \\neq y \\) such that \\( xa \\equiv ya \\pmod{b} \\). Then \\( (x-y)a \\equiv 0 \\pmod{b} \\), implying \\( b \\mid (x-y)a \\). Since \\( a \\) and \\( b \\) are coprime, we must have \\( b \\mid (x-y) \\). However, \\( |x-y| < b \\), so this is only possible if \\( x = y \\), contradicting our assumption. Therefore, all residues must be distinct.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Lemma 1.2.1","id":"18","title":"Lemma 1.2.1"},"19":{"body":"For any integers \\( a \\) and \\( b \\), the equation \\( ax + by = 1 \\) has a solution in integers \\( x \\) and \\( y \\) if and only if \\( a \\) and \\( b \\) are coprime. Proof: (\\( \\Rightarrow \\)) Suppose \\( a \\) and \\( b \\) are not coprime, let \\( d = \\gcd(a,b) > 1 \\). Then \\( d \\mid a \\) and \\( d \\mid b \\), so \\( d \\mid (ax + by) \\) for any integers \\( x \\) and \\( y \\). Thus, \\( ax + by \\neq 1 \\) for any \\( x \\) and \\( y \\). (\\( \\Leftarrow \\)) Suppose \\( a \\) and \\( b \\) are coprime. By the previous lemma, the residues of \\( a \\), \\( 2a \\), ..., \\( (b-1)a \\) modulo \\( b \\) are all distinct. Therefore, there exists an \\( m \\in \\{1, 2, ..., b-1\\} \\) such that \\( ma \\equiv 1 \\pmod{b} \\). This means there exists an integer \\( n \\) such that \\( ma = bn + 1 \\), or equivalently, \\( ma - bn = 1 \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.2","id":"19","title":"Theorem 1.2.2"},"2":{"body":"A mapping \\(\\circ: X \\times X \\rightarrow X\\) is a binary operation on \\(X\\) if for any pair of elements \\((x_1, x_2)\\) in \\(X\\), \\(x_1 \\circ x_2\\) is also in \\(X\\). Example : Addition (+) on the set of integers is a binary operation. For example, \\(5 + 3 = 8\\), and both \\(5, 3, 8\\) are integers, staying within the set of integers.","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Definition: Binary Operation","id":"2","title":"Definition: Binary Operation"},"20":{"body":"For any integers \\( a \\) and \\( b \\), we have: \\( a\\mathbb{Z} + b \\mathbb{Z} = \\gcd(a, b)\\mathbb{Z} \\) Proof: This statement is equivalent to proving that \\( ax + by = c \\) has an integer solution if and only if \\( c \\) is a multiple of \\( \\gcd(a,b) \\). (\\( \\Rightarrow \\)) If \\( ax + by = c \\) for some integers \\( x \\) and \\( y \\), then \\( \\gcd(a,b) \\mid a \\) and \\( \\gcd(a,b) \\mid b \\), so \\( \\gcd(a,b) \\mid (ax + by) = c \\). (\\( \\Leftarrow \\)) Let \\( c = k\\gcd(a,b) \\) for some integer \\( k \\). We can write \\( a = p\\gcd(a,b) \\) and \\( b = q\\gcd(a,b) \\), where \\( p \\) and \\( q \\) are coprime. By the previous theorem, there exist integers \\( m \\) and \\( n \\) such that \\( pm + qn = 1 \\). Multiplying both sides by \\( k\\gcd(a,b) \\), we get: [ akm + bkn = c ] Thus, \\( x = km \\) and \\( y = kn \\) are integer solutions to \\( ax + by = c \\). This theorem implies that for any integers \\( a \\), \\( b \\), and \\( n \\), the equation \\( ax + by = n \\) has an integer solution if and only if \\( \\gcd(a,b) \\mid n \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem: Bézout's Identity","id":"20","title":"Theorem: Bézout's Identity"},"21":{"body":"\\( a + m\\mathbb{Z} \\) is invertible in \\( \\mathbb{Z}/m\\mathbb{Z} \\) if and only if \\( \\gcd(a,m) = 1 \\). Proof: (\\( \\Rightarrow \\)) Suppose \\( a + m\\mathbb{Z} \\) is invertible in \\( \\mathbb{Z}/m\\mathbb{Z} \\). Then there exists an integer \\( x \\) such that \\( ax \\equiv 1 \\pmod{m} \\). Let \\( g = \\gcd(a,m) \\). Since \\( g \\mid ax \\) and \\( g \\mid (ax - 1) \\), we must have \\( g \\mid 1 \\), so \\( g = 1 \\). (\\( \\Leftarrow \\)) Suppose \\( \\gcd(a,m) = 1 \\). By Bézout's identity, there exist integers \\( x \\) and \\( y \\) such that \\( ax + my = 1 \\). Thus, \\( x + m\\mathbb{Z} \\) is the multiplicative inverse of \\( a + m\\mathbb{Z} \\) in \\( \\mathbb{Z}/m\\mathbb{Z} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.3","id":"21","title":"Theorem 1.2.3"},"22":{"body":"\\( (\\mathbb{Z} / m \\mathbb{Z}, +, \\cdot) \\) is a commutative ring where \\( 1 + m \\mathbb{Z} \\) is the multiplicative identity element. This ring is called the residue class ring modulo \\( m \\). Example: \\( \\mathbb{Z}/4\\mathbb{Z} = \\{0 + 4\\mathbb{Z}, 1 + 4\\mathbb{Z}, 2 + 4\\mathbb{Z}, 3 + 4\\mathbb{Z}\\} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Residue Class Ring","id":"22","title":"Definition: Residue Class Ring"},"23":{"body":"A residue class \\( a + m\\mathbb{Z} \\) is called primitive if \\( \\gcd(a, m) = 1 \\). Example: In \\( \\mathbb{Z}/6\\mathbb{Z} \\), the primitive residue classes are \\( 1 + 6\\mathbb{Z} \\) and \\( 5 + 6\\mathbb{Z} \\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Primitive Residue Class","id":"23","title":"Definition: Primitive Residue Class"},"24":{"body":"A residue ring \\( \\mathbb{Z} / m\\mathbb{Z} \\) is a field if and only if \\( m \\) is a prime number. Proof: TBD Example: For \\( m = 5 \\), \\( \\mathbb{Z}/5\\mathbb{Z} = \\{0 + 5\\mathbb{Z}, 1 + 5\\mathbb{Z}, 2 + 5\\mathbb{Z}, 3 + 5\\mathbb{Z}, 4 + 5\\mathbb{Z}\\} \\) forms a field because 5 is a prime number, and every non-zero element has a multiplicative inverse.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.4","id":"24","title":"Theorem 1.2.4"},"25":{"body":"The group of all primitive residue classes modulo \\( m \\) is called the primitive residue class group, denoted by \\( (\\mathbb{Z}/m\\mathbb{Z})^{\\times} \\). Example: For \\( m = 8 \\), the set of all primitive residue classes is \\( (\\mathbb{Z}/8\\mathbb{Z})^{\\times} = \\{1 + 8\\mathbb{Z}, 3 + 8\\mathbb{Z}, 5 + 8\\mathbb{Z}, 7 + 8\\mathbb{Z}\\} \\). These are the integers less than 8 that are coprime to 8 (i.e., \\(\\gcd(a, 8) = 1\\)). Contrast this with \\( m = 9 \\). The primitive residue class group is \\((\\mathbb{Z}/9\\mathbb{Z})^{\\times} = \\{1 + 9\\mathbb{Z}, 2 + 9\\mathbb{Z}, 4 + 9\\mathbb{Z}, 5 + 9\\mathbb{Z}, 7 + 9\\mathbb{Z}, 8 + 9\\mathbb{Z}\\}\\), as these are the integers less than 9 that are coprime to 9.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Primitive Residue Class Group","id":"25","title":"Definition: Primitive Residue Class Group"},"26":{"body":"Euler's totient function \\(\\phi(m)\\) is equal to the order of the primitive residue class group modulo \\(m\\), which is the number of integers less than \\(m\\) and coprime to \\(m\\). Example : For \\(m = 12\\), \\(\\phi(12) = 4\\) because there are 4 integers less than 12 that are coprime to 12: \\({1, 5, 7, 11}\\). For \\(m = 10\\), \\(\\phi(10) = 4\\), as there are also 4 integers less than 10 that are coprime to 10: \\({1, 3, 7, 9}\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Euler's Totient Function","id":"26","title":"Definition: Euler's Totient Function"},"27":{"body":"The order of \\(g \\in G\\) is the smallest natural number \\(e\\) such that \\(g^{e} = 1\\). We denote it as \\(\\text{order}_g(g)\\)or \\(\\text{order } g\\). Example : In \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), the element 3 has order 6 because \\(3^6 \\bmod 7 = 1\\). In other words, \\(3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\bmod 7 = 1\\), and 6 is the smallest such exponent.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Order of an Element within a Group","id":"27","title":"Definition: Order of an Element within a Group"},"28":{"body":"A subset \\(U \\subseteq G\\) is a subgroup of \\(G\\) if \\(U\\) itself is a group under the operation of \\(G\\). Example : Consider \\((\\mathbb{Z}/8\\mathbb{Z})^{\\times} = \\{1 + 8\\mathbb{Z}, 3+ 8\\mathbb{Z}, 5+ 8\\mathbb{Z}, 7+ 8\\mathbb{Z}\\}\\) under multiplication modulo 8. The subset \\({1+ 8\\mathbb{Z}, 7+ 8\\mathbb{Z}}\\) forms a subgroup because it satisfies the group properties: closed under multiplication, contains the identity element (1), and every element has an inverse (\\(7 \\times 7 \\equiv 1 \\bmod 8\\)).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Subgroup","id":"28","title":"Definition: Subgroup"},"29":{"body":"The set \\(\\{g^{k} : k \\in \\mathbb{Z}\\}\\), for some element \\(g \\in G\\), forms a subgroup of \\(G\\) and is called the subgroup generated by \\(g\\), denoted by \\(\\langle g \\rangle\\). Example : Consider the group \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times} = \\{1+ 7\\mathbb{Z}, 2+ 7\\mathbb{Z}, 3+ 7\\mathbb{Z}, 4+ 7\\mathbb{Z}, 5+ 7\\mathbb{Z}, 6+ 7\\mathbb{Z}\\}\\) under multiplication modulo 7. If we take \\(g = 3\\), then \\(\\langle 3 +7\\mathbb{Z} \\rangle =\\) \\(\\{3^1+7\\mathbb{Z}, 3^2+7\\mathbb{Z}, 3^3+7\\mathbb{Z}, 3^4+7\\mathbb{Z}, 3^5+7\\mathbb{Z}, 3^6+7\\mathbb{Z}\\} \\bmod 7 =\\) \\(\\{3+7\\mathbb{Z}, 2+7\\mathbb{Z}, 6+7\\mathbb{Z}, 4+7\\mathbb{Z}, 5+7\\mathbb{Z}, 1+7\\mathbb{Z}\\}\\), which forms a subgroup generated by 3. This subgroup contains all elements of \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), making 3 a generator of the entire group. If \\(g\\) has a finite order \\(e\\), we have that \\(\\langle g \\rangle = \\{g^{k}: 0 \\leq k \\leq e\\}\\), meaning \\(e\\) is the order of \\(\\langle g \\rangle\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Subgroup Generated by \\(g\\)","id":"29","title":"Definition: Subgroup Generated by \\(g\\)"},"3":{"body":"A binary operation \\(\\circ\\) is associative if \\((a \\circ b) \\circ c = a \\circ (b \\circ c)\\) for all \\(a, b, c \\in X\\). Example : Multiplication of real numbers is associative: \\((2 \\times 3) \\times 4 = 2 \\times (3 \\times 4) = 24\\). In a modular context, we also have addition modulo \\(n\\) being associative. For example, for \\(n = 5\\), \\((2 + 3) \\bmod 5 + 4 \\bmod 5 = 2 + (3 \\bmod 5 + 4) \\bmod 5 = 4\\).","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Definition: Associative Property","id":"3","title":"Definition: Associative Property"},"30":{"body":"A group \\(G\\) is called a cyclic group if there exists an element \\(g \\in G\\) such that \\(G = \\langle g \\rangle\\). In this case, \\(g\\) is called a generator of \\(G\\). Example : The group \\((\\mathbb{Z}/6\\mathbb{Z})^{\\times} = \\{1+6\\mathbb{Z}, 5+6\\mathbb{Z}\\}\\) under multiplication modulo 6 is a cyclic group. In this case, both 1 and 5 are generators of the group because \\(\\langle 5 +6\\mathbb{Z} \\rangle = \\{(5^1 \\bmod 6)+6\\mathbb{Z} = 5 +6\\mathbb{Z}, (5^2 \\bmod 6)+6\\mathbb{Z} = 1+6\\mathbb{Z}\\}\\). Since 5 generates all the elements of the group, \\(G\\) is cyclic.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Cyclic Group","id":"30","title":"Definition: Cyclic Group"},"31":{"body":"If \\(G\\) is a finite cyclic group, it has \\(\\phi(|G|)\\)generators, and each generator has order \\(|G|\\). Proof : TBD Example : Consider the group \\((\\mathbb{Z}/8\\mathbb{Z})^{\\times} = \\{1+8\\mathbb{Z}, 3+8\\mathbb{Z}, 5+8\\mathbb{Z}, 7+8\\mathbb{Z}\\}\\). This group is cyclic, and \\(\\phi(8) = 4\\). The generators of this group are \\(\\{1+8\\mathbb{Z}, 3+8\\mathbb{Z}, 5+8\\mathbb{Z}, 7+8\\mathbb{Z}\\}\\), each of which generates the entire group when raised to successive powers modulo 8. Each generator has the same order, which is \\(|G| = 4\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.5","id":"31","title":"Theorem 1.2.5"},"32":{"body":"If \\(G\\) is a finite cyclic group, the order of any subgroup of \\(G\\) divides the order of \\(G\\). Proof : TBD Example : Consider the cyclic group \\((\\mathbb{Z}/6\\mathbb{Z})^{\\times} = \\{1+6\\mathbb{Z}, 5+6\\mathbb{Z}\\}\\) under multiplication modulo 6. If we take the subgroup \\(\\langle 5+6\\mathbb{Z} \\rangle = \\{1+6\\mathbb{Z}, 5+6\\mathbb{Z}\\}\\), this is a subgroup of order 2, and 2 divides the order of the original group, which is 6. This theorem generalizes this property: for any subgroup of a cyclic group, its order divides the order of the group.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.6","id":"32","title":"Theorem 1.2.6"},"33":{"body":"If \\(\\gcd(a, m) = 1\\), then \\(a^{\\phi(m)} \\equiv 1 \\pmod{m}\\). Proof : TBD Example : Take \\(a = 2\\) and \\(m = 5\\). Since \\(\\gcd(2, 5) = 1\\), Fermat's Little Theorem tells us that \\(2^{\\phi(5)} = 2^4 \\equiv 1 \\bmod 5\\). Indeed, \\(2^4 = 16\\) and \\(16 \\bmod 5 = 1\\). This theorem suggests that \\(a^{\\phi(m) - 1} + m \\mathbb{Z}\\) is the inverse residue class of \\(a + m \\mathbb{Z}\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem: Fermat's Little Theorem","id":"33","title":"Theorem: Fermat's Little Theorem"},"34":{"body":"The order of any element in a group divides the order of the group. Proof : TBD Example : In the group \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), consider the element \\(3 + 7\\mathbb{Z}\\). The order of \\(3 + 7\\mathbb{Z}\\) is 6, as \\(3^6 \\equiv 1 \\bmod 7\\). The order of the group itself is also 6, and indeed, the order of the element divides the order of the group.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem 1.2.7","id":"34","title":"Theorem 1.2.7"},"35":{"body":"For any element \\(g \\in G\\), we have \\(g^{|G|} = 1\\). Proof : TBD Example : In the group \\((\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\), for any element \\(g\\), such as \\(g = 3 + 7\\mathbb{Z}\\), we have \\(3^6 \\equiv 1 \\bmod 7\\). This holds for any \\(g \\in (\\mathbb{Z}/7\\mathbb{Z})^{\\times}\\) because the order of the group is 6. Thus, \\(g^{|G|} = 1\\) is satisfied.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Theorem: Generalization of Fermat's Little Theorem","id":"35","title":"Theorem: Generalization of Fermat's Little Theorem"},"36":{"body":"Let \\(K\\) be a field.","breadcrumbs":"Basics of Number Theory » Polynomials » Polynomials","id":"36","title":"Polynomials"},"37":{"body":"A univariate polynomial over a commutative ring \\(R\\) with unity \\(1\\) is an expression of the form \\(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0\\), where \\(x\\) is a variable and coefficients \\(a_0, \\ldots, a_n\\) belong to \\(R\\). The set of all polynomials over \\(R\\) in the variable \\(x\\) is denoted as \\(R[x]\\). Example: In \\(\\mathbb{Z}[x]\\), we have polynomials such as \\(2x^3 + x + 1\\), \\(x\\), and \\(1\\). In \\(\\mathbb{R}[x]\\), we have polynomials like \\(\\pi x^2 - \\sqrt{2}x + e\\). /// Polynomial struct representing a polynomial over Field.\n#[derive(Debug, Clone, PartialEq)]\npub struct Polynomial<F: Field> { pub poly: Vec<F>, pub var: String,\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Definition: Polynomial","id":"37","title":"Definition: Polynomial"},"38":{"body":"The degree of a non-zero polynomial \\(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0\\), denoted as \\(\\deg f\\), is the largest integer \\(n\\) such that \\(a_n \\neq 0\\). The zero polynomial is defined to have degree \\(-1\\). Example \\(\\deg(2x^3 + x + 1) = 3\\) \\(\\deg(x) = 1\\) \\(\\deg(1) = 0\\) \\(\\deg(0) = -1\\) impl<F: Field> Polynomial<F> { /// Removes trailing zeroes from a polynomial's coefficients. fn trim_trailing_zeros(poly: Vec<F>) -> Vec<F> { let mut trimmed = poly; while trimmed.last() == Some(&F::zero(None)) { trimmed.pop(); } trimmed } /// Returns the degree of the polynomial. pub fn degree(&self) -> isize { let trimmed = Self::trim_trailing_zeros(self.poly.clone()); if trimmed.is_empty() { -1 } else { (trimmed.len() - 1) as isize } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Definition: Degree","id":"38","title":"Definition: Degree"},"39":{"body":"For polynomials \\(f(x) = \\sum_{i=0}^n a_i x^i\\) and \\(g(x) = \\sum_{i=0}^m b_i x^i\\), their sum is defined as: \\((f + g)(x) = \\sum_{i=0}^{\\max(n,m)} (a_i + b_i) x^i\\), where we set \\(a_i = 0\\) for \\(i > n\\) and \\(b_i = 0\\) for \\(i > m\\). Example: Let \\(f(x) = 2x^2 + 3x + 1\\) and \\(g(x) = x^3 - x + 4\\). Then, \\((f + g)(x) = x^3 + 2x^2 + 2x + 5\\) impl<F: Field> Add for Polynomial<F> { type Output = Self; fn add(self, other: Self) -> Polynomial<F> { let max_len = std::cmp::max(self.poly.len(), other.poly.len()); let mut result = Vec::with_capacity(max_len); let zero = F::zero(None); for i in 0..max_len { let a = self.poly.get(i).unwrap_or(&zero); let b = other.poly.get(i).unwrap_or(&zero); result.push(a.clone() + b.clone()); } Polynomial { poly: Self::trim_trailing_zeros(result), var: self.var.clone(), } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Sum of Polynomials","id":"39","title":"Sum of Polynomials"},"4":{"body":"A binary operation \\(\\circ\\) is commutative if \\(a \\circ b = b \\circ a\\) for all \\(a, b \\in X\\). Example : Addition modulo \\(n\\) is also commutative. For \\(n = 7\\), \\(5 + 3 \\bmod 7 = 3 + 5 \\bmod 7 = 1\\).","breadcrumbs":"Basics of Number Theory » Computation Rule and Properties » Definition: Commutative Property","id":"4","title":"Definition: Commutative Property"},"40":{"body":"For polynomials \\(f(x) = \\sum_{i=0}^n a_i x^i\\) and \\(g(x) = \\sum_{j=0}^m b_j x^j\\), their product is defined as: \\((fg)(x) = \\sum_{k=0}^{n+m} c_k x^k\\), where \\(c_k = \\sum_{i+j=k} a_i b_j\\) Example: Let \\(f(x) = x + 1\\) and \\(g(x) = x^2 - 1\\). Then, \\((fg)(x) = x^3 + x^2 - x - 1\\) impl<F: Field> Mul for Polynomial<F> { type Output = Self; /// Multiplication of two polynomials. fn mul(self, other: Polynomial<F>) -> Polynomial<F> { let mut result = vec![F::zero(None); self.degree() as usize + other.degree() as usize + 1]; for (i, a) in self.poly.iter().enumerate() { for (j, b) in other.poly.iter().enumerate() { result[i + j] = result[i + j].clone() + (a.clone() * b.clone()); } } Polynomial { poly: Self::trim_trailing_zeros(result), var: self.var.clone(), } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Product of polynomials","id":"40","title":"Product of polynomials"},"41":{"body":"Let \\(f, g \\in K[x]\\) be non-zero polynomials. Then, \\(\\deg(fg) = \\deg f + \\deg g\\). Example: Let \\(f(x) = x^2 + 1\\) and \\(g(x) = x^3 - x\\) in \\(\\mathbb{R}[x]\\). Then, \\(\\deg(fg) = \\deg(x^5 - x^3 + x^2 + 1) = 5 = 2 + 3 = \\deg f + \\deg g\\) We can also define division in the polynomial ring \\(K[x]\\).","breadcrumbs":"Basics of Number Theory » Polynomials » Lemma","id":"41","title":"Lemma"},"42":{"body":"Let \\(f, g \\in K[x]\\), with \\(g \\neq 0\\). There exist unique polynomials \\(q, r \\in K[x]\\) that satisfy \\(f = qg + r\\) and either \\(\\deg r < \\deg g\\) or \\(r = 0\\). Proof TBD \\(q\\) is called the quotient of \\(f\\) divided by \\(g\\), and \\(r\\) is called the remainder; we write \\(r = f \\bmod g\\). Example: In \\(\\mathbb{R}[x]\\), let \\(f(x) = x^3 + 2x^2 - x + 3\\) and \\(g(x) = x^2 + 1\\). Then \\(f = qg + r\\) where \\(q(x) = x + 2\\) and \\(r(x) = -3x + 1\\). impl<F: Field> Div for Polynomial<F> { type Output = Self; /// Division of two polynomials, returns quotient. fn div(self, other: Polynomial<F>) -> Polynomial<F> { let mut remainder_coeffs = Self::trim_trailing_zeros(self.poly.clone()); let divisor_coeffs = Self::trim_trailing_zeros(other.poly.clone()); let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse(); let mut quotient = vec![F::zero(None); self.degree() as usize - other.degree() as usize + 1]; let mut i = 0_i32; while remainder_coeffs.len() >= divisor_coeffs.len() { let lead_term = remainder_coeffs.last().unwrap().clone() * divisor_lead_inv.clone(); let deg_diff = remainder_coeffs.len() - divisor_coeffs.len(); quotient[deg_diff] = lead_term.clone(); for i in 0..divisor_coeffs.len() { remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i].clone() - (lead_term.clone() * divisor_coeffs[i].clone()); } remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs); i += 1; } Polynomial { poly: Self::trim_trailing_zeros(quotient), var: self.var.clone(), } }\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Theorem","id":"42","title":"Theorem"},"43":{"body":"Let \\(f \\in K[x]\\) be a non-zero polynomial, and \\(a \\in K\\) such that \\(f(a) = 0\\). Then, there exists a polynomial \\(q \\in K[x]\\) such that \\(f(x) = (x - a)q(x)\\). In other words, \\((x - a)\\) is a factor of \\(f(x)\\). Example: Let \\(f(x) = x^2 + 1 \\in (\\mathbb{Z}/2\\mathbb{Z})[x]\\). We have \\(f(1) = 1^2 + 1 = 0\\) in \\(\\mathbb{Z}/2\\mathbb{Z}\\), and indeed: \\(x^2 + 1 = (x - 1)^2 = x^2 - 2x + 1 = x^2 + 1\\) in \\((\\mathbb{Z}/2\\mathbb{Z})[x]\\)","breadcrumbs":"Basics of Number Theory » Polynomials » Corollary","id":"43","title":"Corollary"},"44":{"body":"A \\(n\\)-degre polynomial \\(P(x)\\) that goes through different \\(n + 1\\) points \\(\\{(x_1, y_1), (x_2, y_2), \\cdots (x_{n + 1}, y_{n + 1})\\}\\) is uniquely represented as follows: \\begin{align*} P(x) = \\sum^{n+1}_{i=1} y_i \\frac{f_i(x)}{f_i(x_i)} \\end{align*} , where \\(f_i(x) = \\prod_{k \\neq i} (x - x_k)\\) Proof TBD For example, the quadratic polynomial that goes through \\(\\{(1, 0), (2, 3), (3, 8)\\}\\) is as follows: \\begin{align*} P(x) = 0 \\frac{(x - 2)(x - 3)}{(1 - 2) (1 - 3)} + 3 \\frac{(x - 1)(x - 3)}{(2 - 1) (2 - 3)} + 8 \\frac{(x - 1)(x - 2)}{(3 - 1) (3 - 2)} = x^{2} - 1 \\end{align*} Note that Lagrange interpolation finds the lowest degree of interpolating polynomial for the given vector. pub fn interpolate(x_values: &[F], y_values: &[F]) -> Polynomial<F> { let mut lagrange_polys = vec![]; let numerators = Polynomial::from_monomials(x_values); for j in 0..x_values.len() { let mut denominator = F::one(None); for i in 0..x_values.len() { if i != j { denominator = denominator * (x_values[j].clone() - x_values[i].clone()); } } let cur_poly = numerators .clone() .div(Polynomial::from_monomials(&[x_values[j].clone()]).scalar_mul(&denominator)); lagrange_polys.push(cur_poly); } let mut result = Polynomial { poly: vec![], var: \"x\".to_string(), }; for (j, lagrange_poly) in lagrange_polys.iter().enumerate() { result = result + lagrange_poly.scalar_mul(&y_values[j]); } result\n}","breadcrumbs":"Basics of Number Theory » Polynomials » Theorem: Lagrange Interpolation","id":"44","title":"Theorem: Lagrange Interpolation"},"45":{"body":"Let \\(L(v)\\) and \\(L(w)\\) be the polynomial resulting from Lagrange Interpolation on the output (\\(y\\)) vector \\(v\\) and \\(w\\) for the same inputs (\\(x\\)). Then, the following properties hold: Additivity: \\(L(v + w) = L(v) + L(w)\\) for any vectors \\(v\\) and \\(w\\) Scalar multiplication: \\(L(\\gamma v) = \\gamma L(v)\\) for any scalar \\(\\gamma\\) and vector \\(v\\) Proof Let \\(v = (v_1, \\ldots, v_n)\\) and \\(w = (w_1, \\ldots, w_n)\\) be vectors, and \\(x_1, \\ldots, x_n\\) be the interpolation points. \\begin{align*} L(v + w) &= \\sum_{i=1}^n (v_i + w_i) \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = \\sum_{i=1}^n v_i \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} + \\sum_{i=1}^n w_i \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = L(v) + L(w) \\\\ L(\\gamma v) &= \\sum_{i=1}^n (\\gamma v_i) \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = \\gamma \\sum_{i=1}^n v_i \\prod_{j \\neq i} \\frac{x - x_j}{x_i - x_j} = \\gamma L(v) \\end{align*}","breadcrumbs":"Basics of Number Theory » Polynomials » Proposition: Homomorphisms of Lagrange Interpolation","id":"45","title":"Proposition: Homomorphisms of Lagrange Interpolation"},"46":{"body":"We will now discuss the construction of finite fields with \\(p^n\\) elements, where \\(p\\) is a prime number and \\(n\\) is a positive integer. These fields are also known as Galois fields, denoted as \\(GF(p^n)\\). It is evident that \\(\\mathbb{Z}/p\\mathbb{Z}\\) is isomorphic to \\(GF(p)\\).","breadcrumbs":"Basics of Number Theory » Galoid Field » Galois Field","id":"46","title":"Galois Field"},"47":{"body":"A polynomial \\(f \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\) of degree \\(n\\) is called irreducible over \\(\\mathbb{Z}/p\\mathbb{Z}\\) if it cannot be factored as a product of two polynomials of lower degree in \\((\\mathbb{Z}/p\\mathbb{Z})[X]\\). Example: In \\((\\mathbb{Z}/2\\mathbb{Z})[X]\\): \\(X^2 + X + 1\\) is irreducible \\(X^2 + 1 = (X + 1)^2\\) is reducible To construct \\(GF(p^n)\\), we use an irreducible polynomial of degree \\(n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\).","breadcrumbs":"Basics of Number Theory » Galoid Field » Definition: Irreducible Polynomial","id":"47","title":"Definition: Irreducible Polynomial"},"48":{"body":"For \\(f, g \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\), the residue class of \\(g \\bmod f\\) is the set of all polynomials \\(h \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\) such that \\(h \\equiv g \\pmod{f}\\). This class is denoted as: \\[ g + f(\\mathbb{Z}/p\\mathbb{Z})[X] = \\{g + hf : h \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\} \\] Example: In \\((\\mathbb{Z}/2\\mathbb{Z})[X]\\), with \\(f(X) = X^2 + X + 1\\), the residue classes \\(\\bmod f\\) are: \\(0 + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{0, X^2 + X + 1, X^2 + X, X^2 + 1, X^2, X + 1, X, 1\\}\\) \\(1 + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{1, X^2 + X, X^2 + 1, X^2, X + 1, X, 0, X^2 + X + 1\\}\\) \\(X + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{X, X^2 + 1, X^2, X^2 + X + 1, X + 1, 1, X^2 + X, 0\\}\\) \\((X + 1) + f(\\mathbb{Z}/2\\mathbb{Z})[X] = \\{X + 1, X^2, X^2 + X + 1, X^2 + X, 1, 0, X^2 + 1, X\\}\\) These four residue classes form \\(GF(4)\\).","breadcrumbs":"Basics of Number Theory » Galoid Field » Definition: Residue Class modulo a Polynomial","id":"48","title":"Definition: Residue Class modulo a Polynomial"},"49":{"body":"If \\(f \\in (\\mathbb{Z}/p\\mathbb{Z})[X]\\) is an irreducible polynomial of degree \\(n\\), then the residue ring \\((\\mathbb{Z}/p\\mathbb{Z})[X]/(f)\\) is a field with \\(p^n\\) elements, isomorphic to \\(GF(p^n)\\). Proof (Outline) The irreducibility of \\(f\\) ensures that \\((f)\\) is a maximal ideal in \\((\\mathbb{Z}/p\\mathbb{Z})[X]\\), making the quotient ring a field. The number of elements is \\(p^n\\) because there are \\(p^n\\) polynomials of degree less than \\(n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\). This construction allows us to represent elements of \\(GF(p^n)\\) as polynomials of degree less than \\(n\\) over \\(\\mathbb{Z}/p\\mathbb{Z}\\). Addition is performed coefficient-wise modulo \\(p\\), while multiplication is performed modulo the irreducible polynomial \\(f\\). Example: To construct \\(GF(8)\\), we can use the irreducible polynomial \\(f(X) = X^3 + X + 1\\) over \\(\\mathbb{Z}/2\\mathbb{Z}\\). The elements of \\(GF(8)\\) are represented by: [ \\{0, 1, X, X+1, X^2, X^2+1, X^2+X, X^2+X+1\\} ] For instance, multiplication in \\(GF(8)\\): [ (X^2 + 1)(X + 1) = X^3 + X^2 + X + 1 \\equiv X^2 \\pmod{X^3 + X + 1} ]","breadcrumbs":"Basics of Number Theory » Galoid Field » Theorem:","id":"49","title":"Theorem:"},"5":{"body":"","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Semigroup, Group, Ring","id":"5","title":"Semigroup, Group, Ring"},"50":{"body":"Let \\(\\mathbb{F}\\) be a field and \\(P: F^m \\rightarrow \\mathbb{F}\\) and \\(Q: \\mathbb{F}^m \\rightarrow \\mathbb{F}\\) be two distinct multivariate polynomials of total degree at most \\(n\\). For any finite subset \\(\\mathbb{S} \\subseteq \\mathbb{F}\\), we have: \\begin{align} Pr_{u \\sim \\mathbb{S}^{m}}[P(u) = Q(u)] \\leq \\frac{n}{|\\mathbb{S}|} \\end{align} , where \\(u\\) is drawn uniformly at random from \\(\\mathbb{S}^m\\). Proof TBD This lemma states that if \\(\\mathbb{S}\\) is sufficiently large and \\(n\\) is relatively small, the probability that the two different polynomials return the same value for a randomly chosen input is negligibly small. In other words, if we observe \\(P(u) = Q(u)\\) for a random input \\(u\\), we can conclude with high probability that \\(P\\) and \\(Q\\) are identical polynomials.","breadcrumbs":"Basics of Number Theory » Galoid Field » Lemma: Schwartz - Zippel Lemma","id":"50","title":"Lemma: Schwartz - Zippel Lemma"},"51":{"body":"","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Elliptic curve","id":"51","title":"Elliptic curve"},"52":{"body":"An elliptic curve \\(E\\) over a finite field \\(\\mathbb{F}_{p}\\) is defined by the equation: \\begin{equation} y^2 = x^3 + a x + b \\end{equation} , where \\(a, b \\in \\mathbb{F}_{p}\\) and the discriminant \\(\\Delta_{E} = 4a^3 + 27b^2 \\neq 0\\). use crate::modules::field::Field; #[derive(Debug, Clone, PartialEq)]\npub struct EllipticCurve<F: Field> { pub a: F, pub b: F,\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Elliptic Curve","id":"52","title":"Definition: Elliptic Curve"},"53":{"body":"An \\(\\mathbb{F}_{p}\\)-rational point on an elliptic curve \\(E\\) is a point \\((x, y)\\) where both \\(x\\) and \\(y\\) are elements of \\(\\mathbb{F}_{p}\\) and satisfy the curve equation, or the point at infinity \\(\\mathcal{O}\\). Example: For \\(E: y^2 = x^3 + 3x + 4\\) over \\(\\mathbb{F}_7\\), the \\(\\mathbb{F}_{7}\\)-rational points are: \\(\\{(0, 2), (0, 5), (1, 1), (1, 6), (2, 2), (2, 5), (5, 1), (5, 6), \\mathcal{O}\\}\\)","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: \\(\\mathbb{F}_{p}\\)-Rational Point","id":"53","title":"Definition: \\(\\mathbb{F}_{p}\\)-Rational Point"},"54":{"body":"The point at infinity denoted \\(\\mathcal{O}\\), is a special point on the elliptic curve that serves as the identity element for the group operation. It can be visualized as the point where all vertical lines on the curve meet. #[derive(Debug, Clone, PartialEq)]\npub struct EllipticCurvePoint<F: Field> { pub x: Option<F>, pub y: Option<F>, pub curve: EllipticCurve<F>,\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: The point at infinity","id":"54","title":"Definition: The point at infinity"},"55":{"body":"For an elliptic curve \\(E: y^2 = x^3 + ax + b\\), the addition of points \\(P\\) and \\(Q\\) to get \\(R = P + Q\\) is defined as follows: If \\(P = \\mathcal{O}\\), \\(R = Q\\). If \\(Q = \\mathcal{O}\\), \\(R = P\\). Otherwise, let \\(P = (x_P, y_P), Q = (x_Q, y_Q)\\). Then: If \\(y_P = -y_Q\\), \\(R = \\mathcal{O}\\) If \\(y_P \\neq -y_Q\\), \\(R = (x_R = \\lambda^2 - x_P - x_Q, y_R = \\lambda(x_P - x_R) - y_P)\\), where \\(\\lambda = \\) \\( \\begin{cases} \\frac{y_P - y_Q}{x_P - x_Q} \\quad &\\hbox{If } (x_P \\neq x_Q) \\\\ \\frac{3^{2}_{P} + a}{2y_P} \\quad &\\hbox{Otherwise} \\end{cases}\\) Example: On \\(E: y^2 = x^3 + 2x + 3\\) over \\(\\mathbb{F}_{7}\\), let \\(P = (5, 1)\\) and \\(Q = (4, 4)\\). Then, \\(P + Q = (0, 5)\\), where \\(\\lambda = \\frac{1 - 4}{5 - 4} \\equiv 4 \\bmod 7\\). impl<F: Field> EllipticCurvePoint<F> { fn new(x: F, y: F, curve: EllipticCurve<F>) -> Self { EllipticCurvePoint { x: Some(x), y: Some(y), curve: curve, } } pub fn point_at_infinity(curve: EllipticCurve<F>) -> Self { EllipticCurvePoint { x: None, y: None, curve: curve, } } pub fn is_point_at_infinity(&self) -> bool { self.x.is_none() || self.y.is_none() } pub fn line_slope(&self, other: Self) -> F { let x1 = self.x.clone().unwrap(); let y1 = self.y.clone().unwrap(); let x2 = other.x.clone().unwrap(); let y2 = other.y.clone().unwrap(); if self.x.clone() == other.x.clone() { ((x1.clone() * x1.clone()) * (3_i64) + self.curve.a.clone()) / (y1.clone() * (2_i64)) } else { (y2.clone() - y1.clone()) / (x2.clone() - x1.clone()) } }\n} impl<F: Field> Add for EllipticCurvePoint<F> { type Output = Self; fn add(self, other: Self) -> Self { if self.is_point_at_infinity() { return other; } if other.is_point_at_infinity() { return self; } let m = self.line_slope(other.clone()); if self.x == other.x { if self.y != other.y { return EllipticCurvePoint::point_at_infinity(self.curve.clone()); } else { let x1 = self.x.clone().unwrap(); let y1 = self.y.clone().unwrap(); let x3 = m.clone() * m.clone() - x1.clone() - x1.clone(); let y3 = m * (x1 - x3.clone()) - y1; return EllipticCurvePoint::new(x3, y3, self.curve.clone()); } } else { let x1 = self.x.clone().unwrap(); let y1 = self.y.clone().unwrap(); let x2 = other.x.clone().unwrap(); let x3 = m.clone() * m.clone() - x1.clone() - x2.clone(); let y3 = m * (x1 - x3.clone()) - y1; return EllipticCurvePoint::new(x3, y3, self.curve.clone()); } }\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Addition on Elliptic Curve","id":"55","title":"Definition: Addition on Elliptic Curve"},"56":{"body":"The Mordell-Weil group of an elliptic curve \\(E\\) is the group of rational points on \\(E\\) under the addition operation defined above. Example: For \\(E: y^2 = x^3 + x + 6\\) over \\(\\mathbb{F}_{11}\\), the Mordell-Weil group is the set of all \\(\\mathbb{F}_{11}\\)-rational points on \\(E\\) with the elliptic curve addition operation.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Mordell-Weil Group","id":"56","title":"Definition: Mordell-Weil Group"},"57":{"body":"The group order of an elliptic curve \\(E\\) over \\(\\mathbb{F}_{p}\\), denoted \\(\\#E(\\mathbb{F}_{p})\\), is the number of \\(\\mathbb{F}_{p}\\)-rational points on \\(E\\), including the point at infinity. Example: For \\(E: y^2 = x^3 + x + 6\\) over \\(\\mathbb{F}_{11}\\), \\(\\#E(\\mathbb{F}_{11}) = 13\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Group Order","id":"57","title":"Definition: Group Order"},"58":{"body":"Let \\(\\#E(\\mathbb{F}_{p})\\) be the group order of the elliptic curve \\(E\\) over \\(\\mathbb{F}_{p}\\). Then: \\begin{equation} p + 1 - 2 \\sqrt{p} \\leq \\#E \\leq p + 1 + 2 \\sqrt{p} \\end{equation} Example: For an elliptic curve over \\(\\mathbb{F}_{23}\\), the Hasse-Weil theorem guarantees that: \\begin{equation*} 23 + 1 - 2 \\sqrt{23} \\simeq 14.42 \\#E(\\mathbb{F}_{23}) \\geq 23 + 1 + 2 \\sqrt{23} \\simeq 33.58 \\end{equation*}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem: Hasse-Weil","id":"58","title":"Theorem: Hasse-Weil"},"59":{"body":"The order of a point \\(P\\) on an elliptic curve is the smallest positive integer \\(n\\) such that \\(nP = \\mathcal{O}\\) (where \\(nP\\) denotes \\(P\\) added to itself \\(n\\) times). We also denote the set of points of order \\(n\\), also called \\textit{torsion} group, by \\begin{equation} E[n] = \\{P \\in E: [n]P = \\mathcal{O}\\} \\end{equation} Example: On \\(E: y^2 = x^3 + 2x + 2\\) over \\(\\mathbb{F}_{17}\\), the point \\(P = (5, 1)\\) has order 18 because \\(18P = \\mathcal{O}\\), and no smaller positive multiple of \\(P\\) equals \\(\\mathcal{O}\\). The intuitive view is that if you continue to add points to themselves (doubling, tripling, etc.), the lines drawn between the prior point and the next point will eventually become more vertical. When the line becomes vertical, it does not intersect the elliptic curve at any finite point. In elliptic curve geometry, a vertical line is considered to \"intersect\" the curve at a special place called the \"point at infinity,\" This point is like a north pole in geographic terms: no matter which direction you go, if the line becomes vertical (reaching infinitely high), it converges to this point.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Point Order","id":"59","title":"Definition: Point Order"},"6":{"body":"A pair \\((H, \\circ)\\), where \\(H\\) is a non-empty set and \\(\\circ\\) is an associative binary operation on \\(H\\), is called a semigroup. Example : The set of positive integers under multiplication modulo \\(n\\) forms a semigroup. For instance, with \\(n = 6\\), the elements \\(\\{1, 2, 3, 4, 5\\}\\) under multiplication modulo 6 form a semigroup, since multiplication modulo 6 is associative.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Semigroup","id":"6","title":"Definition: Semigroup"},"60":{"body":"Let \\(F\\) and \\(L\\) be fields. If \\(F \\subseteq L\\) and the operations of \\(F\\) are the same as those of \\(L\\), we call \\(L\\) a field extension of \\(F\\). This is denoted as \\(L/F\\). A field extension \\(L/F\\) naturally gives \\(L\\) the structure of a vector space over \\(F\\). The dimension of this vector space is called the degree of the extension. Examples: \\(\\mathbb{C}/\\mathbb{R}\\) is a field extension with basis \\(\\{1, i\\}\\) and degree 2. \\(\\mathbb{R}/\\mathbb{Q}\\) is an infinite degree extension. \\(\\mathbb{Q}(\\sqrt{2})/\\mathbb{Q}\\) is a degree 2 extension with basis \\(\\{1, \\sqrt{2}\\}\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Field Extension","id":"60","title":"Definition: Field Extension"},"61":{"body":"A field extension \\(L/K\\) is called algebraic if every element \\(\\alpha \\in L\\) is algebraic over \\(K\\), i.e., \\(\\alpha\\) is the root of some non-zero polynomial with coefficients in \\(K\\). For an algebraic element \\(\\alpha \\in L\\) over \\(K\\), we denote by \\(K(\\alpha)\\) the smallest field containing both \\(K\\) and \\(\\alpha\\). Example: \\(\\mathbb{C}/\\mathbb{R}\\) is algebraic: any \\(z = a + bi \\in \\mathbb{C}\\) is a root of \\(x^2 - 2ax + (a^2 + b^2) \\in \\mathbb{R}[x]\\). \\(\\mathbb{Q}(\\sqrt[3]{2})/\\mathbb{Q}\\) is algebraic: \\(\\sqrt[3]{2}\\) is a root of \\(x^3 - 2 \\in \\mathbb{Q}[x]\\). \\(\\mathbb{R}/\\mathbb{Q}\\) is not algebraic (a field extension that is not algebraic is called \\textit{transcendental}).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Algebraic Extension","id":"61","title":"Definition: Algebraic Extension"},"62":{"body":"Let \\(K\\) be a field and \\(X\\) be indeterminate. The field of rational functions over \\(K\\), denoted \\(K(X)\\), is defined as: \\begin{equation*} K(X) = \\left\\{ \\frac{f(X)}{g(X)} \\middle|\\ f(X), g(X) \\in K[X], g(X) \\neq 0 \\right\\} \\end{equation*} where \\(K[X]\\) is the ring of polynomials in \\(X\\) with coefficients in \\(K\\). \\(K(X)\\) can be viewed as the field obtained by adjoining a letter \\(X\\) to \\(K\\). This construction generalizes to multiple variables, e.g., \\(K(X,Y)\\). The concept of a function field naturally arises in the context of algebraic curves, particularly elliptic curves. Intuitively, the function field encapsulates the algebraic structure of rational functions on the curve. We first construct the coordinate ring for an elliptic curve \\(E: y^2 = x^3 + ax + b\\). Consider functions \\(X: E \\to K\\) and \\(Y: E \\to K\\) that extract the \\(x\\) and \\(y\\) coordinates, respectively, from an arbitrary point \\(P \\in E\\). These functions generate the polynomial ring \\(K[X,Y]\\), subject to the relation \\(Y^2 = X^3 + aX + b\\). To put it simply, the function field is a field that consists of all functions that determine the value based on the point on the curve.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Field of Rational Functions","id":"62","title":"Definition: Field of Rational Functions"},"63":{"body":"The coordinate ring of an elliptic curve \\(E: y^2 = x^3 + ax + b\\) over a field \\(K\\) is defined as: \\begin{equation} K[E] = K[X, Y]/(Y^2 - X^3 - aX - b) \\end{equation} In other words, we can view \\(K[E]\\) as a ring representing all polynomial functions on \\(E\\). Recall that \\(K[X, Y]\\) is the polynomial ring in two variables \\(X\\) and \\(Y\\) over the field K, meaning that it contains all polynomials in \\(X\\) and \\(Y\\) with coefficients from \\(K\\). Then, the notation \\(K[X, Y]/(Y^2 - X^3 - aX - b)\\) denotes the quotient ring obtained by taking \\(K[X, Y]\\) and \"modding out\" by the ideal \\((Y^2 - X^3 - aX - b)\\). Example For example, for an elliptic curve \\(E: y^2 = x^3 - x\\) over \\(\\mathbb{Q}\\), some elements of the coordinate ring \\(\\mathbb{Q}[E]\\) include: Constants: \\(3, -2, \\frac{1}{7}, \\ldots\\) Linear functions: \\(X, Y, 2X+3Y, \\ldots\\) Quadratic functions: \\(X^2, XY, Y^2 (= X^3 - X), \\ldots\\) Higher-degree functions: \\(X^3, X^2Y, XY^2 (= X^4 - X^2), \\ldots\\) Then, the function field is defined as follows:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Coordinate Ring of an Elliptic Curve","id":"63","title":"Definition: Coordinate Ring of an Elliptic Curve"},"64":{"body":"Let \\(E: y^2 = x^3 + ax + b\\) be an elliptic curve over a field \\(K\\). The function field of \\(E\\), denoted \\(K(E)\\), is defined as: \\begin{equation} K(E) = \\left\\{\\frac{f}{g} ,\\middle|, f, g \\in K[E], g \\neq 0 \\right\\} \\end{equation} where \\(K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)\\) is the coordinate ring of \\(E\\). \\(K(E)\\) can be viewed as the field of all rational functions on \\(E\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Function Field of an Elliptic Curve","id":"64","title":"Definition: Function Field of an Elliptic Curve"},"65":{"body":"Let \\(h \\in K(E)\\) be a non-zero function. A point \\(P \\in E\\) is called a zero of \\(h\\) if \\(h(P) = 0\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Zero of a Function","id":"65","title":"Definition: Zero of a Function"},"66":{"body":"Let \\(h \\in K(E)\\) be a non-zero function. A point \\(P \\in E\\) is called a pole of \\(h\\) if \\(h\\) is not defined at \\(P\\) or, equivalently if \\(1/h\\) has a zero at \\(P\\). Example Consider the elliptic curve \\(E: Y^2 = X^3 - X\\) over a field \\(K\\) of characteristic \\(\\neq 2, 3\\). Let \\(P_{-1} = (-1, 0)\\), \\(P_0 = (0, 0)\\), and \\(P_1 = (1, 0)\\) be the points where \\(Y = 0\\). The function \\(Y \\in K(E)\\) has three simple zeros: \\(P_{-1}\\), \\(P_0\\), and \\(P_1\\). The function \\(X \\in K(E)\\) has a double zero at \\(P_0\\) (since \\(P_0 = -P_0\\)). The function \\(X - 1 \\in K(E)\\) has a simple zero at \\(P_1\\). The function \\(X^2 - 1 \\in K(E)\\) has two simple zeros at \\(P_{-1}\\) and \\(P_1\\). The function \\(\\frac{Y}{X} \\in K(E)\\) has a simple zero at \\(P_{-1}\\) and a simple pole at \\(P_0\\). An important property of functions in \\(K(E)\\) is that they have the same number of zeros and poles when counted with multiplicity. This is a consequence of a more general result known as the Degree-Genus Formula.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Pole of a Function","id":"66","title":"Definition: Pole of a Function"},"67":{"body":"Let \\(f \\in K(E)\\) be a non-zero rational function on an elliptic curve \\(E\\). Then: \\begin{equation} \\sum_{P \\in E} ord_{P(f)} = 0 \\end{equation} , where \\(ord_{P(f)}\\) denotes the order of \\(f\\) at \\(P\\), which is positive for zeros and negative for poles. This theorem implies that the total number of zeros (counting multiplicity) equals the total number of poles for any non-zero function in \\(K(E)\\). We now introduce a powerful tool for analyzing functions on elliptic curves: the concept of divisors.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem: Degree-Genus Formula for Elliptic Curves","id":"67","title":"Theorem: Degree-Genus Formula for Elliptic Curves"},"68":{"body":"Let \\(E: Y^2 = X^3 + AX + B\\) be an elliptic curve over a field \\(K\\), and let \\(f \\in K(E)\\) be a non-zero rational function on \\(E\\). The divisor of \\(f\\), denoted \\(div(f)\\), is defined as: \\begin{equation} div(f) = \\sum_{P \\in E} ord_P(f) [P] \\end{equation} , where \\(ord_P(f)\\) is the order of \\(f\\) at \\(P\\) (positive for zeros, negative for poles), and the sum is taken over all points \\(P \\in E\\), including the point at infinity. This sum has only finitely many non-zero terms. Note that this \\(\\sum_{P \\in E}\\) is a symbolic summation, and we do not calculate the concrete value of a divisor. Example Consider the elliptic curve \\(E: Y^2 = X^3 - X\\) over \\(\\mathbb{Q}\\). For \\(f = X\\), we have \\(div(X) = 2[(0,0)] - 2[\\mathcal{O}]\\). For \\(g = Y\\), we have \\(div(Y) = [(1,0)] + [(0,0)] + [(-1,0)] - 3[\\mathcal{O}]\\). For \\(h = \\frac{X-1}{Y}\\), we have \\(div(h) = [(1,0)] - [(-1,0)]\\). The concept of divisors can be extended to the elliptic curve itself:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Divisor of a Function on an Elliptic Curve","id":"68","title":"Definition: Divisor of a Function on an Elliptic Curve"},"69":{"body":"A divisor \\(D\\) on an elliptic curve \\(E\\) is a formal sum \\begin{equation} D = \\sum_{P \\in E} n_P [P] \\end{equation} where \\(n_P \\in \\mathbb{Z}\\) and \\(n_P = 0\\) for all but finitely many \\(P\\). Example On the curve \\(E: Y^2 = X^3 - X\\): \\(D_1 = 3[(0,0)] - 2[(1,1)] - [(2,\\sqrt{6})]\\) is a divisor. \\(D_2 = [(1,0)] + [(-1,0)] - 2[\\mathcal{O}]\\) is a divisor. \\(D_3 = \\sum_{P \\in E[2]} [P] - 4[\\mathcal{O}]\\) is a divisor (where \\(E[2]\\) are the 2-torsion points). To quantify the properties of divisors, we introduce two important metrics:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Divisor on an Elliptic Curve","id":"69","title":"Definition: Divisor on an Elliptic Curve"},"7":{"body":"A semigroup whose operation is commutative is called an abelian semigroup. Example : The set of natural numbers under addition modulo \\(n\\) forms an abelian semigroup. For \\(n = 7\\), addition modulo 7 is both associative and commutative, so it is an abelian semigroup.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Abelian Semigroup","id":"7","title":"Definition: Abelian Semigroup"},"70":{"body":"The degree of a divisor \\(D = \\sum_{P \\in E} n_P [P]\\) is defined as: \\begin{equation} deg(D) = \\sum_{P \\in E} n_P \\end{equation}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Degree of a Divisor","id":"70","title":"Definition: Degree of a Divisor"},"71":{"body":"The sum of a divisor \\(D = \\sum_{P \\in E} n_P [P]\\) is defined as: \\begin{equation} Sum(D) = \\sum_{P \\in E} n_P P \\end{equation} where \\(n_P P\\) denotes the point addition of \\(P\\) to itself \\(n_P\\) times in the group law of \\(E\\). Example For the divisors in the previous example: \\(deg(D_1) = 3 - 2 - 1 = 0\\) \\(deg(D_2) = 1 + 1 - 2 = 0\\) \\(deg(D_3) = 4 - 4 = 0\\) \\(Sum(D_2) = (1,0) + (-1,0) - 2\\mathcal{O} = \\mathcal{O}\\) (since \\((1,0)\\) and \\((-1,0)\\) are 2-torsion points) The following theorem characterizes divisors of functions and provides a criterion for when a divisor is the divisor of a function:","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Sum of a Divisor","id":"71","title":"Definition: Sum of a Divisor"},"72":{"body":"Let \\(E\\) be an elliptic curve over a field \\(K\\). If \\(f, f' \\in K(E)\\) are non-zero rational functions on \\(E\\) with \\(div(f) = div(f')\\), then there exists a non-zero constant \\(c \\in K^ \\) such that \\(f = cf'\\).* A divisor \\(D\\) on \\(E\\) is the divisor of a rational function on \\(E\\) if and only if \\(deg(D) = 0\\) and \\(Sum(D) = \\mathcal{O}\\). Example On \\(E: Y^2 = X^3 - X\\): The function \\(f = \\frac{Y}{X}\\) has \\(div(f) = [(1,0)] + [(-1,0)] - 2[(0,0)]\\). Note that \\(deg(div(f)) = 0\\) and \\(Sum(div(f)) = (1,0) + (-1,0) - 2(0,0) = \\mathcal{O}\\). The divisor \\(D = 2[(1,1)] - [(2,\\sqrt{6})] - [(0,0)]\\) has \\(deg(D) = 0\\), but \\(Sum(D) \\neq \\mathcal{O}\\). Therefore, \\(D\\) is not the divisor of any rational function on \\(E\\).","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem","id":"72","title":"Theorem"},"73":{"body":"Let \\(G_1\\) and \\(G_2\\) be cyclic groups under addition, both of prime order \\(p\\), with generators \\(P\\) and \\(Q\\) respectively: \\begin{align} G_1 &= \\{0, P, 2P, ..., (p-1)P\\} \\ G_2 &= \\{0, Q, 2Q, ..., (p-1)Q\\} \\end{align} Let \\(G_T\\) be a cyclic group under multiplication, also of order \\(p\\). A pairing is a map \\(e: G_1 \\times G_2 \\rightarrow G_T\\) that satisfies the following bilinear property: \\begin{equation} e(aP, bQ) = e(P, Q)^{ab} \\end{equation} for all \\(a, b \\in \\mathbb{Z}_p\\). Imagine \\(G_1\\) represents length, \\(G_2\\) represents width, and \\(G_T\\) represents area. The pairing function \\(e\\) is like calculating the area: If you double the length and triple the width, the area becomes six times larger: \\(e(2P, 3Q) = e(P, Q)^{6}\\) The Weil pairing is one of the bilinear pairings for elliptic curves. We begin with its formal definition.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: Pairing","id":"73","title":"Definition: Pairing"},"74":{"body":"Let \\(E\\) be an elliptic curve and \\(n\\) be a positive integer. For points \\(P, Q \\in E[n]\\), where \\(E[n]\\) denotes the \\(n\\)-torsion subgroup of \\(E\\), we define the Weil pairing \\(e_n(P, Q)\\) as follows: Let \\(f_P\\) and \\(f_Q\\) be rational functions on \\(E\\) satisfying: \\begin{align} div(f_P) &= n[P] - n[\\mathcal{O}] \\ div(f_Q) &= n[Q] - n[\\mathcal{O}] \\end{align} Then, for an arbitrary point \\(S \\in E\\) such that \\(S \\notin \\{\\mathcal{O}, P, -Q, P-Q\\}\\), the Weil pairing is given by: \\begin{equation} e_n(P, Q) = \\frac{f_P(Q + S)}{f_P(S)} /\\ \\frac{f_Q(P - S)}{f_Q(-S)} \\end{equation} We introduce a crucial theorem about a specific rational function on elliptic curves to construct the functions required for the Weil pairing.","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Definition: The Weil Pairing","id":"74","title":"Definition: The Weil Pairing"},"75":{"body":"Let \\(E\\) be an elliptic curve over a field \\(K\\), and let \\(P = (x_P, y_P)\\) and \\(Q = (x_Q, y_Q)\\) be non-zero points on \\(E\\). Define \\(\\lambda\\) as: \\begin{equation} \\lambda = \\begin{cases} \\hbox{slope of the line through \\(P\\) and \\(Q\\)} &\\quad \\hbox{if \\(P \\neq Q\\)} \\\\ \\hbox{slope of the tangent line to \\(E\\) at \\(P\\)} &\\quad \\hbox{if \\(P = Q\\)} \\\\ \\infty &\\quad \\hbox{if the line is vertical} \\end{cases} \\end{equation} Then, the function \\(g_{P,Q}: E \\to K\\) defined by: \\begin{equation} g_{P,Q} = \\begin{cases} \\frac{y - y_P - \\lambda(x - x_P)}{x + x_P + x_Q - \\lambda^2} &\\quad \\hbox{if } \\lambda \\neq \\infty \\\\ x - x_P &\\quad \\hbox{if } \\lambda = \\infty \\end{cases} \\end{equation} has the following divisor: \\begin{equation} div(g_{P,Q}) = [P] + [Q] - [P + Q] - [\\mathcal{O}] \\end{equation} Proof: We consider two cases based on the value of \\(\\lambda\\). Case 1: \\(\\lambda \\neq \\infty\\) Let \\(y = \\lambda x + v\\) be the line through \\(P\\) and \\(Q\\) (or the tangent line at \\(P\\) if \\(P = Q\\)). This line intersects \\(E\\) at three points: \\(P\\), \\(Q\\), and \\(-P-Q\\). Thus, \\begin{equation} div(y - \\lambda x - v) = [P] + [Q] + [-P - Q] - 3[\\mathcal{O}] \\end{equation} Vertical lines intersect \\(E\\) at points and their negatives, so: \\begin{equation} div(x - x_{P+Q}) = [P + Q] + [-P - Q] - 2[\\mathcal{O}] \\end{equation} It follows that \\(g_{P,Q} = \\frac{y - \\lambda x - v}{x - x_{P+Q}}\\) has the desired divisor. Case 2: \\(\\lambda = \\infty\\) In this case, \\(P + Q = \\mathcal{O}\\), so we want \\(g_{P,Q}\\) to have divisor \\([P] + [-P] - 2[\\mathcal{O}]\\). The function \\(x - x_P\\) has this divisor. \\end{proof}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem","id":"75","title":"Theorem"},"76":{"body":"Let \\(m \\geq 1\\) and write its binary expansion as: \\begin{equation} m = m_0 + m_1 \\cdot 2 + m_2 \\cdot 2^2 + \\cdots + m_{n-1} \\cdot 2^{n-1} \\end{equation} where \\(m_i \\in \\{0, 1\\}\\) and \\(m_{n-1} \\neq 0\\). The following algorithm, using the function \\(g_{P,Q}\\) defined in the previous theorem, returns a function \\(f_P\\) whose divisor satisfies: \\begin{equation} div(f_P) = m[P] - m[P] - (m - 1)[\\mathcal{O}] \\end{equation} =========================== Miller's Algorithm Set \\(T = P\\) and \\(f = 1\\) For \\(i \\gets n-2 \\cdots 0\\) do Set \\(f = f^2 \\cdot g_{T, T}\\) Set \\(T = 2T\\) If \\(m_i = 1\\) then Set \\(f = f \\cdot g_{T, P}\\) Set \\(T = T + P\\) End if End for Return \\(f\\) =========================== Proof: TBD pub fn get_lambda<F: Field>( p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, r: EllipticCurvePoint<F>,\n) -> F { let p_x = p.x.clone().unwrap(); let p_y = p.y.clone().unwrap(); let q_x = q.x.clone().unwrap(); // let q_y = q.y.clone().unwrap(); let r_x = r.x.clone().unwrap(); let r_y = r.y.clone().unwrap(); if (p == q && p_y.clone() == F::zero()) || (p != q && p_x.clone() == q_x.clone()) { return r_x.clone() - p_x.clone(); } let slope = p.line_slope(q.clone()); let numerator = r_y.clone() - p_y.clone() - slope.clone() * (r_x.clone() - p_x.clone()); let denominator = r_x.clone() + p_x.clone() + q_x.clone() - slope.clone() * slope.clone(); return numerator / denominator;\n} pub fn miller<F: Field>(p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, m: BigInt) -> F { if p == q { F::one(); } let mut f = F::one(); let mut t = p.clone(); for i in (1..m.bits()).rev() { f = (f.clone() * f.clone()) * (get_lambda(t.clone(), t.clone(), q.clone())); t = t.clone() + t.clone(); if m.bit(i) { f = f * (get_lambda(t.clone(), p.clone(), q.clone())); t = t.clone() + p.clone(); } } f\n} pub fn weil_pairing<F: Field>( p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, m: BigInt, s: Option<EllipticCurvePoint<F>>,\n) -> F { let s_value = s.unwrap(); let fp_qs = miller(p.clone(), q.clone() + s_value.clone(), m.clone()); let fp_s = miller(p.clone(), s_value.clone(), m.clone()); let fq_ps = miller(q.clone(), p.clone() - s_value.clone(), m.clone()); let fq_s = miller(q.clone(), -s_value.clone(), m.clone()); return (fp_qs / fp_s) / (fq_ps / fq_s);\n}","breadcrumbs":"Basics of Number Theory » Elliptic Curve » Theorem: Miller's Algorithm","id":"76","title":"Theorem: Miller's Algorithm"},"77":{"body":"","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumptions","id":"77","title":"Assumptions"},"78":{"body":"Let \\(G\\) be a finite cyclic group of order \\(n\\), with \\(\\gamma\\) as its generator and \\(1\\) as the identity element. For any element \\(\\alpha \\in G\\), there is currently no known efficient (polynomial-time) algorithm to compute the smallest non-negative integer \\(x\\) such that \\(\\alpha = \\gamma^{x}\\). The Discrete Logarithm Problem can be thought of as a one-way function. It's easy to compute \\(g^{x}\\) given \\(g\\) and \\(x\\), but it's computationally difficult to find \\(x\\) given \\(g\\) and \\(g^{x}\\).","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumption: Discrete Logarithm Problem","id":"78","title":"Assumption: Discrete Logarithm Problem"},"79":{"body":"Let \\(E\\) be an elliptic curve defined over a finite field \\(\\mathbb{F}_q\\), where \\(q\\) is a prime power. Let \\(P\\) be a point on \\(E\\) of point order \\(n\\), and let \\(\\langle P \\rangle\\) be the cyclic subgroup of \\(E\\) generated by \\(P\\). For any element \\(Q \\in \\langle P \\rangle\\), there is currently no known efficient (polynomial-time) algorithm to compute the unique integer \\(k\\), \\(0 \\leq k < n\\), such that \\(Q = kP\\). This assumption is an elliptic curve version of the Discrete Logarithm Problem.","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumption: Elliptic Curve Discrete Logarithm Problem","id":"79","title":"Assumption: Elliptic Curve Discrete Logarithm Problem"},"8":{"body":"An element \\(e \\in H\\) is an identity element of \\(H\\) if it satisfies \\(e \\circ a = a \\circ e = a\\) for any \\(a \\in H\\). Example : 0 is the identity element for addition modulo \\(n\\). For example, \\(0 + a \\bmod 5 = a + 0 \\bmod 5 = a\\). Similarly, 1 is the identity element for multiplication modulo \\(n\\). For example, \\(1 \\times a \\bmod 7 = a \\times 1 \\bmod 7 = a\\).","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Identity Element","id":"8","title":"Definition: Identity Element"},"80":{"body":"Let \\(G\\) be a cyclic group of prime order \\(q\\) with generator \\(g \\in G\\). For any probabilistic polynomial-time algorithm \\(\\mathcal{A}\\) that outputs: \\begin{equation} \\mathcal{A}(g, g^x) = (h, h') \\quad s.t. \\quad h' = h^x \\end{equation} , there exists an efficient extractor \\(\\mathcal{E}\\) such that: \\begin{equation} \\mathcal{E}(\\mathcal{A}, g, g^x) = y \\quad s.t. \\quad h = g^y \\end{equation} This assumption states that if \\(\\mathcal{A}\\) can compute the pair \\((g^y, g^{xy})\\) from \\((g, g^x)\\), then \\(\\mathcal{A}\\) must \"know\" the value \\(y\\), in the sense that \\(\\mathcal{E}\\) can extract \\(y\\) from \\(\\mathcal{A}\\)'s internal state. The Knowledge of Exponent Assumption is useful for constructing verifiable exponential calculation algorithms. Consider a scenario where Alice has a secret value \\(a\\), and Bob has a secret value \\(b\\). Bob wants to obtain \\(g^{ab}\\). This can be achieved through the following protocol: Verifiable Exponential Calculation Algorithm Bob sends \\((g, g'=g^{b})\\) to Alice Alice sends \\((h=g^{a}, h'=g'^{a})\\) to Bob Bob checks \\(h^{b} = h'\\). Thanks to the Discrete Logarithm Assumption and the Knowledge of Exponent Assumption, the following properties hold: Bob cannot derive \\(a\\) from \\((h, h')\\). Alice cannot derive \\(b\\) from \\((g, g')\\). Alice cannot generate \\((t, t')\\) such that \\(t \\neq h\\) and \\(t^{b} = t'\\). If \\(h^{b} = h'\\), Bob can conclude that \\(h\\) is the power of \\(g\\).","breadcrumbs":"Basics of Number Theory » Useful Assumptions » Assumption: Knowledge of Exponent Assumption","id":"80","title":"Assumption: Knowledge of Exponent Assumption"},"81":{"body":"","breadcrumbs":"Basics of zk-SNARKs » Basics of zk-SNARK","id":"81","title":"Basics of zk-SNARK"},"82":{"body":"The ultimate goal of Zero-Knowledge Proofs (ZKP) is to allow the prover to demonstrate their knowledge to the verifier without revealing any additional information. This knowledge typically involves solving complex problems, such as finding a secret input value that corresponds to a given public hash. ZKP protocols usually convert these statements into polynomial constraints. This process is often called \\textit{arithmetization}. To make the protocol flexible, we need to encode this knowledge in a specific format, and one common approach is using Boolean circuits. It's well-known that problems in P (those solvable in polynomial time) and NP (those where the solution can be verified in polynomial time) can be represented as Boolean circuits. This means adopting Boolean circuits allows us to handle both P and NP problems. However, Boolean circuits are often large and inefficient. Even a simple operation, like adding two 256-bit integers, can require hundreds of Boolean operators. In contrast, arithmetic circuits—essentially systems of equations involving addition, multiplication, and equality—offer a much more compact representation. Additionally, any Boolean circuit can be converted into an arithmetic circuit. For instance, the Boolean expression \\(z = x \\land y\\) can be represented as \\(x(x-1) = 0\\), \\(y(y-1) = 0\\), and \\(z = xy\\) in an arithmetic circuit. Furthermore, as we'll see in this section, converting arithmetic circuits into polynomial constraints allows for much faster evaluation.","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » Arithmetization","id":"82","title":"Arithmetization"},"83":{"body":"There are many formats to represent arithmetic circuits, and one of the most popular ones is R1CS (Rank-1 Constraint System), which represents arithmetic circuits as \\underline{a set of equality constraints, each involving only one multiplication}. In an arithmetic circuit, we call the concrete values assigned to the variables within the constraints witness. We first provide the formal definition of R1CS as follows:","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » Rank-1 Constraint System (R1CS)","id":"83","title":"Rank-1 Constraint System (R1CS)"},"84":{"body":"An R1CS structure \\(\\mathcal{S}\\) consists of: Size bounds \\(m, d, \\ell \\in \\mathbb{N}\\) where \\(d > \\ell\\) Three matrices \\(O, L, R \\in \\mathbb{F}^{m\\ times d}\\) with at most \\(\\Omega(\\max(m, d))\\) non-zero entries in total An R1CS instance includes a public input \\(p \\in \\mathbb{F}^\\ell\\), while an R1CS witness is a vector \\(w \\in \\mathbb{F}^{d - \\ell - 1}\\). A structure-instance tuple \\((S, p)\\) is satisfied by a witness \\(w\\) if: \\begin{equation} (L \\cdot a) \\circ (R \\cdot a) - O \\cdot a = \\mathbf{0} \\end{equation} where \\(a = (1, w, p) \\in \\mathbb{F}^d\\), \\(\\cdot\\) denotes matrix-vector multiplication, and \\(\\circ\\) is the Hadamard product. The intuitive interpretation of each matrix is as follows: \\(L\\): Encodes the left input of each gate \\(R\\): Encodes the right input of each gate \\(O\\): Encodes the output of each gate The leading 1 in the witness vector allows for constant terms Single Multiplication Let's consider a simple example where we want to prove \\(z = x \\cdot y\\), with \\(z = 3690\\), \\(x = 82\\), and \\(y = 45\\). \\textbf{Witness vector}: \\((1, z, x, y) = (1, 3690, 82, 45)\\) \\textbf{Number of witnesses}: \\(m = 4\\) \\textbf{Number of constraints}: \\(d = 1\\) The R1CS constraint for \\(z = x \\cdot y\\) is satisfied when: \\begin{align*} &(\\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\cdot a) \\circ (\\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix} \\cdot a) - \\begin{bmatrix} 0 & 1 & 0 & 0 \\end{bmatrix} \\cdot a \\\\ =&(\\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\cdot \\begin{bmatrix} 1 \\\\ 3690 \\\\ 82 \\\\ 45 \\end{bmatrix}) \\circ (\\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix} \\cdot \\begin{bmatrix} 1 \\\\ 3690 \\\\ 82 \\\\ 45 \\end{bmatrix}) - \\begin{bmatrix} 0 & 1 & 0 & 0 \\end{bmatrix} \\cdot \\begin{bmatrix} 1 \\\\ 3690 \\\\ 82 \\\\ 45 \\end{bmatrix} \\\\ =& 82 \\cdot 45 - 3690 \\\\ =& 3690 - 3690 \\\\ =& 0 \\end{align*} This example demonstrates how R1CS encodes a simple multiplication constraint: \\(L = \\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix}\\) selects \\(x\\) (left input) \\(R = \\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix}\\) selects \\(y\\) (right input) \\(O = \\begin{bmatrix} 0 & 1 & 0 & 0 \\end{bmatrix}\\) selects \\(z\\) (output) Multiple Constraints Let's examine a more complex example: \\(r = a \\cdot b \\cdot c \\cdot d\\). Since R1CS requires that each constraint contain only one multiplication, we need to break this down into multiple constraints: \\begin{align*} v_1 &= a \\cdot b \\\\ v_2 &= c \\cdot d \\\\ r &= v_1 \\cdot v_2 \\end{align*} Note that alternative representations are possible, such as \\(v_1 = ab, v_2 = v_1c, r = v_2d\\). In this example, we use 7 variables \\((r, a, b, c, d, v_1, v_2)\\), so the dimension of the witness vector will be \\(m = 8\\) (including the constant 1). We have three constraints, so \\(n = 3\\). To construct the matrices \\(L\\), \\(R\\), and \\(O\\), we can interpret the constraints as linear combinations: \\begin{align*} v_1 &= (0 \\cdot 1 + 0 \\cdot r + 1 \\cdot a + 0 \\cdot b + 0 \\cdot c + 0 \\cdot d + 0 \\cdot v_1 + 0 \\cdot v_2) \\cdot b \\\\ v_2 &= (0 \\cdot 1 + 0 \\cdot r + 0 \\cdot a + 0 \\cdot b + 1 \\cdot c + 0 \\cdot d + 0 \\cdot v_1 + 0 \\cdot v_2) \\cdot d \\\\ r &= (0 \\cdot 1 + 0 \\cdot r + 0 \\cdot a + 0 \\cdot b + 0 \\cdot c + 0 \\cdot d + 1 \\cdot v_1 + 0 \\cdot v_2) \\cdot v_2 \\end{align*} Thus, we can construct \\(L\\), \\(R\\), and \\(O\\) as follows: \\begin{equation*} L = \\begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\end{bmatrix} \\end{equation*} \\begin{equation*} R = \\begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\end{bmatrix} \\end{equation*} \\begin{equation*} O = \\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\end{bmatrix} \\end{equation*} Where the columns in each matrix correspond to \\((1, r, a, b, c, d, v_1, v_2)\\). Addition with a Constant Let's examine the case \\(z = x \\cdot y + 3\\). We can represent this as \\(-3 + z = x \\cdot y\\). For the witness vector \\((1, z, x, y)\\), we have: \\begin{align*} L &= \\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\\\ R &= \\begin{bmatrix} 0 & 0 & 0 & 1 \\end{bmatrix} \\\\ O &= \\begin{bmatrix} -3 & 1 & 0 & 0 \\end{bmatrix} \\end{align*} Note that the constant 3 appears in the \\(O\\) matrix with a negative sign, effectively moving it to the left side of the equation Multiplication with a Constant Now, let's consider \\(z = 3x^2 + y\\). The requirement of \"one multiplication per constraint\" doesn't apply to multiplication with a constant, as we can treat it as repeated addition. \\begin{align*} L &= \\begin{bmatrix} 0 & 0 & 3 & 0 \\end{bmatrix} \\\\ R &= \\begin{bmatrix} 0 & 0 & 1 & 0 \\end{bmatrix} \\\\ O &= \\begin{bmatrix} 0 & 1 & 0 & -1 \\end{bmatrix} \\end{align*}","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » Definition: R1CS","id":"84","title":"Definition: R1CS"},"85":{"body":"Recall that the prover aims to demonstrate knowledge of a witness \\(w\\) without revealing it. This is equivalent to knowing a vector \\(a\\) that satisfies \\((L \\cdot a) \\circ (R \\cdot a) = O \\cdot a\\), where \\(\\circ\\) denotes the Hadamard (element-wise) product. However, evaluating this equivalence directly requires \\(\\Omega(d)\\) operations, where \\(d\\) is the number of rows. To improve efficiency, we can convert this matrix comparison to a polynomial comparison, leveraging the Schwartz-Zippel Lemma, which allows us to check polynomial equality with \\(\\Omega(1)\\) evaluations. Let's consider a simpler example to illustrate this concept. Suppose we want to test the equivalence \\(Av = Bu\\), where: \\begin{align*} A = \\begin{bmatrix} 2 & 5 \\\\ 3 & 1 \\\\ \\end{bmatrix}, B = \\begin{bmatrix} 4 & 1 \\\\ 2 & 3 \\\\ \\end{bmatrix}, v = \\begin{bmatrix} 3 \\\\ 1 \\end{bmatrix}, u = \\begin{bmatrix} 2 \\\\ 2 \\end{bmatrix} \\end{align*} The equivalence check can be represented as: \\begin{equation*} \\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix} \\cdot 3 + \\begin{bmatrix} 5 \\\\ 1 \\end{bmatrix} \\cdot 1 = \\begin{bmatrix} 4 \\\\ 2 \\end{bmatrix} \\cdot 2 + \\begin{bmatrix} 1 \\\\ 3 \\end{bmatrix} \\cdot 2 \\end{equation*} This matrix-vector equality check is equivalent to the following polynomial equality check: \\begin{equation*} 3 \\cdot L([(1, 2), (2, 3)]) + 1 \\cdot L([(1, 5), (2, 1)]) = 2 \\cdot L([(1, 4), (2, 2)]) + 2 \\cdot L([(1, 1), (2, 3)]) \\end{equation*} where \\(L\\) denotes Lagrange Interpolation. In \\(\\mathbb{F}_{11}\\) (field with 11 elements), we have: \\begin{align*} L([(1, 2), (2, 3)]) &= x + 1 \\\\ L([(1, 5), (2, 1)]) &= 7x + 9 \\\\ L([(1, 4), (2, 2)]) &= 9x + 6 \\\\ L([(1, 1), (2, 3)]) &= 2x + 10 \\end{align*} The Schwartz-Zippel Lemma states that we need only one evaluation at a random point to check the equivalence of polynomials with high probability. However, the homomorphic property for multiplication doesn't hold for Lagrange Interpolation. Thus, we don't have \\(\\ell(x) \\cdot r(x) = o(x)\\), where \\(\\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) are the polynomials resulting from the Lagrange interpolation of \\(L \\cdot a\\), \\(R \\cdot a\\), and \\(O \\cdot a\\) respectively. While \\(\\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) are of degree at most \\(d-1\\), \\(\\ell(x) \\cdot r(x)\\) is of degree at most \\(2d-2\\). To address this discrepancy, we introduce a degree \\(d\\) polynomial \\(t(x) = \\prod_{i=1}^{d} (x - i)\\). We can then rewrite the equation as: \\begin{equation} \\ell(x) \\cdot r(x) = o(x) + h(x) \\cdot t(x) \\end{equation} where \\(h(x) = \\frac{\\ell(x) \\cdot r(x) - o(x)}{t(x)}\\). This formulation allows us to maintain the desired polynomial relationships while accounting for the degree differences.","breadcrumbs":"Basics of zk-SNARKs » Arithmetization » uadratic Arithmetic Program (QAP)","id":"85","title":"uadratic Arithmetic Program (QAP)"},"86":{"body":"Before dealing with all of \\(\\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) at once, we design a protocol that allows the Prover \\(\\mathcal{A}\\) to convince the Verifier \\(\\mathcal{B}\\) that \\(\\mathcal{A}\\) knows a specific polynomial. Let's denote this polynomial of degree \\(n\\) with coefficients in a finite field as: \\begin{equation} P(x) = c_0 + c_1 x + c_2 x^{2} + \\cdots c_n x^{n} \\end{equation} Assume \\(P(x)\\) has \\(n\\) roots, \\(a_1, a_2, \\ldots, a_n \\in \\mathbb{F}\\), such that \\(P(x) = (x - a_1)(x - a_2)\\cdots(x - a_n)\\). The Verifier \\(\\mathcal{B}\\) knows \\(d < n\\) roots of \\(P(x)\\), namely \\(a_1, a_2, \\ldots, a_d\\). Let \\(T(x) = (x - a_1)(x - a_2)\\cdots(x - a_d)\\). Note that the Prover also knows \\(T(x)\\). The Prover's objective is to convince the Verifier that \\(\\mathcal{A}\\) knows a polynomial \\(H(x) = \\frac{P(x)}{T(x)}\\).","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » Proving Single Polynomial","id":"86","title":"Proving Single Polynomial"},"87":{"body":"The simplest approach to prove that \\(\\mathcal{A}\\) knows \\(H(x)\\) is as follows: Naive Approach \\(\\mathcal{B}\\) sends all possible values in \\(\\mathbb{F}\\) to \\(\\mathcal{A}\\). \\(\\mathcal{A}\\) computes and sends all possible outputs of \\(H(x)\\) and \\(P(x)\\). \\(\\mathcal{B}\\) checks whether \\(H(a)T(a) = P(a)\\) holds for any \\(a\\) in \\(\\mathbb{F}\\). This protocol is highly inefficient, requiring \\(\\mathcal{O}(|\\mathbb{F}|)\\) evaluations and communications.","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » Naive Approach","id":"87","title":"Naive Approach"},"88":{"body":"Instead of evaluating polynomials at all values in \\(\\mathbb{F}\\), we can leverage the Schwartz-Zippel Lemma: if \\(H(s) = \\frac{P(s)}{T(s)}\\) or equivalently \\(H(s)T(s) = P(s)\\) for a random element \\(s\\), we can conclude that \\(H(x) = \\frac{P(x)}{T(x)}\\) with high probability. Thus, the Prover \\(\\mathcal{A}\\) only needs to send evaluations of \\(P(s)\\) and \\(H(s)\\) for a random input \\(s\\) received from \\(\\mathcal{B}\\). \\(+\\) Schwartz-Zippel Lemma \\(\\mathcal{B}\\) draws random \\(s\\) from \\(\\mathbb{F}\\) and sends it to \\(\\mathcal{A}\\). \\(\\mathcal{A}\\) computes \\(h = H(s)\\) and \\(p = P(s)\\) and send them to \\(\\mathcal{B}\\). \\(\\mathcal{B}\\) checks whether \\(p = t h\\), where \\(t\\) denotes \\(T(s)\\). This protocol is efficient, requiring only a constant number of evaluations and communications. However, it is vulnerable to a malicious prover who could send an arbitrary value \\(h'\\) and the corresponding \\(p'\\) such that \\(p' = h't\\).","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » \\(+\\) Schwartz-Zippel Lemma","id":"88","title":"\\(+\\) Schwartz-Zippel Lemma"},"89":{"body":"To address this vulnerability, the Verifier must hide the randomly chosen input \\(s\\) from the Prover. This can be achieved using the discrete logarithm assumption: it is computationally hard to determine \\(s\\) from \\(\\alpha\\), where \\(\\alpha = g^s \\bmod p\\). Thus, it's safe for the Verifier to send \\(\\alpha\\), as the Prover cannot easily derive \\(s\\) from it. An interesting property of polynomial exponentiation is: \\begin{align} g^{P(x)} &= g^{c_0 + c_1 x + c_2 x^{2} + \\cdots c_n x^{n}} = g^{c_0} (g^{x})^{c_1} (g^{(x^2)})^{c_2} \\cdots (g^{(x^n)})^{c_n} \\end{align} Instead of sending \\(s\\), the Verifier can send \\(g\\) and \\(\\alpha_{i} = g^{(s^i)}\\) for \\(i = 1, \\cdots n\\). BE CAREFUL THAT \\(g^{(s^i)} \\neq (g^s)^i\\) . The Prover can still evaluate \\(g^p = g^{P(s)}\\) using these powers of \\(g\\): \\begin{equation} g^{p} = g^{P(s)} = g^{c_0} \\alpha_{1}^{c_1} (\\alpha_{2})^{c_2} \\cdots (\\alpha_{n})^{c_n} \\end{equation} Similarly, the Prover can evaluate \\(g^h = g^{H(s)}\\). The Verifier can then check \\(p = ht \\iff g^p = (g^h)^t\\). \\(+\\) Discrete Logarithm Assumption \\(\\mathcal{B}\\) randomly draw \\(s\\) from \\(\\mathbb{F}\\). \\(\\mathcal{B}\\) computes and sends \\(\\{\\alpha, \\alpha_2, ..., \\alpha_{n}\\}\\), where \\(\\alpha_i= g^{(s^{i})}\\). \\(\\mathcal{A}\\) computes and sends \\(u = g^{p}\\) and \\(v = g^{h}\\). \\(\\mathcal{B}\\) checks whether \\(u = v^{t}\\). This approach prevents the Prover from obtaining \\(s\\) or \\(t = T(s)\\), making it impossible to send fake \\((h', p')\\) such that \\(p' = h't\\). However, this protocol still has a flaw. Since the Prover can compute \\(g^t = \\alpha_{c_1}(\\alpha_2)^{c_2}\\cdots(\\alpha_d)^{c_d}\\), they could send fake values \\(((g^{t})^{z}, g^{z})\\) instead of \\((g^p, g^h)\\) for an arbitrary value \\(z\\). The verifier's check would still pass, and they could not detect this deception.","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » \\(+\\) Discrete Logarithm Assumption","id":"89","title":"\\(+\\) Discrete Logarithm Assumption"},"9":{"body":"A semigroup with an identity element is called a monoid. Example : The set of non-negative integers under addition modulo \\(n\\) forms a monoid. For \\(n = 5\\), the set \\(\\{0, 1, 2, 3, 4\\}\\) under addition modulo 5 forms a monoid with 0 as the identity element.","breadcrumbs":"Basics of Number Theory » Semigroup, Group, Ring » Definition: Monoid","id":"9","title":"Definition: Monoid"},"90":{"body":"To address the vulnerability where the verifier \\(\\mathcal{B}\\) cannot distinguish if \\(v (= g^h)\\) from the prover is a power of \\(\\alpha_i = g^{(s^i)}\\), we can employ the Knowledge of Exponent Assumption. This approach involves the following steps: \\(\\mathcal{B}\\) sends both \\(\\alpha_i\\) and \\(\\alpha'_i = \\alpha_i^r\\) for a new random value \\(r\\). The prover returns \\(a = (\\alpha_i)^{c_i}\\) and \\(a' = (\\alpha'_i)^{c_i}\\) for \\(i = 1, ..., n\\). \\(\\mathcal{B}\\) can conclude that \\(a\\) is a power of \\(\\alpha_i\\) if \\(a^r = a'\\). Based on this assumption, we can design an improved protocol: \\(+\\) Knowledge of Exponent Assumption \\(\\mathcal{B}\\) randomly selects \\(s\\) and \\(r\\) from field \\(\\mathbb{F}\\). \\(\\mathcal{B}\\) computes and sends \\({\\alpha_1, \\alpha_2, ..., \\alpha_{n}}\\) and \\({\\alpha'_1, \\alpha'_2, ..., \\alpha'_{n}}\\), where \\(\\alpha_i = g^{(s^i)}\\) and \\(\\alpha' = \\alpha_{r} = g^{(s^{i})r}\\). \\(\\mathcal{A}\\) computes and sends \\(u = g^{p}\\), \\(v = g^{h}\\), and \\(w = g^{p'}\\), where \\(g^{p'} = g^{P(sr)}\\). \\(\\mathcal{B}\\) checks whether \\(u^{r} = w\\). \\(\\mathcal{B}\\) checks whether \\(u = v^{t}\\). The prover can compute \\(g^{p'} = g^{P(sr)} = \\alpha'^{c_1} (\\alpha'^{2})^{c_2} \\cdots (\\alpha'^{n})^{c_n}\\) using powers of \\(\\alpha'\\). This protocol now satisfies the properties of a SNARK: completeness, soundness, and efficiency.","breadcrumbs":"Basics of zk-SNARKs » Proving Single Polynomial » \\(+\\) Knowledge of Exponent Assumption","id":"90","title":"\\(+\\) Knowledge of Exponent Assumption"},"91":{"body":"To transform the above protocol into a zk-SNARK, we need to ensure that the verifier cannot learn anything about \\(P(x)\\) from the prover's information. 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-                <ol class="chapter"><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html" class="active"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html" class="active"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
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+                        <h1 id="arithmetization"><a class="header" href="#arithmetization">Arithmetization</a></h1>
+<p>The ultimate goal of Zero-Knowledge Proofs (ZKP) is to allow the prover to demonstrate their knowledge to the verifier without revealing any additional information. This knowledge typically involves solving complex problems, such as finding a secret input value that corresponds to a given public hash. ZKP protocols usually convert these statements into polynomial constraints. This process is often called \textit{arithmetization}.</p>
+<p>To make the protocol flexible, we need to encode this knowledge in a specific format, and one common approach is using Boolean circuits. It's well-known that problems in P (those solvable in polynomial time) and NP (those where the solution can be verified in polynomial time) can be represented as Boolean circuits. This means adopting Boolean circuits allows us to handle both P and NP problems.</p>
+<p>However, Boolean circuits are often large and inefficient. Even a simple operation, like adding two 256-bit integers, can require hundreds of Boolean operators. In contrast, arithmetic circuits—essentially systems of equations involving addition, multiplication, and equality—offer a much more compact representation. Additionally, any Boolean circuit can be converted into an arithmetic circuit. For instance, the Boolean expression \(z = x \land y\) can be represented as \(x(x-1) = 0\), \(y(y-1) = 0\), and \(z = xy\) in an arithmetic circuit. Furthermore, as we'll see in this section, converting arithmetic circuits into polynomial constraints allows for much faster evaluation.</p>
+<h2 id="rank-1-constraint-system-r1cs"><a class="header" href="#rank-1-constraint-system-r1cs">Rank-1 Constraint System (R1CS)</a></h2>
+<p>There are many formats to represent arithmetic circuits, and one of the most popular ones is R1CS (Rank-1 Constraint System), which represents arithmetic circuits as \underline{a set of equality constraints, each involving only one multiplication}. In an arithmetic circuit, we call the concrete values assigned to the variables within the constraints witness. We first provide the formal definition of R1CS as follows:</p>
+<h3 id="definition-r1cs"><a class="header" href="#definition-r1cs">Definition: R1CS</a></h3>
+<p>An R1CS structure \(\mathcal{S}\) consists of:</p>
+<ul>
+<li>Size bounds \(m, d, \ell \in \mathbb{N}\) where \(d &gt; \ell\)</li>
+<li>Three matrices \(O, L, R \in \mathbb{F}^{m\ times d}\) with at most \(\Omega(\max(m, d))\) non-zero entries in total</li>
+</ul>
+<p>An R1CS instance includes a public input \(p \in \mathbb{F}^\ell\), while an R1CS witness is a vector \(w \in \mathbb{F}^{d - \ell - 1}\).
+A structure-instance tuple \((S, p)\) is satisfied by a witness \(w\) if:
+\begin{equation}
+(L \cdot a) \circ (R \cdot a) - O \cdot a = \mathbf{0}
+\end{equation}
+where \(a = (1, w, p) \in \mathbb{F}^d\), \(\cdot\) denotes matrix-vector multiplication, and \(\circ\) is the Hadamard product.</p>
+<p>The intuitive interpretation of each matrix is as follows:</p>
+<ul>
+<li>\(L\): Encodes the left input of each gate</li>
+<li>\(R\): Encodes the right input of each gate</li>
+<li>\(O\): Encodes the output of each gate</li>
+<li>The leading 1 in the witness vector allows for constant terms</li>
+</ul>
+<p><strong>Single Multiplication</strong></p>
+<p>Let's consider a simple example where we want to prove \(z = x \cdot y\), with \(z = 3690\), \(x = 82\), and \(y = 45\).</p>
+<ul>
+<li>\textbf{Witness vector}: \((1, z, x, y) = (1, 3690, 82, 45)\)</li>
+<li>\textbf{Number of witnesses}: \(m = 4\)</li>
+<li>\textbf{Number of constraints}: \(d = 1\)</li>
+</ul>
+<p>The R1CS constraint for \(z = x \cdot y\) is satisfied when:</p>
+<p>\begin{align*}
+&amp;(\begin{bmatrix} 0 &amp; 0 &amp; 1 &amp; 0 \end{bmatrix} \cdot a) \circ (\begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix} \cdot a) - \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; 0 \end{bmatrix} \cdot a \\
+=&amp;(\begin{bmatrix} 0 &amp; 0 &amp; 1 &amp; 0 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\ 3690 \\ 82 \\ 45
+\end{bmatrix}) \circ (\begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\ 3690 \\ 82 \\ 45
+\end{bmatrix}) - \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; 0 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\ 3690 \\ 82 \\ 45
+\end{bmatrix} \\
+=&amp; 82 \cdot 45 - 3690 \\
+=&amp; 3690 - 3690 \\
+=&amp; 0
+\end{align*}</p>
+<p>This example demonstrates how R1CS encodes a simple multiplication constraint:</p>
+<ul>
+<li>\(L = \begin{bmatrix} 0 &amp; 0 &amp; 1 &amp; 0 \end{bmatrix}\) selects \(x\) (left input)</li>
+<li>\(R = \begin{bmatrix} 0 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix}\) selects \(y\) (right input)</li>
+<li>\(O = \begin{bmatrix} 0 &amp; 1 &amp; 0 &amp; 0 \end{bmatrix}\) selects \(z\) (output)</li>
+</ul>
+<p><strong>Multiple Constraints</strong></p>
+<p>Let's examine a more complex example: \(r = a \cdot b \cdot c \cdot d\). Since R1CS requires that each constraint contain only one multiplication, we need to break this down into multiple constraints:</p>
+<p>\begin{align*}
+v_1 &amp;= a \cdot b \\
+v_2 &amp;= c \cdot d \\
+r &amp;= v_1 \cdot v_2
+\end{align*}</p>
+<p>Note that alternative representations are possible, such as \(v_1 = ab, v_2 = v_1c, r = v_2d\). In this example, we use 7 variables \((r, a, b, c, d, v_1, v_2)\), so the dimension of the witness vector will be \(m = 8\) (including the constant 1). We have three constraints, so \(n = 3\).
+To construct the matrices \(L\), \(R\), and \(O\), we can interpret the constraints as linear combinations:</p>
+<p>\begin{align*}
+v_1 &amp;= (0 \cdot 1 + 0 \cdot r + 1 \cdot a + 0 \cdot b + 0 \cdot c + 0 \cdot d + 0 \cdot v_1 + 0 \cdot v_2) \cdot b \\
+v_2 &amp;= (0 \cdot 1 + 0 \cdot r + 0 \cdot a + 0 \cdot b + 1 \cdot c + 0 \cdot d + 0 \cdot v_1 + 0 \cdot v_2) \cdot d \\
+r &amp;= (0 \cdot 1 + 0 \cdot r + 0 \cdot a + 0 \cdot b + 0 \cdot c + 0 \cdot d + 1 \cdot v_1 + 0 \cdot v_2) \cdot v_2
+\end{align*}</p>
+<p>Thus, we can construct \(L\), \(R\), and \(O\) as follows:</p>
+<p>\begin{equation*}
+L = \begin{bmatrix}
+0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0
+\end{bmatrix}
+\end{equation*}
+\begin{equation*}
+R = \begin{bmatrix}
+0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1
+\end{bmatrix}
+\end{equation*}
+\begin{equation*}
+O = \begin{bmatrix}
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 \\
+0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 \\
+0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0
+\end{bmatrix}
+\end{equation*}</p>
+<p>Where the columns in each matrix correspond to \((1, r, a, b, c, d, v_1, v_2)\).</p>
+<p><strong>Addition with a Constant</strong></p>
+<p>Let's examine the case \(z = x \cdot y + 3\). We can represent this as \(-3 + z = x \cdot y\). For the witness vector \((1, z, x, y)\), we have:</p>
+<p>\begin{align*}
+L &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 1 &amp; 0
+\end{bmatrix} \\
+R &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 0 &amp; 1
+\end{bmatrix} \\
+O &amp;= \begin{bmatrix}
+-3 &amp; 1 &amp; 0 &amp; 0
+\end{bmatrix}
+\end{align*}</p>
+<p>Note that the constant 3 appears in the \(O\) matrix with a negative sign, effectively moving it to the left side of the equation</p>
+<p><strong>Multiplication with a Constant</strong></p>
+<p>Now, let's consider \(z = 3x^2 + y\). The requirement of "one multiplication per constraint" doesn't apply to multiplication with a constant, as we can treat it as repeated addition.</p>
+<p>\begin{align*}
+L &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 3 &amp; 0
+\end{bmatrix} \\
+R &amp;= \begin{bmatrix}
+0 &amp; 0 &amp; 1 &amp; 0
+\end{bmatrix} \\
+O &amp;= \begin{bmatrix}
+0 &amp; 1 &amp; 0 &amp; -1
+\end{bmatrix}
+\end{align*}</p>
+<h2 id="uadratic-arithmetic-program-qap"><a class="header" href="#uadratic-arithmetic-program-qap">uadratic Arithmetic Program (QAP)</a></h2>
+<p>Recall that the prover aims to demonstrate knowledge of a witness \(w\) without revealing it. This is equivalent to knowing a vector \(a\) that satisfies \((L \cdot a) \circ (R \cdot a) = O \cdot a\), where \(\circ\) denotes the Hadamard (element-wise) product. However, evaluating this equivalence directly requires \(\Omega(d)\) operations, where \(d\) is the number of rows. To improve efficiency, we can convert this matrix comparison to a polynomial comparison, leveraging the Schwartz-Zippel Lemma, which allows us to check polynomial equality with \(\Omega(1)\) evaluations.</p>
+<p>Let's consider a simpler example to illustrate this concept. Suppose we want to test the equivalence \(Av = Bu\), where:</p>
+<p>\begin{align*}
+A = \begin{bmatrix}
+2 &amp; 5 \\
+3 &amp; 1 \\
+\end{bmatrix},
+B = \begin{bmatrix}
+4 &amp; 1 \\
+2 &amp; 3 \\
+\end{bmatrix},
+v = \begin{bmatrix}
+3 \\ 1
+\end{bmatrix},
+u = \begin{bmatrix}
+2 \\ 2
+\end{bmatrix}
+\end{align*}</p>
+<p>The equivalence check can be represented as:</p>
+<p>\begin{equation*}
+\begin{bmatrix}
+2 \\ 3
+\end{bmatrix} \cdot 3 + \begin{bmatrix}
+5 \\ 1
+\end{bmatrix} \cdot 1 = \begin{bmatrix}
+4 \\ 2
+\end{bmatrix} \cdot 2 + \begin{bmatrix}
+1 \\ 3
+\end{bmatrix} \cdot 2
+\end{equation*}</p>
+<p>This matrix-vector equality check is equivalent to the following polynomial equality check:</p>
+<p>\begin{equation*}
+3 \cdot L([(1, 2), (2, 3)]) + 1 \cdot L([(1, 5), (2, 1)]) = 2 \cdot L([(1, 4), (2, 2)]) + 2 \cdot L([(1, 1), (2, 3)])
+\end{equation*}</p>
+<p>where \(L\) denotes Lagrange Interpolation. In \(\mathbb{F}_{11}\) (field with 11 elements), we have:</p>
+<p>\begin{align*}
+L([(1, 2), (2, 3)]) &amp;= x + 1 \\
+L([(1, 5), (2, 1)]) &amp;= 7x + 9 \\
+L([(1, 4), (2, 2)]) &amp;= 9x + 6 \\
+L([(1, 1), (2, 3)]) &amp;= 2x + 10
+\end{align*}</p>
+<p>The Schwartz-Zippel Lemma states that we need only one evaluation at a random point to check the equivalence of polynomials with high probability.</p>
+<p>However, the homomorphic property for multiplication doesn't hold for Lagrange Interpolation. Thus, we don't have \(\ell(x) \cdot r(x) = o(x)\), where \(\ell(x)\), \(r(x)\), and \(o(x)\) are the polynomials resulting from the Lagrange interpolation of \(L \cdot a\), \(R \cdot a\), and \(O \cdot a\) respectively.</p>
+<p>While \(\ell(x)\), \(r(x)\), and \(o(x)\) are of degree at most \(d-1\), \(\ell(x) \cdot r(x)\) is of degree at most \(2d-2\).</p>
+<p>To address this discrepancy, we introduce a degree \(d\) polynomial \(t(x) = \prod_{i=1}^{d} (x - i)\). We can then rewrite the equation as:</p>
+<p>\begin{equation}
+\ell(x) \cdot r(x) = o(x) + h(x) \cdot t(x)
+\end{equation}</p>
+<p>where \(h(x) = \frac{\ell(x) \cdot r(x) - o(x)}{t(x)}\). This formulation allows us to maintain the desired polynomial relationships while accounting for the degree differences.</p>
+
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+                        <h1 id="proving-single-polynomial"><a class="header" href="#proving-single-polynomial">Proving Single Polynomial</a></h1>
+<p>Before dealing with all of \(\ell(x)\), \(r(x)\), and \(o(x)\) at once, we design a protocol that allows the Prover \(\mathcal{A}\) to convince the Verifier \(\mathcal{B}\) that \(\mathcal{A}\) knows a specific polynomial. Let's denote this polynomial of degree \(n\) with coefficients in a finite field as:</p>
+<p>\begin{equation}
+P(x) = c_0 + c_1 x + c_2 x^{2} + \cdots c_n x^{n}
+\end{equation}</p>
+<p>Assume \(P(x)\) has \(n\) roots, \(a_1, a_2, \ldots, a_n \in \mathbb{F}\), such that \(P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)\). The Verifier \(\mathcal{B}\) knows \(d &lt; n\) roots of \(P(x)\), namely \(a_1, a_2, \ldots, a_d\). Let \(T(x) = (x - a_1)(x - a_2)\cdots(x - a_d)\). Note that the Prover also knows \(T(x)\).</p>
+<p>The Prover's objective is to convince the Verifier that \(\mathcal{A}\) knows a polynomial \(H(x) = \frac{P(x)}{T(x)}\).</p>
+<h3 id="naive-approach"><a class="header" href="#naive-approach">Naive Approach</a></h3>
+<p>The simplest approach to prove that \(\mathcal{A}\) knows \(H(x)\) is as follows:</p>
+<p><em>Naive Approach</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) sends all possible values in \(\mathbb{F}\) to \(\mathcal{A}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends all possible outputs of \(H(x)\) and \(P(x)\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(H(a)T(a) = P(a)\) holds for any \(a\) in \(\mathbb{F}\).</em></li>
+</ul>
+<p>This protocol is highly inefficient, requiring \(\mathcal{O}(|\mathbb{F}|)\) evaluations and communications.</p>
+<h3 id="-schwartz-zippel-lemma"><a class="header" href="#-schwartz-zippel-lemma">\(+\) Schwartz-Zippel Lemma</a></h3>
+<p>Instead of evaluating polynomials at all values in \(\mathbb{F}\), we can leverage the Schwartz-Zippel Lemma: if \(H(s) = \frac{P(s)}{T(s)}\) or equivalently \(H(s)T(s) = P(s)\) for a random element \(s\), we can conclude that \(H(x) = \frac{P(x)}{T(x)}\) with high probability. Thus, the Prover \(\mathcal{A}\) only needs to send evaluations of \(P(s)\) and \(H(s)\) for a random input \(s\) received from \(\mathcal{B}\).</p>
+<p><em>\(+\) Schwartz-Zippel Lemma</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) draws random \(s\) from \(\mathbb{F}\) and sends it to \(\mathcal{A}\).</em></li>
+<li><em>\(\mathcal{A}\) computes \(h = H(s)\) and \(p = P(s)\) and send them to \(\mathcal{B}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(p = t h\), where \(t\) denotes \(T(s)\).</em></li>
+</ul>
+<p>This protocol is efficient, requiring only a constant number of evaluations and communications. However, it is vulnerable to a malicious prover who could send an arbitrary value \(h'\) and the corresponding \(p'\) such that \(p' = h't\).</p>
+<h3 id="-discrete-logarithm-assumption"><a class="header" href="#-discrete-logarithm-assumption">\(+\) Discrete Logarithm Assumption</a></h3>
+<p>To address this vulnerability, the Verifier must hide the randomly chosen input \(s\) from the Prover. This can be achieved using the discrete logarithm assumption: it is computationally hard to determine \(s\) from \(\alpha\), where \(\alpha = g^s \bmod p\). Thus, it's safe for the Verifier to send \(\alpha\), as the Prover cannot easily derive \(s\) from it.</p>
+<p>An interesting property of polynomial exponentiation is:</p>
+<p>\begin{align}
+g^{P(x)} &amp;= g^{c_0 + c_1 x + c_2 x^{2} + \cdots c_n x^{n}} = g^{c_0} (g^{x})^{c_1}  (g^{(x^2)})^{c_2} \cdots (g^{(x^n)})^{c_n}
+\end{align}</p>
+<p>Instead of sending \(s\), the Verifier can send \(g\) and \(\alpha_{i} = g^{(s^i)}\) for \(i = 1, \cdots n\). BE CAREFUL THAT <strong>\(g^{(s^i)} \neq (g^s)^i\)</strong>. The Prover can still evaluate \(g^p = g^{P(s)}\) using these powers of \(g\):</p>
+<p>\begin{equation}
+g^{p} = g^{P(s)} = g^{c_0} \alpha_{1}^{c_1} (\alpha_{2})^{c_2} \cdots (\alpha_{n})^{c_n}
+\end{equation}</p>
+<p>Similarly, the Prover can evaluate \(g^h = g^{H(s)}\). The Verifier can then check \(p = ht \iff g^p = (g^h)^t\).</p>
+<p><em>\(+\) Discrete Logarithm Assumption</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) randomly draw \(s\) from \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{B}\) computes and sends \(\{\alpha, \alpha_2, ..., \alpha_{n}\}\), where \(\alpha_i= g^{(s^{i})}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends \(u = g^{p}\) and \(v = g^{h}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u = v^{t}\).</em></li>
+</ul>
+<p>This approach prevents the Prover from obtaining \(s\) or \(t = T(s)\), making it impossible to send fake \((h', p')\) such that \(p' = h't\).</p>
+<p>However, this protocol still has a flaw. Since the Prover can compute \(g^t = \alpha_{c_1}(\alpha_2)^{c_2}\cdots(\alpha_d)^{c_d}\), they could send fake values \(((g^{t})^{z}, g^{z})\) instead of \((g^p, g^h)\) for an arbitrary value \(z\). The verifier's check would still pass, and they could not detect this deception.</p>
+<h3 id="-knowledge-of-exponent-assumption"><a class="header" href="#-knowledge-of-exponent-assumption">\(+\) Knowledge of Exponent Assumption</a></h3>
+<p>To address the vulnerability where the verifier \(\mathcal{B}\) cannot distinguish if \(v (= g^h)\) from the prover is a power of \(\alpha_i = g^{(s^i)}\), we can employ the Knowledge of Exponent Assumption. This approach involves the following steps:</p>
+<ul>
+<li>\(\mathcal{B}\) sends both \(\alpha_i\) and \(\alpha'_i = \alpha_i^r\) for a new random value \(r\).</li>
+<li>The prover returns \(a = (\alpha_i)^{c_i}\) and \(a' = (\alpha'_i)^{c_i}\) for \(i = 1, ..., n\).</li>
+<li>\(\mathcal{B}\) can conclude that \(a\) is a power of \(\alpha_i\) if \(a^r = a'\).</li>
+</ul>
+<p>Based on this assumption, we can design an improved protocol:</p>
+<p><em>\(+\) Knowledge of Exponent Assumption</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) randomly selects \(s\) and \(r\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{B}\) computes and sends \({\alpha_1, \alpha_2, ..., \alpha_{n}}\) and \({\alpha'_1, \alpha'_2, ..., \alpha'_{n}}\), where \(\alpha_i = g^{(s^i)}\) and \(\alpha' = \alpha_{r} = g^{(s^{i})r}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends \(u = g^{p}\), \(v = g^{h}\), and \(w = g^{p'}\), where \(g^{p'} = g^{P(sr)}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u^{r} = w\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u = v^{t}\).</em></li>
+</ul>
+<p>The prover can compute \(g^{p'} = g^{P(sr)} = \alpha'^{c_1} (\alpha'^{2})^{c_2} \cdots (\alpha'^{n})^{c_n}\) using powers of \(\alpha'\). This protocol now satisfies the properties of a SNARK: completeness, soundness, and efficiency.</p>
+<h3 id="-zero-knowledge"><a class="header" href="#-zero-knowledge">\(+\) Zero Knowledge</a></h3>
+<p>To transform the above protocol into a zk-SNARK, we need to ensure that the verifier cannot learn anything about \(P(x)\) from the prover's information. This is achieved by having the prover obfuscate all information with a random secret value \(\delta\):</p>
+<p><em>\(+\) Zero Knowledge</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) randomly selects \(s\) and \(r\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{B}\) computes and sends \(\{\alpha_1, \alpha_2, ..., \alpha_{n}\}\) and \(\{\alpha_1', \alpha'_2, ..., \alpha'_{n}\}\), where \(\alpha_i = g^{(s^{i})}\) and \(\alpha_i' = \alpha_i^{r} = g^{(s^{i})r}\).</em></li>
+<li><em>\(\mathcal{A}\) randomly selects \(\delta\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{A}\) computes and sends \(u' = (g^{p})^{\delta}\), \(v' = (g^{h})^{\delta}\), and \(w' = (g^{p'})^{\delta}\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u'^{r} = w'\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u' = v'^{t}\).</em></li>
+</ul>
+<p>By introducing the random value \(\delta\), the verifier can no longer learn anything about \(p\), \(h\), or \(w\), thus achieving zero knowledge.</p>
+<h3 id="-non-interactivity"><a class="header" href="#-non-interactivity">\(+\) Non-interactivity</a></h3>
+<p>The previously described protocol requires each verifier to generate unique random values, which becomes inefficient when a prover needs to demonstrate knowledge to multiple verifiers. To address this, we aim to eliminate the interaction between the prover and verifier. One effective solution is the use of a trusted setup.</p>
+<p><em>\(+\) Non-Interactive: Trusted Setup</em></p>
+<ul>
+<li><em><strong>Secret Seed:</strong> A trusted third party generates the random values \(s\) and \(r\)</em></li>
+<li><em><strong>Proof Key:</strong> Provided to the prover</em>
+<ul>
+<li><em>\(\{\alpha_1, \alpha_2, ..., \alpha_{n}\}\), where \(\alpha_{i} = g^{(s^i)}\)</em></li>
+<li><em>\(\{\alpha'_1, \alpha'_2, ..., \alpha'_{n}\}\), where \(\alpha_i' = g^{(s^{i})r}\)</em></li>
+</ul>
+</li>
+<li><em><strong>Verification Key:</strong> Distributed to verifiers</em>
+<ul>
+<li><em>\(g^{r}\)</em></li>
+<li><em>\(g^{T(s)}\)</em></li>
+</ul>
+</li>
+<li><em>After distribution, the original \(s\) and \(r\) values are securely destroyed.</em></li>
+</ul>
+<p>Then, the non-interactive protocol consists of two main parts: proof generation and verification.</p>
+<p><em>\(+\) Non-Interactive: Proof</em></p>
+<ul>
+<li><em>\(\mathcal{A}\) receives the proof key</em></li>
+<li><em>\(\mathcal{A}\) randomly selects \(\delta\) from field \(\mathbb{F}\).</em></li>
+<li><em>\(\mathcal{A}\) broadcast the proof \(\pi = (u' = (g^{p})^{\delta}, v' = (g^{h})^{\delta}, w' = (g^{p'})^{\delta})\)</em></li>
+</ul>
+<p>Since \(r\) is not shared and already destroyed, the verifier \(\mathcal{B}\) cannot calculate \(u'^{r}\) to check \(u'^{r} = w'\). Instead, the verifier can use a paring with bilinear mapping; \(u'^{r} = w'\) is equivalent to \(e(u' = (g^{p})^{\delta}, g^{r}) = e(w'=(g^{p'})^{\delta}, g)\).</p>
+<p><em>\(+\) Non-Interactive: Verification</em></p>
+<ul>
+<li><em>\(\mathcal{B}\) receives the verification key.</em></li>
+<li><em>\(\mathcal{B}\) receives the proof \(\pi\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(e(u', g^{r}) = e(w', g)\).</em></li>
+<li><em>\(\mathcal{B}\) checks whether \(u' = v'^{t}\).</em></li>
+</ul>
+
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diff --git a/docs/book/book/zkstark/index.html b/docs/book/book/zkstark/index.html
index dfe6671..2c0551e 100644
--- a/docs/book/book/zkstark/index.html
+++ b/docs/book/book/zkstark/index.html
@@ -29,6 +29,7 @@
         <link rel="stylesheet" href="../ayu-highlight.css">
 
         <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="../theme/css/custom.css">
 
         <!-- MathJax -->
         <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
@@ -90,7 +91,7 @@
 
         <nav id="sidebar" class="sidebar" aria-label="Table of contents">
             <div class="sidebar-scrollbox">
-                <ol class="chapter"><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="../zkstark/index.html" class="active"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html" class="active"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
             </div>
             <div id="sidebar-resize-handle" class="sidebar-resize-handle">
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@@ -180,7 +181,7 @@ <h1 class="menu-title">MyZKP: Building Zero Knowledge Proof from Scratch</h1>
 
                     <nav class="nav-wrapper" aria-label="Page navigation">
                         <!-- Mobile navigation buttons -->
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                                 <i class="fa fa-angle-left"></i>
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@@ -194,7 +195,7 @@ <h1 class="menu-title">MyZKP: Building Zero Knowledge Proof from Scratch</h1>
             </div>
 
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-                    <a rel="prev" href="../zksnark/index.html" class="nav-chapters previous" title="Previous chapter" aria-label="Previous chapter" aria-keyshortcuts="Left">
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diff --git a/docs/book/book/zkvm/index.html b/docs/book/book/zkvm/index.html
index aed8d60..11de47d 100644
--- a/docs/book/book/zkvm/index.html
+++ b/docs/book/book/zkvm/index.html
@@ -29,6 +29,7 @@
         <link rel="stylesheet" href="../ayu-highlight.css">
 
         <!-- Custom theme stylesheets -->
+        <link rel="stylesheet" href="../theme/css/custom.css">
 
         <!-- MathJax -->
         <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
@@ -90,7 +91,7 @@
 
         <nav id="sidebar" class="sidebar" aria-label="Table of contents">
             <div class="sidebar-scrollbox">
-                <ol class="chapter"><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">1.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">1.1.</strong> Computation Rule and Properties</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">2.</strong> Basics of zk-SNARKs</a></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html" class="active"><strong aria-hidden="true">4.</strong> Basics of zkVM</a></li></ol>
+                <ol class="chapter"><li class="chapter-item expanded "><a href="../index.html"><strong aria-hidden="true">1.</strong> Introduction</a></li><li class="chapter-item expanded "><a href="../number_theory/index.html"><strong aria-hidden="true">2.</strong> Basics of Number Theory</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../number_theory/subsec1.html"><strong aria-hidden="true">2.1.</strong> Computation Rule and Properties</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec2.html"><strong aria-hidden="true">2.2.</strong> Semigroup, Group, Ring</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec3.html"><strong aria-hidden="true">2.3.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec4.html"><strong aria-hidden="true">2.4.</strong> Galoid Field</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec5.html"><strong aria-hidden="true">2.5.</strong> Elliptic Curve</a></li><li class="chapter-item expanded "><a href="../number_theory/subsec6.html"><strong aria-hidden="true">2.6.</strong> Useful Assumptions</a></li></ol></li><li class="chapter-item expanded "><a href="../zksnark/index.html"><strong aria-hidden="true">3.</strong> Basics of zk-SNARKs</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../zksnark/subsec2.html"><strong aria-hidden="true">3.1.</strong> Arithmetization</a></li><li class="chapter-item expanded "><a href="../zksnark/subsec3.html"><strong aria-hidden="true">3.2.</strong> Proving Single Polynomial</a></li></ol></li><li class="chapter-item expanded "><a href="../zkstark/index.html"><strong aria-hidden="true">4.</strong> Basics of zk-STARKS</a></li><li class="chapter-item expanded "><a href="../zkvm/index.html" class="active"><strong aria-hidden="true">5.</strong> Basics of zkVM</a></li></ol>
             </div>
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                 <div class="sidebar-resize-indicator"></div>
diff --git a/docs/book/src/README.md b/docs/book/src/README.md
new file mode 100644
index 0000000..e69de29
diff --git a/docs/book/src/SUMMARY.md b/docs/book/src/SUMMARY.md
index df0dacb..af51594 100644
--- a/docs/book/src/SUMMARY.md
+++ b/docs/book/src/SUMMARY.md
@@ -1,7 +1,15 @@
 # Summary
 
+- [Introduction](./README.md)
 - [Basics of Number Theory](./number_theory/README.md)
   - [Computation Rule and Properties](./number_theory/subsec1.md)
+  - [Semigroup, Group, Ring](./number_theory/subsec2.md)
+  - [Polynomials](./number_theory/subsec3.md)
+  - [Galoid Field](./number_theory/subsec4.md)
+  - [Elliptic Curve](./number_theory/subsec5.md)
+  - [Useful Assumptions](./number_theory/subsec6.md)
 - [Basics of zk-SNARKs](./zksnark/README.md)
+  - [Arithmetization](./zksnark/subsec2.md)
+  - [Proving Single Polynomial](./zksnark/subsec3.md)
 - [Basics of zk-STARKS](./zkstark/README.md)
 - [Basics of zkVM](./zkvm/README.md)
diff --git a/docs/book/src/number_theory/subsec1.md b/docs/book/src/number_theory/subsec1.md
index 910d1bf..10a4251 100644
--- a/docs/book/src/number_theory/subsec1.md
+++ b/docs/book/src/number_theory/subsec1.md
@@ -1,28 +1,32 @@
-## Computation Rule and Properties
+# Computation Rule and Properties
+
+
+### Definition: Binary Operation
 
 ---
 
-***Definition: Binary Operation***
+*A mapping \\(\circ: X \times X \rightarrow X\\) is a binary operation on \\(X\\) if for any pair of elements \\((x_1, x_2)\\) in \\(X\\), \\(x_1 \circ x_2\\) is also in \\(X\\).*
 
-A mapping \\(\circ: X \times X \rightarrow X\\) is a binary operation on \\(X\\) if for any pair of elements \\((x_1, x_2)\\) in \\(X\\), \\(x_1 \circ x_2\\) is also in \\(X\\).
+---
 
 **Example**: Addition (+) on the set of integers is a binary operation. For example, \\(5 + 3 = 8\\), and both \\(5, 3, 8\\) are integers, staying within the set of integers.
 
-----
+### Definition: Associative Property
 
-***Definition: Associative Property***
+---
 
-A binary operation \\(\circ\\) is associative if \\((a \circ b) \circ c = a \circ (b \circ c)\\) for all \\(a, b, c \in X\\).
+*A binary operation \\(\circ\\) is associative if \\((a \circ b) \circ c = a \circ (b \circ c)\\) for all \\(a, b, c \in X\\).*
 
+---
 
 **Example**: Multiplication of real numbers is associative: \\((2 \times 3) \times 4 = 2 \times (3 \times 4) = 24\\). In a modular context, we also have addition modulo \\(n\\) being associative. For example, for \\(n = 5\\), \\((2 + 3) \bmod 5 + 4 \bmod 5 = 2 + (3 \bmod 5 + 4) \bmod 5 = 4\\).
 
+### Definition: Commutative Property
+
 ---
 
-***Definition: Commutative Property***
+*A binary operation \\(\circ\\) is commutative if \\(a \circ b = b \circ a\\) for all \\(a, b \in X\\).*
 
-A binary operation \\(\circ\\) is commutative if \\(a \circ b = b \circ a\\) for all \\(a, b \in X\\).
+---
 
 **Example**: Addition modulo \\(n\\) is also commutative. For \\(n = 7\\), \\(5 + 3 \bmod 7 = 3 + 5 \bmod 7 = 1\\).
-
----
diff --git a/docs/book/src/number_theory/subsec2.md b/docs/book/src/number_theory/subsec2.md
new file mode 100644
index 0000000..d161f4c
--- /dev/null
+++ b/docs/book/src/number_theory/subsec2.md
@@ -0,0 +1,374 @@
+# Semigroup, Group, Ring
+
+### Definition: Semigroup
+
+---
+
+*A pair \\((H, \circ)\\), where \\(H\\) is a non-empty set and \\(\circ\\) is an associative binary operation on \\(H\\), is called a semigroup.*
+
+---
+
+**Example**: The set of positive integers under multiplication modulo \\(n\\) forms a semigroup. For instance, with \\(n = 6\\), the elements \\(\\{1, 2, 3, 4, 5\\}\\) under multiplication modulo 6 form a semigroup, since multiplication modulo 6 is associative.
+
+
+### Definition: Abelian Semigroup
+
+---
+
+*A semigroup whose operation is commutative is called an abelian semigroup.*
+
+---
+
+**Example**: The set of natural numbers under addition modulo \\(n\\) forms an abelian semigroup. For \\(n = 7\\), addition modulo 7 is both associative and commutative, so it is an abelian semigroup.
+
+
+### Definition: Identity Element
+
+---
+
+*An element \\(e \in H\\) is an identity element of \\(H\\) if it satisfies \\(e \circ a = a \circ e = a\\) for any \\(a \in H\\).*
+
+---
+
+**Example**: 0 is the identity element for addition modulo \\(n\\). For example, \\(0 + a \bmod 5 = a + 0 \bmod 5 = a\\). Similarly, 1 is the identity element for multiplication modulo \\(n\\). For example, \\(1 \times a \bmod 7 = a \times 1 \bmod 7 = a\\).
+
+### Definition: Monoid
+
+---
+
+*A semigroup with an identity element is called a monoid.*
+
+---
+
+**Example**: The set of non-negative integers under addition modulo \\(n\\) forms a monoid. For \\(n = 5\\), the set \\(\\{0, 1, 2, 3, 4\\}\\) under addition modulo 5 forms a monoid with 0 as the identity element.
+
+### Definition: Inverse
+
+---
+
+*For an element \\(a \in H\\), an element \\(b \in H\\) is an inverse of \\(a\\) if \\(a \circ b = b \circ a = e\\), where \\(e\\) is the identity element.*
+
+---
+
+**Example**: In modulo \\(n\\) arithmetic (addition), the inverse of an element exists if it can cancel itself out to yield the identity element. In the set of integers modulo 7, the inverse of 3 is 5, because \\(3 \times 5 \bmod 7 = 1\\), where 1 is the identity element for multiplication.
+
+### Definition: Group
+
+---
+
+*A monoid in which every element has an inverse is called a group.*
+
+---
+
+**Example**: The set of integers modulo a prime \\(p\\) under multiplication forms a group (Can you prove it?). For instance, in \\(\mathbb{Z}/5\mathbb{Z}\\), every non-zero element \\(\\{1 + 5\mathbb{Z}, 2 + 5\mathbb{Z}, 3 + 5\mathbb{Z}, 4 + 5\mathbb{Z}\\}\\) has an inverse, making it a group.
+
+### Definition: Order of a Group
+
+---
+
+*The order of a group is the number of elements in the group.*
+
+---
+
+**Example**: The group of integers modulo 4 under addition has order 4, because the set of elements is \\(\\{0, 1, 2, 3\\}\\).
+
+
+### Definition: Ring
+
+---
+
+*A triple \\((R, +, \cdot)\\) is a ring if \\((R, +)\\) is an abelian group, \\((R, \cdot)\\) is a semigroup, and the distributive property holds: \\(x \cdot (y + z) = (x \cdot y) + (x \cdot z)\\) and \\((x + y) \cdot z = (x \cdot z) + (y \cdot z)\\) for all \\(x, y, z \in R\\).*
+
+---
+
+**Example**: The set of integers with usual addition and multiplication modulo \\(n\\) forms a ring. For example, in \\(\mathbb{Z}/6\mathbb{Z}\\), addition and multiplication modulo 6 form a ring.
+
+```rust
+use std::fmt;
+use std::ops::{Add, Mul, Neg, Sub};
+
+pub trait Ring:
+    Sized
+    + Clone
+    + PartialEq
+    + fmt::Display
+    + Add<Output = Self>
+    + Sub<Output = Self>
+    + Mul<Output = Self>
+    + Mul<i64, Output = Self>
+    + Neg<Output = Self>
+{
+    // A ring is an algebraic structure with addition and multiplication
+    fn zero() -> Self;
+    fn one() -> Self;
+}
+```
+
+### Definition: Communicative Ring
+
+---
+
+*A ring is called a commutative ring if its multiplication operation is commutative.*
+
+---
+
+**Example** The set of real numbers under usual addition and multiplication forms a commutative ring.
+
+### Definition: Field
+
+---
+
+*A commutative ring with a multiplicative identity element where every non-zero element has a multiplicative inverse is called a field.*
+
+---
+
+**Example** The set of rational numbers under usual addition and multiplication forms a field.
+
+```rust
+use crate::modules::ring::Ring;
+
+pub trait Field: Ring + Div<Output = Self> {
+    fn inverse(&self) -> Self;
+}
+```
+
+
+### Definition: Residue Class
+
+---
+
+*The residue class of \\( a \\) modulo \\( m \\), denoted as \\( a + m\mathbb{Z} \\), is the set \\( \\{b : b \equiv a \pmod{m}\\\} \\).*
+
+---
+
+**Example:** For \\( m = 3 \\), the residue class of 2 is \\( 2 + 3\mathbb{Z} = \\{\ldots, -4, -1, 2, 5, 8, \ldots\\} \\).
+
+### Definition: Inverse of Residue Class
+
+---
+
+*We denote the set of all residue classes modulo \\( m \\) as \\( \mathbb{Z} / m\mathbb{Z} \\). We say that \\( a + m\mathbb{Z} \\) is invertible in \\( \mathbb{Z} / m\mathbb{Z} \\) if and only if there exists a solution for \\( ax \equiv 1 \pmod{m} \\).*
+
+---
+
+**Example:** In \\( \mathbb{Z}/5\mathbb{Z} \\), \\( 3 + 5\mathbb{Z} \\) is invertible because \\( \gcd(3, 5) = 1 \\) (since \\( 3\cdot2 \equiv 1 \pmod{5} \\)). However, in \\( \mathbb{Z}/6\mathbb{Z} \\), \\( 3 + 6\mathbb{Z} \\) is not invertible because \\( \gcd(3, 6) = 3 \neq 1 \\).
+
+### Lemma 1.2.1 
+
+---
+
+*If \\( a \\) and \\( b \\) are coprime, the residues of \\( a \\), \\( 2a \\), \\( 3a \\), ..., \\( (b-1)a \\) modulo \\( b \\) are all distinct.*
+
+---
+
+**Proof:** Suppose, for contradiction, that there exist \\( x, y \in \\{1, 2, \dots, b-1\\} \\) with \\( x \neq y \\) such that \\( xa \equiv ya \pmod{b} \\). Then \\( (x-y)a \equiv 0 \pmod{b} \\), implying \\( b \mid (x-y)a \\). Since \\( a \\) and \\( b \\) are coprime, we must have \\( b \mid (x-y) \\). However, \\( |x-y| < b \\), so this is only possible if \\( x = y \\), contradicting our assumption. Therefore, all residues must be distinct.
+
+### Theorem 1.2.2
+
+---
+
+*For any integers \\( a \\) and \\( b \\), the equation \\( ax + by = 1 \\) has a solution in integers \\( x \\) and \\( y \\) if and only if \\( a \\) and \\( b \\) are coprime.*
+
+---
+
+**Proof:**  
+
+- **(\\( \Rightarrow \\))** Suppose \\( a \\) and \\( b \\) are not coprime, let \\( d = \gcd(a,b) > 1 \\). Then \\( d \mid a \\) and \\( d \mid b \\), so \\( d \mid (ax + by) \\) for any integers \\( x \\) and \\( y \\). Thus, \\( ax + by \neq 1 \\) for any \\( x \\) and \\( y \\).
+- **(\\( \Leftarrow \\))** Suppose \\( a \\) and \\( b \\) are coprime. By the previous lemma, the residues of \\( a \\), \\( 2a \\), ..., \\( (b-1)a \\) modulo \\( b \\) are all distinct. Therefore, there exists an \\( m \in \\{1, 2, ..., b-1\\} \\) such that \\( ma \equiv 1 \pmod{b} \\). This means there exists an integer \\( n \\) such that \\( ma = bn + 1 \\), or equivalently, \\( ma - bn = 1 \\).
+
+### Theorem: Bézout's Identity
+
+---
+
+*For any integers \\( a \\) and \\( b \\), we have:
+\\(
+a\mathbb{Z} + b \mathbb{Z} = \gcd(a, b)\mathbb{Z}
+\\)*
+
+---
+
+**Proof:** 
+
+This statement is equivalent to proving that \\( ax + by = c \\) has an integer solution if and only if \\( c \\) is a multiple of \\( \gcd(a,b) \\).
+
+- **(\\( \Rightarrow \\))** If \\( ax + by = c \\) for some integers \\( x \\) and \\( y \\), then \\( \gcd(a,b) \mid a \\) and \\( \gcd(a,b) \mid b \\), so \\( \gcd(a,b) \mid (ax + by) = c \\).
+- **(\\( \Leftarrow \\))** Let \\( c = k\gcd(a,b) \\) for some integer \\( k \\). We can write \\( a = p\gcd(a,b) \\) and \\( b = q\gcd(a,b) \\), where \\( p \\) and \\( q \\) are coprime. By the previous theorem, there exist integers \\( m \\) and \\( n \\) such that \\( pm + qn = 1 \\). Multiplying both sides by \\( k\gcd(a,b) \\), we get:
+  \[
+  akm + bkn = c
+  \]
+  Thus, \\( x = km \\) and \\( y = kn \\) are integer solutions to \\( ax + by = c \\).
+
+This theorem implies that for any integers \\( a \\), \\( b \\), and \\( n \\), the equation \\( ax + by = n \\) has an integer solution if and only if \\( \gcd(a,b) \mid n \\).
+
+### Theorem 1.2.3
+
+---
+
+*\\( a + m\mathbb{Z} \\) is invertible in \\( \mathbb{Z}/m\mathbb{Z} \\) if and only if \\( \gcd(a,m) = 1 \\).*
+
+---
+
+**Proof:**  
+
+- **(\\( \Rightarrow \\))** Suppose \\( a + m\mathbb{Z} \\) is invertible in \\( \mathbb{Z}/m\mathbb{Z} \\). Then there exists an integer \\( x \\) such that \\( ax \equiv 1 \pmod{m} \\). Let \\( g = \gcd(a,m) \\). Since \\( g \mid ax \\) and \\( g \mid (ax - 1) \\), we must have \\( g \mid 1 \\), so \\( g = 1 \\).
+- **(\\( \Leftarrow \\))** Suppose \\( \gcd(a,m) = 1 \\). By Bézout's identity, there exist integers \\( x \\) and \\( y \\) such that \\( ax + my = 1 \\). Thus, \\( x + m\mathbb{Z} \\) is the multiplicative inverse of \\( a + m\mathbb{Z} \\) in \\( \mathbb{Z}/m\mathbb{Z} \\).
+
+### Definition: Residue Class Ring
+
+*\\( (\mathbb{Z} / m \mathbb{Z}, +, \cdot) \\) is a commutative ring where \\( 1 + m \mathbb{Z} \\) is the multiplicative identity element. This ring is called the residue class ring modulo \\( m \\).*
+
+**Example:** \\( \mathbb{Z}/4\mathbb{Z} = \\{0 + 4\mathbb{Z}, 1 + 4\mathbb{Z}, 2 + 4\mathbb{Z}, 3 + 4\mathbb{Z}\\} \\).
+
+### Definition: Primitive Residue Class
+
+---
+
+*A residue class \\( a + m\mathbb{Z} \\) is called primitive if \\( \gcd(a, m) = 1 \\).*
+
+---
+
+**Example:** In \\( \mathbb{Z}/6\mathbb{Z} \\), the primitive residue classes are \\( 1 + 6\mathbb{Z} \\) and \\( 5 + 6\mathbb{Z} \\).
+
+### Theorem 1.2.4
+
+---
+
+*A residue ring \\( \mathbb{Z} / m\mathbb{Z} \\) is a field if and only if \\( m \\) is a prime number.*
+
+---
+
+**Proof:** TBD
+
+**Example:** For \\( m = 5 \\), \\( \mathbb{Z}/5\mathbb{Z} = \\{0 + 5\mathbb{Z}, 1 + 5\mathbb{Z}, 2 + 5\mathbb{Z}, 3 + 5\mathbb{Z}, 4 + 5\mathbb{Z}\\} \\) forms a field because 5 is a prime number, and every non-zero element has a multiplicative inverse.
+
+### Definition: Primitive Residue Class Group
+
+---
+
+*The group of all primitive residue classes modulo \\( m \\) is called the primitive residue class group, denoted by \\( (\mathbb{Z}/m\mathbb{Z})^{\times} \\).*
+
+---
+
+**Example:** For \\( m = 8 \\), the set of all primitive residue classes is \\( (\mathbb{Z}/8\mathbb{Z})^{\times} = \\{1 + 8\mathbb{Z}, 3 + 8\mathbb{Z}, 5 + 8\mathbb{Z}, 7 + 8\mathbb{Z}\\} \\). These are the integers less than 8 that are coprime to 8 (i.e., \\(\gcd(a, 8) = 1\\)).
+
+Contrast this with \\( m = 9 \\). The primitive residue class group is \\((\mathbb{Z}/9\mathbb{Z})^{\times} = \\{1 + 9\mathbb{Z}, 2 + 9\mathbb{Z}, 4 + 9\mathbb{Z}, 5 + 9\mathbb{Z}, 7 + 9\mathbb{Z}, 8 + 9\mathbb{Z}\\}\\), as these are the integers less than 9 that are coprime to 9.
+
+### Definition: Euler's Totient Function
+
+---
+
+*Euler's totient function \\(\phi(m)\\) is equal to the order of the primitive residue class group modulo \\(m\\), which is the number of integers less than \\(m\\) and coprime to \\(m\\).*
+
+---
+
+**Example**: For \\(m = 12\\), \\(\phi(12) = 4\\) because there are 4 integers less than 12 that are coprime to 12: \\(\{1, 5, 7, 11\}\\).
+
+For \\(m = 10\\), \\(\phi(10) = 4\\), as there are also 4 integers less than 10 that are coprime to 10: \\(\{1, 3, 7, 9\}\\).
+
+### Definition: Order of an Element within a Group
+
+---
+
+*The order of \\(g \in G\\) is the smallest natural number \\(e\\) such that \\(g^{e} = 1\\). We denote it as \\(\text{order}_g(g)\\)or \\(\text{order } g\\).*
+
+---
+
+**Example**: In \\((\mathbb{Z}/7\mathbb{Z})^{\times}\\), the element 3 has order 6 because \\(3^6 \bmod 7 = 1\\). In other words, \\(3 \times 3 \times 3 \times 3 \times 3 \times 3 \bmod 7 = 1\\), and 6 is the smallest such exponent.
+
+### Definition: Subgroup
+
+---
+
+*A subset \\(U \subseteq G\\) is a subgroup of \\(G\\) if \\(U\\) itself is a group under the operation of \\(G\\).*
+
+---
+
+**Example**: Consider \\((\mathbb{Z}/8\mathbb{Z})^{\times} = \\{1 + 8\mathbb{Z}, 3+ 8\mathbb{Z}, 5+ 8\mathbb{Z}, 7+ 8\mathbb{Z}\\}\\) under multiplication modulo 8. The subset \\(\{1+ 8\mathbb{Z}, 7+ 8\mathbb{Z}\}\\) forms a subgroup because it satisfies the group properties: closed under multiplication, contains the identity element (1), and every element has an inverse (\\(7 \times 7 \equiv 1 \bmod 8\\)).
+
+### Definition: Subgroup Generated by \\(g\\)
+
+---
+
+*The set \\(\\{g^{k} : k \in \mathbb{Z}\\}\\), for some element \\(g \in G\\), forms a subgroup of \\(G\\) and is called the subgroup generated by \\(g\\), denoted by \\(\langle g \rangle\\).*
+
+---
+
+**Example**: Consider the group \\((\mathbb{Z}/7\mathbb{Z})^{\times} = \\{1+ 7\mathbb{Z}, 2+ 7\mathbb{Z}, 3+ 7\mathbb{Z}, 4+ 7\mathbb{Z}, 5+ 7\mathbb{Z}, 6+ 7\mathbb{Z}\\}\\) under multiplication modulo 7. If we take \\(g = 3\\), then \\(\langle 3 +7\mathbb{Z} \rangle =\\) \\(\\{3^1+7\mathbb{Z}, 3^2+7\mathbb{Z}, 3^3+7\mathbb{Z}, 3^4+7\mathbb{Z}, 3^5+7\mathbb{Z}, 3^6+7\mathbb{Z}\\} \bmod 7 =\\) \\(\\{3+7\mathbb{Z}, 2+7\mathbb{Z}, 6+7\mathbb{Z}, 4+7\mathbb{Z}, 5+7\mathbb{Z}, 1+7\mathbb{Z}\\}\\), which forms a subgroup generated by 3. This subgroup contains all elements of \\((\mathbb{Z}/7\mathbb{Z})^{\times}\\), making 3 a generator of the entire group.
+
+If \\(g\\) has a finite order \\(e\\), we have that \\(\langle g \rangle = \\{g^{k}: 0 \leq k \leq e\\}\\), meaning \\(e\\) is the order of \\(\langle g \rangle\\).
+
+### Definition: Cyclic Group
+
+---
+
+*A group \\(G\\) is called a cyclic group if there exists an element \\(g \in G\\) such that \\(G = \langle g \rangle\\). In this case, \\(g\\) is called a generator of \\(G\\).*
+
+---
+
+**Example**: The group \\((\mathbb{Z}/6\mathbb{Z})^{\times} = \\{1+6\mathbb{Z}, 5+6\mathbb{Z}\\}\\) under multiplication modulo 6 is a cyclic group. In this case, both 1 and 5 are generators of the group because \\(\langle 5 +6\mathbb{Z} \rangle = \\{(5^1 \bmod 6)+6\mathbb{Z} = 5 +6\mathbb{Z}, (5^2 \bmod 6)+6\mathbb{Z} = 1+6\mathbb{Z}\\}\\). Since 5 generates all the elements of the group, \\(G\\) is cyclic.
+
+### Theorem 1.2.5
+
+---
+
+*If \\(G\\) is a finite cyclic group, it has \\(\phi(|G|)\\)generators, and each generator has order \\(|G|\\).*
+
+---
+
+**Proof**: TBD
+
+**Example**: Consider the group \\((\mathbb{Z}/8\mathbb{Z})^{\times} = \\{1+8\mathbb{Z}, 3+8\mathbb{Z}, 5+8\mathbb{Z}, 7+8\mathbb{Z}\\}\\). This group is cyclic, and \\(\phi(8) = 4\\). The generators of this group are \\(\\{1+8\mathbb{Z}, 3+8\mathbb{Z}, 5+8\mathbb{Z}, 7+8\mathbb{Z}\\}\\), each of which generates the entire group when raised to successive powers modulo 8. Each generator has the same order, which is \\(|G| = 4\\).
+
+### Theorem 1.2.6
+
+---
+
+*If \\(G\\) is a finite cyclic group, the order of any subgroup of \\(G\\) divides the order of \\(G\\).*
+
+---
+
+**Proof**: TBD
+
+**Example**: Consider the cyclic group \\((\mathbb{Z}/6\mathbb{Z})^{\times} = \\{1+6\mathbb{Z}, 5+6\mathbb{Z}\\}\\) under multiplication modulo 6. If we take the subgroup \\(\langle 5+6\mathbb{Z} \rangle = \\{1+6\mathbb{Z}, 5+6\mathbb{Z}\\}\\), this is a subgroup of order 2, and 2 divides the order of the original group, which is 6. This theorem generalizes this property: for any subgroup of a cyclic group, its order divides the order of the group.
+
+### Theorem: Fermat's Little Theorem
+
+---
+
+*If \\(\gcd(a, m) = 1\\), then \\(a^{\phi(m)} \equiv 1 \pmod{m}\\).*
+
+---
+
+**Proof**: TBD
+
+**Example**: Take \\(a = 2\\) and \\(m = 5\\). Since \\(\gcd(2, 5) = 1\\), Fermat's Little Theorem tells us that \\(2^{\phi(5)} = 2^4 \equiv 1 \bmod 5\\). Indeed, \\(2^4 = 16\\) and \\(16 \bmod 5 = 1\\).
+
+This theorem suggests that \\(a^{\phi(m) - 1} + m \mathbb{Z}\\) is the inverse residue class of \\(a + m \mathbb{Z}\\).
+
+### Theorem 1.2.7
+
+---
+
+*The order of any element in a group divides the order of the group.*
+
+---
+
+**Proof**: TBD
+
+**Example**: In the group \\((\mathbb{Z}/7\mathbb{Z})^{\times}\\), consider the element \\(3 + 7\mathbb{Z}\\). The order of \\(3 + 7\mathbb{Z}\\) is 6, as \\(3^6 \equiv 1 \bmod 7\\). The order of the group itself is also 6, and indeed, the order of the element divides the order of the group.
+
+### Theorem: Generalization of Fermat's Little Theorem
+
+---
+
+*For any element \\(g \in G\\), we have \\(g^{|G|} = 1\\).*
+
+---
+
+**Proof**: TBD
+
+**Example**: In the group \\((\mathbb{Z}/7\mathbb{Z})^{\times}\\), for any element \\(g\\), such as \\(g = 3 + 7\mathbb{Z}\\), we have \\(3^6 \equiv 1 \bmod 7\\). This holds for any \\(g \in (\mathbb{Z}/7\mathbb{Z})^{\times}\\) because the order of the group is 6. Thus, \\(g^{|G|} = 1\\) is satisfied.
+
+
+
+
diff --git a/docs/book/src/number_theory/subsec3.md b/docs/book/src/number_theory/subsec3.md
new file mode 100644
index 0000000..39839a5
--- /dev/null
+++ b/docs/book/src/number_theory/subsec3.md
@@ -0,0 +1,234 @@
+# Polynomials
+
+Let \\(K\\) be a field.
+
+### Definition: Polynomial
+
+*A univariate polynomial over a commutative ring \\(R\\) with unity \\(1\\) is an expression of the form \\(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\\), where \\(x\\) is a variable and coefficients \\(a_0, \ldots, a_n\\) belong to \\(R\\). The set of all polynomials over \\(R\\) in the variable \\(x\\) is denoted as \\(R[x]\\).*
+
+**Example:** In \\(\mathbb{Z}[x]\\), we have polynomials such as \\(2x^3 + x + 1\\), \\(x\\), and \\(1\\). In \\(\mathbb{R}[x]\\), we have polynomials like \\(\pi x^2 - \sqrt{2}x + e\\).
+
+```rust
+/// Polynomial struct representing a polynomial over Field.
+#[derive(Debug, Clone, PartialEq)]
+pub struct Polynomial<F: Field> {
+    pub poly: Vec<F>,
+    pub var: String,
+}
+```
+
+### Definition: Degree
+
+*The degree of a non-zero polynomial \\(f(x) = a_n x^{n} + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\\), denoted as \\(\deg f\\), is the largest integer \\(n\\) such that \\(a_n \neq 0\\). The zero polynomial is defined to have degree \\(-1\\).*
+
+**Example**
+
+- \\(\deg(2x^3 + x + 1) = 3\\)
+- \\(\deg(x) = 1\\)
+- \\(\deg(1) = 0\\)
+- \\(\deg(0) = -1\\)
+
+```rust
+impl<F: Field> Polynomial<F> {
+    /// Removes trailing zeroes from a polynomial's coefficients.
+    fn trim_trailing_zeros(poly: Vec<F>) -> Vec<F> {
+        let mut trimmed = poly;
+        while trimmed.last() == Some(&F::zero(None)) {
+            trimmed.pop();
+        }
+        trimmed
+    }
+
+    /// Returns the degree of the polynomial.
+    pub fn degree(&self) -> isize {
+        let trimmed = Self::trim_trailing_zeros(self.poly.clone());
+        if trimmed.is_empty() {
+            -1
+        } else {
+            (trimmed.len() - 1) as isize
+        }
+    }
+}
+```
+
+### Sum of Polynomials
+
+*For polynomials \\(f(x) = \sum_{i=0}^n a_i x^i\\) and \\(g(x) = \sum_{i=0}^m b_i x^i\\), their sum is defined as: \\((f + g)(x) = \sum_{i=0}^{\max(n,m)} (a_i + b_i) x^i\\), where we set \\(a_i = 0\\) for \\(i > n\\) and \\(b_i = 0\\) for \\(i > m\\).*
+
+**Example:** Let \\(f(x) = 2x^2 + 3x + 1\\) and \\(g(x) = x^3 - x + 4\\). Then, \\((f + g)(x) = x^3 + 2x^2 + 2x + 5\\)
+
+```rust
+impl<F: Field> Add for Polynomial<F> {
+    type Output = Self;
+
+    fn add(self, other: Self) -> Polynomial<F> {
+        let max_len = std::cmp::max(self.poly.len(), other.poly.len());
+        let mut result = Vec::with_capacity(max_len);
+
+        let zero = F::zero(None);
+
+        for i in 0..max_len {
+            let a = self.poly.get(i).unwrap_or(&zero);
+            let b = other.poly.get(i).unwrap_or(&zero);
+            result.push(a.clone() + b.clone());
+        }
+        Polynomial {
+            poly: Self::trim_trailing_zeros(result),
+            var: self.var.clone(),
+        }
+    }
+}
+```
+
+### Product of polynomials
+
+*For polynomials \\(f(x) = \sum_{i=0}^n a_i x^i\\) and \\(g(x) = \sum_{j=0}^m b_j x^j\\), their product is defined as: \\((fg)(x) = \sum_{k=0}^{n+m} c_k x^k\\), where \\(c_k = \sum_{i+j=k} a_i b_j\\)*
+
+**Example:** Let \\(f(x) = x + 1\\) and \\(g(x) = x^2 - 1\\). Then, \\((fg)(x) = x^3 + x^2 - x - 1\\)
+
+```rust
+impl<F: Field> Mul for Polynomial<F> {
+    type Output = Self;
+
+    /// Multiplication of two polynomials.
+    fn mul(self, other: Polynomial<F>) -> Polynomial<F> {
+        let mut result = vec![F::zero(None); self.degree() as usize + other.degree() as usize + 1];
+
+        for (i, a) in self.poly.iter().enumerate() {
+            for (j, b) in other.poly.iter().enumerate() {
+                result[i + j] = result[i + j].clone() + (a.clone() * b.clone());
+            }
+        }
+        Polynomial {
+            poly: Self::trim_trailing_zeros(result),
+            var: self.var.clone(),
+        }
+    }
+}
+```
+
+### Lemma
+
+*Let \\(f, g \in K[x]\\) be non-zero polynomials. Then, \\(\deg(fg) = \deg f + \deg g\\).*
+
+
+**Example:** Let \\(f(x) = x^2 + 1\\) and \\(g(x) = x^3 - x\\) in \\(\mathbb{R}[x]\\). Then, \\(\deg(fg) = \deg(x^5 - x^3 + x^2 + 1) = 5 = 2 + 3 = \deg f + \deg g\\)
+
+We can also define division in the polynomial ring \\(K[x]\\).
+
+### Theorem
+*Let \\(f, g \in K[x]\\), with \\(g \neq 0\\). There exist unique polynomials \\(q, r \in K[x]\\) that satisfy \\(f = qg + r\\) and either \\(\deg r < \deg g\\) or \\(r = 0\\).*
+
+**Proof**
+    TBD
+
+\\(q\\) is called the quotient of \\(f\\) divided by \\(g\\), and \\(r\\) is called the remainder; we write \\(r = f \bmod g\\).
+
+**Example:** In \\(\mathbb{R}[x]\\), let \\(f(x) = x^3 + 2x^2 - x + 3\\) and \\(g(x) = x^2 + 1\\).  Then \\(f = qg + r\\) where \\(q(x) = x + 2\\) and \\(r(x) = -3x + 1\\).
+
+
+```rust
+impl<F: Field> Div for Polynomial<F> {
+    type Output = Self;
+
+    /// Division of two polynomials, returns quotient.
+    fn div(self, other: Polynomial<F>) -> Polynomial<F> {
+        let mut remainder_coeffs = Self::trim_trailing_zeros(self.poly.clone());
+        let divisor_coeffs = Self::trim_trailing_zeros(other.poly.clone());
+        let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse();
+
+        let mut quotient =
+            vec![F::zero(None); self.degree() as usize - other.degree() as usize + 1];
+
+        let mut i = 0_i32;
+        while remainder_coeffs.len() >= divisor_coeffs.len() {
+            let lead_term = remainder_coeffs.last().unwrap().clone() * divisor_lead_inv.clone();
+            let deg_diff = remainder_coeffs.len() - divisor_coeffs.len();
+            quotient[deg_diff] = lead_term.clone();
+
+            for i in 0..divisor_coeffs.len() {
+                remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i].clone()
+                    - (lead_term.clone() * divisor_coeffs[i].clone());
+            }
+            remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs);
+            i += 1;
+        }
+
+        Polynomial {
+            poly: Self::trim_trailing_zeros(quotient),
+            var: self.var.clone(),
+        }
+    }
+}
+```
+
+### Corollary
+
+*Let \\(f \in K[x]\\) be a non-zero polynomial, and \\(a \in K\\) such that \\(f(a) = 0\\). Then, there exists a polynomial \\(q \in K[x]\\) such that \\(f(x) = (x - a)q(x)\\). In other words, \\((x - a)\\) is a factor of \\(f(x)\\).*
+
+**Example:** Let \\(f(x) = x^2 + 1 \in (\mathbb{Z}/2\mathbb{Z})[x]\\). We have \\(f(1) = 1^2 + 1 = 0\\) in \\(\mathbb{Z}/2\mathbb{Z}\\), and indeed: \\(x^2 + 1 = (x - 1)^2 = x^2 - 2x + 1 = x^2 + 1\\) in \\((\mathbb{Z}/2\mathbb{Z})[x]\\)
+
+### Theorem: Lagrange Interpolation
+
+*A \\(n\\)-degre polynomial \\(P(x)\\) that goes through different \\(n + 1\\) points \\(\\{(x_1, y_1), (x_2, y_2), \cdots (x_{n + 1}, y_{n + 1})\\}\\) is uniquely represented as follows:*
+
+\begin{align*}
+    P(x) = \sum^{n+1}_{i=1} y_i \frac{f_i(x)}{f_i(x_i)}
+\end{align*}
+
+*, where \\(f_i(x) = \prod_{k \neq i} (x - x_k)\\)*
+
+**Proof** TBD
+
+For example, the quadratic polynomial that goes through \\(\\{(1, 0), (2, 3), (3, 8)\\}\\) is as follows:
+
+\begin{align*}
+    P(x) = 0 \frac{(x - 2)(x - 3)}{(1 - 2) (1 - 3)} + 3 \frac{(x - 1)(x - 3)}{(2 - 1) (2 - 3)} + 8 \frac{(x - 1)(x - 2)}{(3 - 1) (3 - 2)} = x^{2} - 1
+\end{align*}
+
+Note that Lagrange interpolation finds the lowest degree of interpolating polynomial for the given vector.
+
+```rust
+pub fn interpolate(x_values: &[F], y_values: &[F]) -> Polynomial<F> {
+    let mut lagrange_polys = vec![];
+    let numerators = Polynomial::from_monomials(x_values);
+
+    for j in 0..x_values.len() {
+        let mut denominator = F::one(None);
+        for i in 0..x_values.len() {
+            if i != j {
+                denominator = denominator * (x_values[j].clone() - x_values[i].clone());
+            }
+        }
+        let cur_poly = numerators
+            .clone()
+            .div(Polynomial::from_monomials(&[x_values[j].clone()]).scalar_mul(&denominator));
+        lagrange_polys.push(cur_poly);
+    }
+
+    let mut result = Polynomial {
+        poly: vec![],
+        var: "x".to_string(),
+    };
+    for (j, lagrange_poly) in lagrange_polys.iter().enumerate() {
+        result = result + lagrange_poly.scalar_mul(&y_values[j]);
+    }
+    result
+}
+```
+
+### Proposition: Homomorphisms of Lagrange Interpolation
+
+*Let \\(L(v)\\) and \\(L(w)\\) be the polynomial resulting from Lagrange Interpolation on the output (\\(y\\)) vector \\(v\\) and \\(w\\) for the same inputs (\\(x\\)). Then, the following properties hold:*
+
+- *Additivity: \\(L(v + w) = L(v) + L(w)\\) for any vectors \\(v\\) and \\(w\\)*
+- *Scalar multiplication: \\(L(\gamma v) = \gamma L(v)\\) for any scalar \\(\gamma\\) and vector \\(v\\)*
+
+**Proof**
+    
+Let \\(v = (v_1, \ldots, v_n)\\) and \\(w = (w_1, \ldots, w_n)\\) be vectors, and \\(x_1, \ldots, x_n\\) be the interpolation points.
+
+\begin{align*}
+    L(v + w) &= \sum_{i=1}^n (v_i + w_i) \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \sum_{i=1}^n v_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} + \sum_{i=1}^n w_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = L(v) + L(w) \\\\
+    L(\gamma v) &= \sum_{i=1}^n (\gamma v_i) \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \gamma \sum_{i=1}^n v_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} = \gamma L(v)
+\end{align*}
diff --git a/docs/book/src/number_theory/subsec4.md b/docs/book/src/number_theory/subsec4.md
new file mode 100644
index 0000000..416c828
--- /dev/null
+++ b/docs/book/src/number_theory/subsec4.md
@@ -0,0 +1,76 @@
+# Galois Field
+
+We will now discuss the construction of finite fields with \\(p^n\\) elements, where \\(p\\) is a prime number and \\(n\\) is a positive integer. These fields are also known as Galois fields, denoted as \\(GF(p^n)\\). It is evident that \\(\mathbb{Z}/p\mathbb{Z}\\) is isomorphic to \\(GF(p)\\).
+
+
+### Definition: Irreducible Polynomial
+
+---
+
+*A polynomial \\(f \in (\mathbb{Z}/p\mathbb{Z})[X]\\) of degree \\(n\\) is called irreducible over \\(\mathbb{Z}/p\mathbb{Z}\\) if it cannot be factored as a product of two polynomials of lower degree in \\((\mathbb{Z}/p\mathbb{Z})[X]\\).*
+
+---
+
+**Example:** In \\((\mathbb{Z}/2\mathbb{Z})[X]\\):
+
+- \\(X^2 + X + 1\\) is irreducible
+- \\(X^2 + 1 = (X + 1)^2\\) is reducible
+
+
+To construct \\(GF(p^n)\\), we use an irreducible polynomial of degree \\(n\\) over \\(\mathbb{Z}/p\mathbb{Z}\\).
+
+### Definition: Residue Class modulo a Polynomial
+
+---
+
+*For \\(f, g \in (\mathbb{Z}/p\mathbb{Z})[X]\\), the residue class of \\(g \bmod f\\) is the set of all polynomials \\(h \in (\mathbb{Z}/p\mathbb{Z})[X]\\) such that \\(h \equiv g \pmod{f}\\). This class is denoted as:*
+
+\\[ g + f(\mathbb{Z}/p\mathbb{Z})[X] = \\{g + hf : h \in (\mathbb{Z}/p\mathbb{Z})[X]\\} \\]
+
+---
+
+**Example:** In \\((\mathbb{Z}/2\mathbb{Z})[X]\\), with \\(f(X) = X^2 + X + 1\\), the residue classes \\(\bmod f\\) are:
+
+- \\(0 + f(\mathbb{Z}/2\mathbb{Z})[X] = \\{0, X^2 + X + 1, X^2 + X, X^2 + 1, X^2, X + 1, X, 1\\}\\)
+- \\(1 + f(\mathbb{Z}/2\mathbb{Z})[X] = \\{1, X^2 + X, X^2 + 1, X^2, X + 1, X, 0, X^2 + X + 1\\}\\)
+- \\(X + f(\mathbb{Z}/2\mathbb{Z})[X] = \\{X, X^2 + 1, X^2, X^2 + X + 1, X + 1, 1, X^2 + X, 0\\}\\)
+- \\((X + 1) + f(\mathbb{Z}/2\mathbb{Z})[X] = \\{X + 1, X^2, X^2 + X + 1, X^2 + X, 1, 0, X^2 + 1, X\\}\\)
+
+These four residue classes form \\(GF(4)\\).
+
+### Theorem:
+
+---
+
+*If \\(f \in (\mathbb{Z}/p\mathbb{Z})[X]\\) is an irreducible polynomial of degree \\(n\\), then the residue ring \\((\mathbb{Z}/p\mathbb{Z})[X]/(f)\\) is a field with \\(p^n\\) elements, isomorphic to \\(GF(p^n)\\).*
+
+---
+
+**Proof** (Outline) The irreducibility of \\(f\\) ensures that \\((f)\\) is a maximal ideal in \\((\mathbb{Z}/p\mathbb{Z})[X]\\), making the quotient ring a field. The number of elements is \\(p^n\\) because there are \\(p^n\\) polynomials of degree less than \\(n\\) over \\(\mathbb{Z}/p\mathbb{Z}\\).
+
+This construction allows us to represent elements of \\(GF(p^n)\\) as polynomials of degree less than \\(n\\) over \\(\mathbb{Z}/p\mathbb{Z}\\). Addition is performed coefficient-wise modulo \\(p\\), while multiplication is performed modulo the irreducible polynomial \\(f\\).
+
+
+**Example:** To construct \\(GF(8)\\), we can use the irreducible polynomial \\(f(X) = X^3 + X + 1\\) over \\(\mathbb{Z}/2\mathbb{Z}\\). The elements of \\(GF(8)\\) are represented by:
+\[ \\{0, 1, X, X+1, X^2, X^2+1, X^2+X, X^2+X+1\\} \]
+For instance, multiplication in \\(GF(8)\\):
+\[ (X^2 + 1)(X + 1) = X^3 + X^2 + X + 1 \equiv X^2 \pmod{X^3 + X + 1} \]
+
+### Lemma: Schwartz - Zippel Lemma
+
+---
+
+*Let \\(\mathbb{F}\\) be a field and \\(P: F^m \rightarrow \mathbb{F}\\) and \\(Q: \mathbb{F}^m \rightarrow \mathbb{F}\\) be two distinct multivariate polynomials of total degree at most \\(n\\). For any finite subset \\(\mathbb{S} \subseteq \mathbb{F}\\), we have:*
+
+\begin{align}
+    Pr_{u \sim \mathbb{S}^{m}}[P(u) = Q(u)] \leq \frac{n}{|\mathbb{S}|}
+\end{align}
+
+*, where \\(u\\) is drawn uniformly at random from \\(\mathbb{S}^m\\).*
+
+---
+
+**Proof** TBD
+
+
+This lemma states that if \\(\mathbb{S}\\) is sufficiently large and \\(n\\) is relatively small, the probability that the two different polynomials return the same value for a randomly chosen input is negligibly small. In other words, if we observe \\(P(u) = Q(u)\\) for a random input \\(u\\), we can conclude with high probability that \\(P\\) and \\(Q\\) are identical polynomials.
diff --git a/docs/book/src/number_theory/subsec5.md b/docs/book/src/number_theory/subsec5.md
new file mode 100644
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+++ b/docs/book/src/number_theory/subsec5.md
@@ -0,0 +1,625 @@
+# Elliptic curve
+
+### Definition: Elliptic Curve
+    
+---
+
+*An elliptic curve \\(E\\) over a finite field \\(\mathbb{F}\_{p}\\) is defined by the equation:*
+
+\begin{equation}
+    y^2 = x^3 + a x + b
+\end{equation}
+    
+*, where \\(a, b \in \mathbb{F}\_{p}\\) and the discriminant \\(\Delta_{E} = 4a^3 + 27b^2 \neq 0\\).*
+
+---
+
+```rust
+use crate::modules::field::Field;
+
+#[derive(Debug, Clone, PartialEq)]
+pub struct EllipticCurve<F: Field> {
+    pub a: F,
+    pub b: F,
+}
+```
+
+### Definition: \\(\mathbb{F}\_{p}\\)-Rational Point
+
+---
+
+*An \\(\mathbb{F}\_{p}\\)-rational point on an elliptic curve \\(E\\) is a point \\((x, y)\\) where both \\(x\\) and \\(y\\) are elements of \\(\mathbb{F}\_{p}\\) and satisfy the curve equation, or the point at infinity \\(\mathcal{O}\\).*
+
+---
+
+Example: For \\(E: y^2 = x^3 + 3x + 4\\) over \\(\mathbb{F}\_7\\), the \\(\mathbb{F}\_{7}\\)-rational points are:
+\\(\\{(0, 2), (0, 5), (1, 1), (1, 6), (2, 2), (2, 5), (5, 1), (5, 6), \mathcal{O}\\}\\)
+
+### Definition: The point at infinity
+
+---
+
+The point at infinity denoted \\(\mathcal{O}\\), is a special point on the elliptic curve that serves as the identity element for the group operation. It can be visualized as the point where all vertical lines on the curve meet.
+
+---
+
+```rust
+#[derive(Debug, Clone, PartialEq)]
+pub struct EllipticCurvePoint<F: Field> {
+    pub x: Option<F>,
+    pub y: Option<F>,
+    pub curve: EllipticCurve<F>,
+}
+```
+
+### Definition: Addition on Elliptic Curve
+
+---
+
+*For an elliptic curve \\(E: y^2 = x^3 + ax + b\\), the addition of points \\(P\\) and \\(Q\\) to get \\(R = P + Q\\) is defined as follows:*
+
+   
+   - *If \\(P = \mathcal{O}\\), \\(R = Q\\).*
+   - *If \\(Q = \mathcal{O}\\), \\(R = P\\).*
+   - *Otherwise, let \\(P = (x_P, y_P), Q = (x_Q, y_Q)\\). Then:*
+       
+       - *If \\(y_P = -y_Q\\), \\(R = \mathcal{O}\\)*
+       - *If \\(y_P \neq -y_Q\\), \\(R = (x_R = \lambda^2 - x_P - x_Q, y_R = \lambda(x_P - x_R) - y_P)\\), where \\(\lambda = \\)* \\( \begin{cases}
+               \frac{y_P - y_Q}{x_P - x_Q} \quad &\hbox{If } (x_P \neq x_Q) \\\\
+               \frac{3^{2}_{P} + a}{2y_P} \quad &\hbox{Otherwise}
+           \end{cases}\\)
+       
+---
+
+
+Example: On \\(E: y^2 = x^3 + 2x + 3\\) over \\(\mathbb{F}_{7}\\), let \\(P = (5, 1)\\) and \\(Q = (4, 4)\\). Then, \\(P + Q = (0, 5)\\), where \\(\lambda = \frac{1 - 4}{5 - 4} \equiv 4 \bmod 7\\).
+
+```rust
+impl<F: Field> EllipticCurvePoint<F> {
+    fn new(x: F, y: F, curve: EllipticCurve<F>) -> Self {
+        EllipticCurvePoint {
+            x: Some(x),
+            y: Some(y),
+            curve: curve,
+        }
+    }
+
+    pub fn point_at_infinity(curve: EllipticCurve<F>) -> Self {
+        EllipticCurvePoint {
+            x: None,
+            y: None,
+            curve: curve,
+        }
+    }
+
+    pub fn is_point_at_infinity(&self) -> bool {
+        self.x.is_none() || self.y.is_none()
+    }
+
+    pub fn line_slope(&self, other: Self) -> F {
+        let x1 = self.x.clone().unwrap();
+        let y1 = self.y.clone().unwrap();
+        let x2 = other.x.clone().unwrap();
+        let y2 = other.y.clone().unwrap();
+
+        if self.x.clone() == other.x.clone() {
+            ((x1.clone() * x1.clone()) * (3_i64) + self.curve.a.clone()) / (y1.clone() * (2_i64))
+        } else {
+            (y2.clone() - y1.clone()) / (x2.clone() - x1.clone())
+        }
+    }
+}
+
+impl<F: Field> Add for EllipticCurvePoint<F> {
+    type Output = Self;
+
+    fn add(self, other: Self) -> Self {
+        if self.is_point_at_infinity() {
+            return other;
+        }
+        if other.is_point_at_infinity() {
+            return self;
+        }
+
+        let m = self.line_slope(other.clone());
+
+        if self.x == other.x {
+            if self.y != other.y {
+                return EllipticCurvePoint::point_at_infinity(self.curve.clone());
+            } else {
+                let x1 = self.x.clone().unwrap();
+                let y1 = self.y.clone().unwrap();
+
+                let x3 = m.clone() * m.clone() - x1.clone() - x1.clone();
+                let y3 = m * (x1 - x3.clone()) - y1;
+
+                return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+            }
+        } else {
+            let x1 = self.x.clone().unwrap();
+            let y1 = self.y.clone().unwrap();
+            let x2 = other.x.clone().unwrap();
+            let x3 = m.clone() * m.clone() - x1.clone() - x2.clone();
+            let y3 = m * (x1 - x3.clone()) - y1;
+
+            return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+        }
+    }
+}
+```
+
+### Definition: Mordell-Weil Group
+
+---
+
+*The Mordell-Weil group of an elliptic curve \\(E\\) is the group of rational points on \\(E\\) under the addition operation defined above.*
+
+---
+
+Example: For \\(E: y^2 = x^3 + x + 6\\) over \\(\mathbb{F}\_{11}\\), the Mordell-Weil group is the set of all \\(\mathbb{F}\_{11}\\)-rational points on \\(E\\) with the elliptic curve addition operation.
+
+### Definition: Group Order
+
+---
+
+*The group order of an elliptic curve \\(E\\) over \\(\mathbb{F}\_{p}\\), denoted \\(\\#E(\mathbb{F}\_{p})\\), is the number of \\(\mathbb{F}\_{p}\\)-rational points on \\(E\\), including the point at infinity.*
+
+---
+
+Example: For \\(E: y^2 = x^3 + x + 6\\) over \\(\mathbb{F}\_{11}\\), \\(\\#E(\mathbb{F}\_{11}) = 13\\).
+
+### Theorem: Hasse-Weil
+
+---
+
+*Let \\(\\#E(\mathbb{F}\_{p})\\) be the group order of the elliptic curve \\(E\\) over \\(\mathbb{F}\_{p}\\). Then:*
+
+\begin{equation}
+    p + 1 - 2 \sqrt{p} \leq \\#E \leq p + 1 + 2 \sqrt{p}
+\end{equation}
+
+---
+
+**Example:** For an elliptic curve over \\(\mathbb{F}_{23}\\), the Hasse-Weil theorem guarantees that: 
+
+\begin{equation*}
+23 + 1 - 2 \sqrt{23} \simeq 14.42 \\#E(\mathbb{F}_{23}) \geq 23 + 1 + 2 \sqrt{23} \simeq 33.58
+\end{equation*}
+
+### Definition: Point Order
+
+
+The order of a point \\(P\\) on an elliptic curve is the smallest positive integer \\(n\\) such that \\(nP = \mathcal{O}\\) (where \\(nP\\) denotes \\(P\\) added to itself \\(n\\) times). We also denote the set of points of order \\(n\\), also called \textit{torsion} group, by 
+
+\begin{equation}
+    E[n] = \\{P \in E: [n]P = \mathcal{O}\\}
+\end{equation}
+
+
+Example: On \\(E: y^2 = x^3 + 2x + 2\\) over \\(\mathbb{F}_{17}\\), the point \\(P = (5, 1)\\) has order 18 because \\(18P = \mathcal{O}\\), and no smaller positive multiple of \\(P\\) equals \\(\mathcal{O}\\).
+
+The intuitive view is that if you continue to add points to themselves (doubling, tripling, etc.), the lines drawn between the prior point and the next point will eventually become more vertical. When the line becomes vertical, it does not intersect the elliptic curve at any finite point. In elliptic curve geometry, a vertical line is considered to "intersect" the curve at a special place called the "point at infinity," This point is like a north pole in geographic terms: no matter which direction you go, if the line becomes vertical (reaching infinitely high), it converges to this point.
+
+### Definition: Field Extension
+
+---
+
+*Let \\(F\\) and \\(L\\) be fields. If \\(F \subseteq L\\) and the operations of \\(F\\) are the same as those of \\(L\\), we call \\(L\\) a field extension of \\(F\\). This is denoted as \\(L/F\\).*
+
+---
+
+A field extension \\(L/F\\) naturally gives \\(L\\) the structure of a vector space over \\(F\\). The dimension of this vector space is called the degree of the extension.
+
+**Examples:**
+
+- \\(\mathbb{C}/\mathbb{R}\\) is a field extension with basis \\(\\{1, i\\}\\) and degree 2.
+- \\(\mathbb{R}/\mathbb{Q}\\) is an infinite degree extension.
+- \\(\mathbb{Q}(\sqrt{2})/\mathbb{Q}\\) is a degree 2 extension with basis \\(\\{1, \sqrt{2}\\}\\).
+
+
+### Definition: Algebraic Extension
+
+---
+
+*A field extension \\(L/K\\) is called algebraic if every element \\(\alpha \in L\\) is algebraic over \\(K\\), i.e., \\(\alpha\\) is the root of some non-zero polynomial with coefficients in \\(K\\).*
+
+*For an algebraic element \\(\alpha \in L\\) over \\(K\\), we denote by \\(K(\alpha)\\) the smallest field containing both \\(K\\) and \\(\alpha\\).*
+
+---
+
+**Example:**
+
+- \\(\mathbb{C}/\mathbb{R}\\) is algebraic: any \\(z = a + bi \in \mathbb{C}\\) is a root of \\(x^2 - 2ax + (a^2 + b^2) \in \mathbb{R}[x]\\).
+- \\(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}\\) is algebraic: \\(\sqrt[3]{2}\\) is a root of \\(x^3 - 2 \in \mathbb{Q}[x]\\).
+- \\(\mathbb{R}/\mathbb{Q}\\) is not algebraic (a field extension that is not algebraic is called \textit{transcendental}).
+
+
+### Definition: Field of Rational Functions
+
+---
+
+*Let \\(K\\) be a field and \\(X\\) be indeterminate. The field of rational functions over \\(K\\), denoted \\(K(X)\\), is defined as:*
+
+\begin{equation*}
+    K(X) = \left\\{ \frac{f(X)}{g(X)} \middle|\ f(X), g(X) \in K[X], g(X) \neq 0 \right\\}
+\end{equation*}
+
+*where \\(K[X]\\) is the ring of polynomials in \\(X\\) with coefficients in \\(K\\).*
+
+---
+
+\\(K(X)\\) can be viewed as the field obtained by adjoining a letter \\(X\\) to \\(K\\). This construction generalizes to multiple variables, e.g., \\(K(X,Y)\\).
+
+The concept of a function field naturally arises in the context of algebraic curves, particularly elliptic curves. Intuitively, the function field encapsulates the algebraic structure of rational functions on the curve.
+
+We first construct the coordinate ring for an elliptic curve \\(E: y^2 = x^3 + ax + b\\). Consider functions \\(X: E \to K\\) and \\(Y: E \to K\\) that extract the \\(x\\) and \\(y\\) coordinates, respectively, from an arbitrary point \\(P \in E\\). These functions generate the polynomial ring \\(K[X,Y]\\), subject to the relation \\(Y^2 = X^3 + aX + b\\).
+
+To put it simply, the function field is a field that consists of all functions that determine the value based on the point on the curve.
+
+### Definition: Coordinate Ring of an Elliptic Curve
+
+---
+
+*The **coordinate ring** of an elliptic curve \\(E: y^2 = x^3 + ax + b\\) over a field \\(K\\) is defined as:*
+\begin{equation}
+    K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)
+\end{equation}
+
+---
+
+In other words, we can view \\(K[E]\\) as a ring representing all polynomial functions on \\(E\\). Recall that \\(K[X, Y]\\) is the polynomial ring in two variables \\(X\\) and \\(Y\\) over the field K, meaning that it contains all polynomials in \\(X\\) and \\(Y\\) with coefficients from \\(K\\). Then, the notation \\(K[X, Y]/(Y^2 - X^3 - aX - b)\\) denotes the quotient ring obtained by taking \\(K[X, Y]\\) and "modding out" by the ideal \\((Y^2 - X^3 - aX - b)\\).
+
+**Example** For example, for an elliptic curve \\(E: y^2 = x^3 - x\\) over \\(\mathbb{Q}\\), some elements of the coordinate ring \\(\mathbb{Q}[E]\\) include:
+
+
+- Constants: \\(3, -2, \frac{1}{7}, \ldots\\)
+- Linear functions: \\(X, Y, 2X+3Y, \ldots\\)
+- Quadratic functions: \\(X^2, XY, Y^2 (= X^3 - X), \ldots\\)
+- Higher-degree functions: \\(X^3, X^2Y, XY^2 (= X^4 - X^2), \ldots\\)
+
+
+Then, the function field is defined as follows:
+
+### Definition: Function Field of an Elliptic Curve
+
+---
+
+*Let \\(E: y^2 = x^3 + ax + b\\) be an elliptic curve over a field \\(K\\). The **function field** of \\(E\\), denoted \\(K(E)\\), is defined as:*
+\begin{equation}
+    K(E) = \left\\{\frac{f}{g} \,\middle|\, f, g \in K[E], g \neq 0 \right\\}
+\end{equation}
+*where \\(K[E] = K[X, Y]/(Y^2 - X^3 - aX - b)\\) is the coordinate ring of \\(E\\).*
+
+---
+
+\\(K(E)\\) can be viewed as the field of all rational functions on \\(E\\).
+
+### Definition: Zero of a Function
+
+---
+
+*Let \\(h \in K(E)\\) be a non-zero function. A point \\(P \in E\\) is called a **zero** of \\(h\\) if \\(h(P) = 0\\).*
+
+---
+
+### Definition: Pole of a Function
+
+---
+
+*Let \\(h \in K(E)\\) be a non-zero function. A point \\(P \in E\\) is called a **pole** of \\(h\\) if \\(h\\) is not defined at \\(P\\) or, equivalently if \\(1/h\\) has a zero at \\(P\\).*
+
+---
+
+**Example** Consider the elliptic curve \\(E: Y^2 = X^3 - X\\) over a field \\(K\\) of characteristic \\(\neq 2, 3\\). Let \\(P_{-1} = (-1, 0)\\), \\(P_0 = (0, 0)\\), and \\(P_1 = (1, 0)\\) be the points where \\(Y = 0\\).
+
+
+- The function \\(Y \in K(E)\\) has three simple zeros: \\(P_{-1}\\), \\(P_0\\), and \\(P_1\\).
+- The function \\(X \in K(E)\\) has a double zero at \\(P_0\\) (since \\(P_0 = -P_0\\)).
+- The function \\(X - 1 \in K(E)\\) has a simple zero at \\(P_1\\).
+- The function \\(X^2 - 1 \in K(E)\\) has two simple zeros at \\(P_{-1}\\) and \\(P_1\\).
+- The function \\(\frac{Y}{X} \in K(E)\\) has a simple zero at \\(P_{-1}\\) and a simple pole at \\(P_0\\).
+
+
+An important property of functions in \\(K(E)\\) is that they have the same number of zeros and poles when counted with multiplicity. This is a consequence of a more general result known as the Degree-Genus Formula.
+
+### Theorem: Degree-Genus Formula for Elliptic Curves
+
+---
+
+*Let \\(f \in K(E)\\) be a non-zero rational function on an elliptic curve \\(E\\). Then:*
+
+\begin{equation}
+    \sum_{P \in E} ord_{P(f)} = 0
+\end{equation}
+
+*, where \\(ord_{P(f)}\\) denotes the order of \\(f\\) at \\(P\\), which is positive for zeros and negative for poles.*
+
+---
+
+This theorem implies that the total number of zeros (counting multiplicity) equals the total number of poles for any non-zero function in \\(K(E)\\).
+
+We now introduce a powerful tool for analyzing functions on elliptic curves: the concept of divisors.
+
+### Definition: Divisor of a Function on an Elliptic Curve
+
+---
+
+*Let \\(E: Y^2 = X^3 + AX + B\\) be an elliptic curve over a field \\(K\\), and let \\(f \in K(E)\\) be a non-zero rational function on \\(E\\). The **divisor** of \\(f\\), denoted \\(div(f)\\), is defined as:*
+\begin{equation}
+    div(f) = \sum_{P \in E} ord_P(f) [P]
+\end{equation}
+*, where \\(ord_P(f)\\) is the order of \\(f\\) at \\(P\\) (positive for zeros, negative for poles), and the sum is taken over all points \\(P \in E\\), including the point at infinity. This sum has only finitely many non-zero terms.*
+
+
+---
+
+Note that this \\(\sum_{P \in E}\\) is a symbolic summation, and we do not calculate the concrete value of a divisor.
+
+**Example** Consider the elliptic curve \\(E: Y^2 = X^3 - X\\) over \\(\mathbb{Q}\\).
+
+- For \\(f = X\\), we have \\(div(X) = 2[(0,0)] - 2[\mathcal{O}]\\).
+- For \\(g = Y\\), we have \\(div(Y) = [(1,0)] + [(0,0)] + [(-1,0)] - 3[\mathcal{O}]\\).
+- For \\(h = \frac{X-1}{Y}\\), we have \\(div(h) = [(1,0)] - [(-1,0)]\\).
+
+
+The concept of divisors can be extended to the elliptic curve itself:
+
+### Definition: Divisor on an Elliptic Curve
+
+---
+
+*A **divisor** \\(D\\) on an elliptic curve \\(E\\) is a formal sum*
+\begin{equation}
+    D = \sum_{P \in E} n_P [P]
+\end{equation}
+*where \\(n_P \in \mathbb{Z}\\) and \\(n_P = 0\\) for all but finitely many \\(P\\).*
+
+---
+
+**Example** On the curve \\(E: Y^2 = X^3 - X\\):
+
+- \\(D_1 = 3[(0,0)] - 2[(1,1)] - [(2,\sqrt{6})]\\) is a divisor.
+- \\(D_2 = [(1,0)] + [(-1,0)] - 2[\mathcal{O}]\\) is a divisor.
+- \\(D_3 = \sum_{P \in E[2]} [P] - 4[\mathcal{O}]\\) is a divisor (where \\(E[2]\\) are the 2-torsion points).
+
+
+To quantify the properties of divisors, we introduce two important metrics:
+
+### Definition: Degree of a Divisor
+
+---
+
+*The **degree** of a divisor \\(D = \sum_{P \in E} n_P [P]\\) is defined as:*
+\begin{equation}
+    deg(D) = \sum_{P \in E} n_P
+\end{equation}
+
+---
+
+### Definition: Sum of a Divisor
+
+---
+
+*The **sum** of a divisor \\(D = \sum_{P \in E} n_P [P]\\) is defined as:*
+\begin{equation}
+    Sum(D) = \sum_{P \in E} n_P P
+\end{equation}
+*where \\(n_P P\\) denotes the point addition of \\(P\\) to itself \\(n_P\\) times in the group law of \\(E\\).*
+
+---
+
+**Example** For the divisors in the previous example:
+
+- \\(deg(D_1) = 3 - 2 - 1 = 0\\)
+- \\(deg(D_2) = 1 + 1 - 2 = 0\\)
+- \\(deg(D_3) = 4 - 4 = 0\\)
+- \\(Sum(D_2) = (1,0) + (-1,0) - 2\mathcal{O} = \mathcal{O}\\) (since \\((1,0)\\) and \\((-1,0)\\) are 2-torsion points)
+
+
+The following theorem characterizes divisors of functions and provides a criterion for when a divisor is the divisor of a function:
+
+### Theorem 
+
+---
+
+*Let \\(E\\) be an elliptic curve over a field \\(K\\).*
+
+
+- *If \\(f, f' \in K(E)\\) are non-zero rational functions on \\(E\\) with \\(div(f) = div(f')\\), then there exists a non-zero constant \\(c \in K^*\\) such that \\(f = cf'\\).*
+- *A divisor \\(D\\) on \\(E\\) is the divisor of a rational function on \\(E\\) if and only if \\(deg(D) = 0\\) and \\(Sum(D) = \mathcal{O}\\).*
+
+
+---
+
+**Example** On \\(E: Y^2 = X^3 - X\\):
+
+- The function \\(f = \frac{Y}{X}\\) has \\(div(f) = [(1,0)] + [(-1,0)] - 2[(0,0)]\\). Note that \\(deg(div(f)) = 0\\) and \\(Sum(div(f)) = (1,0) + (-1,0) - 2(0,0) = \mathcal{O}\\).
+- The divisor \\(D = 2[(1,1)] - [(2,\sqrt{6})] - [(0,0)]\\) has \\(deg(D) = 0\\), but \\(Sum(D) \neq \mathcal{O}\\). Therefore, \\(D\\) is not the divisor of any rational function on \\(E\\).
+
+
+### Definition: Pairing
+
+---
+
+*Let \\(G_1\\) and \\(G_2\\) be cyclic groups under addition, both of prime order \\(p\\), with generators \\(P\\) and \\(Q\\) respectively:*
+
+\begin{align}
+    G_1 &= \\{0, P, 2P, ..., (p-1)P\\} \\
+    G_2 &= \\{0, Q, 2Q, ..., (p-1)Q\\}
+\end{align}
+
+*Let \\(G_T\\) be a cyclic group under multiplication, also of order \\(p\\). A pairing is a map \\(e: G_1 \times G_2 \rightarrow G_T\\) that satisfies the following bilinear property:*
+
+\begin{equation}
+    e(aP, bQ) = e(P, Q)^{ab}
+\end{equation} *for all \\(a, b \in \mathbb{Z}_p\\).*
+
+---
+
+Imagine \\(G_1\\) represents length, \\(G_2\\) represents width, and \\(G_T\\) represents area. The pairing function \\(e\\) is like calculating the area: If you double the length and triple the width, the area becomes six times larger: \\(e(2P, 3Q) = e(P, Q)^{6}\\)
+
+The Weil pairing is one of the bilinear pairings for elliptic curves. We begin with its formal definition.
+
+### Definition: The Weil Pairing
+
+---
+
+*Let \\(E\\) be an elliptic curve and \\(n\\) be a positive integer. For points \\(P, Q \in E[n]\\), where \\(E[n]\\) denotes the \\(n\\)-torsion subgroup of \\(E\\), we define the Weil pairing \\(e_n(P, Q)\\) as follows:*
+
+Let \\(f_P\\) and \\(f_Q\\) be rational functions on \\(E\\) satisfying:
+\begin{align}
+    div(f_P) &= n[P] - n[\mathcal{O}] \\
+    div(f_Q) &= n[Q] - n[\mathcal{O}]
+\end{align}
+
+*Then, for an arbitrary point \\(S \in E\\) such that \\(S \notin \\{\mathcal{O}, P, -Q, P-Q\\}\\), the Weil pairing is given by:*
+
+\begin{equation}
+e_n(P, Q) = \frac{f_P(Q + S)}{f_P(S)} /\ \frac{f_Q(P - S)}{f_Q(-S)}
+\end{equation}
+
+---
+
+We introduce a crucial theorem about a specific rational function on elliptic curves to construct the functions required for the Weil pairing.
+
+### Theorem
+
+---
+
+*Let \\(E\\) be an elliptic curve over a field \\(K\\), and let \\(P = (x_P, y_P)\\) and \\(Q = (x_Q, y_Q)\\) be non-zero points on \\(E\\). Define \\(\lambda\\) as:*
+
+\begin{equation}
+    \lambda = \begin{cases}
+        \hbox{slope of the line through \\(P\\) and \\(Q\\)} &\quad \hbox{if \\(P \neq Q\\)} \\\\
+        \hbox{slope of the tangent line to \\(E\\) at \\(P\\)} &\quad \hbox{if \\(P = Q\\)} \\\\
+        \infty &\quad \hbox{if the line is vertical}
+    \end{cases}
+\end{equation}
+
+*Then, the function \\(g_{P,Q}: E \to K\\) defined by:*
+\begin{equation}
+g_{P,Q} = \begin{cases}
+\frac{y - y_P - \lambda(x - x_P)}{x + x_P + x_Q - \lambda^2} &\quad \hbox{if } \lambda \neq \infty \\\\
+x - x_P &\quad \hbox{if } \lambda = \infty
+\end{cases}
+\end{equation} *has the following divisor:*
+
+\begin{equation}
+div(g_{P,Q}) = [P] + [Q] - [P + Q] - [\mathcal{O}]
+\end{equation}
+
+---
+
+**Proof:** We consider two cases based on the value of \\(\lambda\\).
+
+Case 1: \\(\lambda \neq \infty\\)
+
+Let \\(y = \lambda x + v\\) be the line through \\(P\\) and \\(Q\\) (or the tangent line at \\(P\\) if \\(P = Q\\)). This line intersects \\(E\\) at three points: \\(P\\), \\(Q\\), and \\(-P-Q\\). Thus,
+\begin{equation}
+div(y - \lambda x - v) = [P] + [Q] + [-P - Q] - 3[\mathcal{O}]
+\end{equation}
+Vertical lines intersect \\(E\\) at points and their negatives, so:
+\begin{equation}
+div(x - x_{P+Q}) = [P + Q] + [-P - Q] - 2[\mathcal{O}]
+\end{equation}
+It follows that \\(g_{P,Q} = \frac{y - \lambda x - v}{x - x_{P+Q}}\\) has the desired divisor.
+
+Case 2: \\(\lambda = \infty\\)
+
+In this case, \\(P + Q = \mathcal{O}\\), so we want \\(g_{P,Q}\\) to have divisor \\([P] + [-P] - 2[\mathcal{O}]\\). The function \\(x - x_P\\) has this divisor.
+\end{proof}
+
+### Theorem: Miller's Algorithm
+
+---
+
+*Let \\(m \geq 1\\) and write its binary expansion as:*
+\begin{equation}
+m = m_0 + m_1 \cdot 2 + m_2 \cdot 2^2 + \cdots + m_{n-1} \cdot 2^{n-1}
+\end{equation}
+*where \\(m_i \in \\{0, 1\\}\\) and \\(m_{n-1} \neq 0\\).*
+
+*The following algorithm, using the function \\(g_{P,Q}\\) defined in the previous theorem, returns a function \\(f_P\\) whose divisor satisfies:*
+
+\begin{equation}
+    div(f_P) = m[P] - m[P] - (m - 1)[\mathcal{O}]
+\end{equation}
+
+===========================
+
+**Miller's Algorithm**
+
+1. Set \\(T = P\\) and \\(f = 1\\)
+2. **For** \\(i \gets n-2 \cdots 0\\) **do**
+3. &nbsp;&nbsp;&nbsp;&nbsp;Set \\(f = f^2 \cdot g_{T, T}\\)
+4. &nbsp;&nbsp;&nbsp;&nbsp;Set \\(T = 2T\\)
+5. &nbsp;&nbsp;&nbsp;&nbsp;**If** \\(m_i = 1\\) **then**
+6. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Set \\(f = f \cdot g_{T, P}\\)
+7. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Set \\(T = T + P\\)
+8. &nbsp;&nbsp;&nbsp;&nbsp;**End if**
+9. **End for**
+10. **Return** \\(f\\)
+    
+===========================
+
+---
+
+**Proof:** TBD
+
+```rust
+pub fn get_lambda<F: Field>(
+    p: EllipticCurvePoint<F>,
+    q: EllipticCurvePoint<F>,
+    r: EllipticCurvePoint<F>,
+) -> F {
+    let p_x = p.x.clone().unwrap();
+    let p_y = p.y.clone().unwrap();
+    let q_x = q.x.clone().unwrap();
+    // let q_y = q.y.clone().unwrap();
+    let r_x = r.x.clone().unwrap();
+    let r_y = r.y.clone().unwrap();
+
+    if (p == q && p_y.clone() == F::zero()) || (p != q && p_x.clone() == q_x.clone()) {
+        return r_x.clone() - p_x.clone();
+    }
+    let slope = p.line_slope(q.clone());
+    let numerator = r_y.clone() - p_y.clone() - slope.clone() * (r_x.clone() - p_x.clone());
+    let denominator = r_x.clone() + p_x.clone() + q_x.clone() - slope.clone() * slope.clone();
+    return numerator / denominator;
+}
+
+pub fn miller<F: Field>(p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, m: BigInt) -> F {
+    if p == q {
+        F::one();
+    }
+
+    let mut f = F::one();
+    let mut t = p.clone();
+
+    for i in (1..m.bits()).rev() {
+        f = (f.clone() * f.clone()) * (get_lambda(t.clone(), t.clone(), q.clone()));
+        t = t.clone() + t.clone();
+        if m.bit(i) {
+            f = f * (get_lambda(t.clone(), p.clone(), q.clone()));
+            t = t.clone() + p.clone();
+        }
+    }
+
+    f
+}
+
+pub fn weil_pairing<F: Field>(
+    p: EllipticCurvePoint<F>,
+    q: EllipticCurvePoint<F>,
+    m: BigInt,
+    s: Option<EllipticCurvePoint<F>>,
+) -> F {
+    let s_value = s.unwrap();
+    let fp_qs = miller(p.clone(), q.clone() + s_value.clone(), m.clone());
+    let fp_s = miller(p.clone(), s_value.clone(), m.clone());
+    let fq_ps = miller(q.clone(), p.clone() - s_value.clone(), m.clone());
+    let fq_s = miller(q.clone(), -s_value.clone(), m.clone());
+
+    return (fp_qs / fp_s) / (fq_ps / fq_s);
+}
+```
diff --git a/docs/book/src/number_theory/subsec6.md b/docs/book/src/number_theory/subsec6.md
new file mode 100644
index 0000000..2bccff1
--- /dev/null
+++ b/docs/book/src/number_theory/subsec6.md
@@ -0,0 +1,53 @@
+# Assumptions
+
+### Assumption: Discrete Logarithm Problem 
+
+---
+
+*Let \\(G\\) be a finite cyclic group of order \\(n\\), with \\(\gamma\\) as its generator and \\(1\\) as the identity element. For any element \\(\alpha \in G\\), there is currently no known efficient (polynomial-time) algorithm to compute the smallest non-negative integer \\(x\\) such that \\(\alpha = \gamma^{x}\\).*
+
+---
+
+The Discrete Logarithm Problem can be thought of as a one-way function. It's easy to compute \\(g^{x}\\) given \\(g\\) and \\(x\\), but it's computationally difficult to find \\(x\\) given \\(g\\) and \\(g^{x}\\).
+
+### Assumption: Elliptic Curve Discrete Logarithm Problem
+
+---
+
+*Let \\(E\\) be an elliptic curve defined over a finite field \\(\mathbb{F}_q\\), where \\(q\\) is a prime power. Let \\(P\\) be a point on \\(E\\) of point order \\(n\\), and let \\(\langle P \rangle\\) be the cyclic subgroup of \\(E\\) generated by \\(P\\). For any element \\(Q \in \langle P \rangle\\), there is currently no known efficient (polynomial-time) algorithm to compute the unique integer \\(k\\), \\(0 \leq k < n\\), such that \\(Q = kP\\).*
+
+---
+
+This assumption is an elliptic curve version of the Discrete Logarithm Problem.
+
+### Assumption: Knowledge of Exponent Assumption
+
+---
+
+Let \\(G\\) be a cyclic group of prime order \\(q\\) with generator \\(g \in G\\). For any probabilistic polynomial-time algorithm \\(\mathcal{A}\\) that outputs:
+
+\begin{equation}
+\mathcal{A}(g, g^x) = (h, h') \quad s.t. \quad h' = h^x
+\end{equation}
+, there exists an efficient extractor \\(\mathcal{E}\\) such that:
+\begin{equation}
+\mathcal{E}(\mathcal{A}, g, g^x) = y \quad s.t. \quad h = g^y
+\end{equation}
+
+---
+
+This assumption states that if \\(\mathcal{A}\\) can compute the pair \\((g^y, g^{xy})\\) from \\((g, g^x)\\), then \\(\mathcal{A}\\) must "know" the value \\(y\\), in the sense that \\(\mathcal{E}\\) can extract \\(y\\) from \\(\mathcal{A}\\)'s internal state.
+The Knowledge of Exponent Assumption is useful for constructing verifiable exponential calculation algorithms. Consider a scenario where Alice has a secret value \\(a\\), and Bob has a secret value \\(b\\). Bob wants to obtain \\(g^{ab}\\). This can be achieved through the following protocol:
+
+**Verifiable Exponential Calculation Algorithm**
+
+1. Bob sends \\((g, g'=g^{b})\\) to Alice
+2. Alice sends \\((h=g^{a}, h'=g'^{a})\\) to Bob
+3. Bob checks \\(h^{b} = h'\\).
+
+Thanks to the Discrete Logarithm Assumption and the Knowledge of Exponent Assumption, the following properties hold:
+
+- Bob cannot derive \\(a\\) from \\((h, h')\\).
+- Alice cannot derive \\(b\\) from \\((g, g')\\).
+- Alice cannot generate \\((t, t')\\) such that \\(t \neq h\\) and \\(t^{b} = t'\\).  
+- If \\(h^{b} = h'\\), Bob can conclude that \\(h\\) is the power of \\(g\\).
diff --git a/docs/book/src/zksnark/README.md b/docs/book/src/zksnark/README.md
index e69de29..d922f5f 100644
--- a/docs/book/src/zksnark/README.md
+++ b/docs/book/src/zksnark/README.md
@@ -0,0 +1 @@
+# Basics of zk-SNARK
\ No newline at end of file
diff --git a/docs/book/src/zksnark/subsec1.md b/docs/book/src/zksnark/subsec1.md
new file mode 100644
index 0000000..e69de29
diff --git a/docs/book/src/zksnark/subsec2.md b/docs/book/src/zksnark/subsec2.md
new file mode 100644
index 0000000..015388e
--- /dev/null
+++ b/docs/book/src/zksnark/subsec2.md
@@ -0,0 +1,212 @@
+# Arithmetization
+
+The ultimate goal of Zero-Knowledge Proofs (ZKP) is to allow the prover to demonstrate their knowledge to the verifier without revealing any additional information. This knowledge typically involves solving complex problems, such as finding a secret input value that corresponds to a given public hash. ZKP protocols usually convert these statements into polynomial constraints. This process is often called \textit{arithmetization}.
+
+To make the protocol flexible, we need to encode this knowledge in a specific format, and one common approach is using Boolean circuits. It's well-known that problems in P (those solvable in polynomial time) and NP (those where the solution can be verified in polynomial time) can be represented as Boolean circuits. This means adopting Boolean circuits allows us to handle both P and NP problems.
+
+However, Boolean circuits are often large and inefficient. Even a simple operation, like adding two 256-bit integers, can require hundreds of Boolean operators. In contrast, arithmetic circuits—essentially systems of equations involving addition, multiplication, and equality—offer a much more compact representation. Additionally, any Boolean circuit can be converted into an arithmetic circuit. For instance, the Boolean expression \\(z = x \land y\\) can be represented as \\(x(x-1) = 0\\), \\(y(y-1) = 0\\), and \\(z = xy\\) in an arithmetic circuit. Furthermore, as we'll see in this section, converting arithmetic circuits into polynomial constraints allows for much faster evaluation.
+
+## Rank-1 Constraint System (R1CS)
+
+There are many formats to represent arithmetic circuits, and one of the most popular ones is R1CS (Rank-1 Constraint System), which represents arithmetic circuits as \underline{a set of equality constraints, each involving only one multiplication}. In an arithmetic circuit, we call the concrete values assigned to the variables within the constraints witness. We first provide the formal definition of R1CS as follows:
+
+### Definition: R1CS
+
+An R1CS structure \\(\mathcal{S}\\) consists of:
+
+- Size bounds \\(m, d, \ell \in \mathbb{N}\\) where \\(d > \ell\\)
+- Three matrices \\(O, L, R \in \mathbb{F}^{m\ times d}\\) with at most \\(\Omega(\max(m, d))\\) non-zero entries in total
+
+
+An R1CS instance includes a public input \\(p \in \mathbb{F}^\ell\\), while an R1CS witness is a vector \\(w \in \mathbb{F}^{d - \ell - 1}\\).
+A structure-instance tuple \\((S, p)\\) is satisfied by a witness \\(w\\) if:
+\begin{equation}
+(L \cdot a) \circ (R \cdot a) - O \cdot a = \mathbf{0}
+\end{equation}
+where \\(a = (1, w, p) \in \mathbb{F}^d\\), \\(\cdot\\) denotes matrix-vector multiplication, and \\(\circ\\) is the Hadamard product.
+
+The intuitive interpretation of each matrix is as follows:
+
+
+- \\(L\\): Encodes the left input of each gate
+- \\(R\\): Encodes the right input of each gate
+- \\(O\\): Encodes the output of each gate
+- The leading 1 in the witness vector allows for constant terms
+
+
+**Single Multiplication**
+
+Let's consider a simple example where we want to prove \\(z = x \cdot y\\), with \\(z = 3690\\), \\(x = 82\\), and \\(y = 45\\).
+
+- \textbf{Witness vector}: \\((1, z, x, y) = (1, 3690, 82, 45)\\)
+- \textbf{Number of witnesses}: \\(m = 4\\)
+- \textbf{Number of constraints}: \\(d = 1\\)
+
+
+The R1CS constraint for \\(z = x \cdot y\\) is satisfied when:
+
+\begin{align*}
+&(\begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix} \cdot a) \circ (\begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix} \cdot a) - \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix} \cdot a \\\\
+=&(\begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\\\ 3690 \\\\ 82 \\\\ 45
+\end{bmatrix}) \circ (\begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\\\ 3690 \\\\ 82 \\\\ 45
+\end{bmatrix}) - \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix}
+1 \\\\ 3690 \\\\ 82 \\\\ 45
+\end{bmatrix} \\\\
+=& 82 \cdot 45 - 3690 \\\\
+=& 3690 - 3690 \\\\
+=& 0
+\end{align*}
+
+This example demonstrates how R1CS encodes a simple multiplication constraint:
+
+- \\(L = \begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix}\\) selects \\(x\\) (left input)
+- \\(R = \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}\\) selects \\(y\\) (right input)
+- \\(O = \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix}\\) selects \\(z\\) (output)
+
+
+**Multiple Constraints**
+
+Let's examine a more complex example: \\(r = a \cdot b \cdot c \cdot d\\). Since R1CS requires that each constraint contain only one multiplication, we need to break this down into multiple constraints:
+
+\begin{align*}
+v_1 &= a \cdot b \\\\
+v_2 &= c \cdot d \\\\
+r &= v_1 \cdot v_2
+\end{align*}
+
+Note that alternative representations are possible, such as \\(v_1 = ab, v_2 = v_1c, r = v_2d\\). In this example, we use 7 variables \\((r, a, b, c, d, v_1, v_2)\\), so the dimension of the witness vector will be \\(m = 8\\) (including the constant 1). We have three constraints, so \\(n = 3\\).
+To construct the matrices \\(L\\), \\(R\\), and \\(O\\), we can interpret the constraints as linear combinations:
+
+\begin{align*}
+v_1 &= (0 \cdot 1 + 0 \cdot r + 1 \cdot a + 0 \cdot b + 0 \cdot c + 0 \cdot d + 0 \cdot v_1 + 0 \cdot v_2) \cdot b \\\\
+v_2 &= (0 \cdot 1 + 0 \cdot r + 0 \cdot a + 0 \cdot b + 1 \cdot c + 0 \cdot d + 0 \cdot v_1 + 0 \cdot v_2) \cdot d \\\\
+r &= (0 \cdot 1 + 0 \cdot r + 0 \cdot a + 0 \cdot b + 0 \cdot c + 0 \cdot d + 1 \cdot v_1 + 0 \cdot v_2) \cdot v_2
+\end{align*}
+
+Thus, we can construct \\(L\\), \\(R\\), and \\(O\\) as follows:
+
+\begin{equation*}
+L = \begin{bmatrix}
+0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\
+0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\
+0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
+\end{bmatrix}
+\end{equation*}
+\begin{equation*}
+R = \begin{bmatrix}
+0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\
+0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\
+0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
+\end{bmatrix}
+\end{equation*}
+\begin{equation*}
+O = \begin{bmatrix}
+0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\
+0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\
+0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
+\end{bmatrix}
+\end{equation*}
+
+Where the columns in each matrix correspond to \\((1, r, a, b, c, d, v_1, v_2)\\).
+
+**Addition with a Constant**
+
+Let's examine the case \\(z = x \cdot y + 3\\). We can represent this as \\(-3 + z = x \cdot y\\). For the witness vector \\((1, z, x, y)\\), we have:
+
+\begin{align*}
+L &= \begin{bmatrix}
+0 & 0 & 1 & 0
+\end{bmatrix} \\\\
+R &= \begin{bmatrix}
+0 & 0 & 0 & 1
+\end{bmatrix} \\\\
+O &= \begin{bmatrix}
+-3 & 1 & 0 & 0
+\end{bmatrix}
+\end{align*}
+
+Note that the constant 3 appears in the \\(O\\) matrix with a negative sign, effectively moving it to the left side of the equation
+
+**Multiplication with a Constant**
+
+Now, let's consider \\(z = 3x^2 + y\\). The requirement of "one multiplication per constraint" doesn't apply to multiplication with a constant, as we can treat it as repeated addition.
+
+\begin{align*}
+L &= \begin{bmatrix}
+0 & 0 & 3 & 0
+\end{bmatrix} \\\\
+R &= \begin{bmatrix}
+0 & 0 & 1 & 0
+\end{bmatrix} \\\\
+O &= \begin{bmatrix}
+0 & 1 & 0 & -1
+\end{bmatrix}
+\end{align*}
+
+## uadratic Arithmetic Program (QAP)
+
+Recall that the prover aims to demonstrate knowledge of a witness \\(w\\) without revealing it. This is equivalent to knowing a vector \\(a\\) that satisfies \\((L \cdot a) \circ (R \cdot a) = O \cdot a\\), where \\(\circ\\) denotes the Hadamard (element-wise) product. However, evaluating this equivalence directly requires \\(\Omega(d)\\) operations, where \\(d\\) is the number of rows. To improve efficiency, we can convert this matrix comparison to a polynomial comparison, leveraging the Schwartz-Zippel Lemma, which allows us to check polynomial equality with \\(\Omega(1)\\) evaluations.
+
+Let's consider a simpler example to illustrate this concept. Suppose we want to test the equivalence \\(Av = Bu\\), where:
+
+\begin{align*}
+A = \begin{bmatrix}
+2 & 5 \\\\
+3 & 1 \\\\
+\end{bmatrix},
+B = \begin{bmatrix}
+4 & 1 \\\\
+2 & 3 \\\\
+\end{bmatrix},
+v = \begin{bmatrix}
+3 \\\\ 1
+\end{bmatrix},
+u = \begin{bmatrix}
+2 \\\\ 2
+\end{bmatrix}
+\end{align*}
+
+The equivalence check can be represented as:
+
+\begin{equation*}
+\begin{bmatrix}
+2 \\\\ 3
+\end{bmatrix} \cdot 3 + \begin{bmatrix}
+5 \\\\ 1
+\end{bmatrix} \cdot 1 = \begin{bmatrix}
+4 \\\\ 2
+\end{bmatrix} \cdot 2 + \begin{bmatrix}
+1 \\\\ 3
+\end{bmatrix} \cdot 2
+\end{equation*}
+
+This matrix-vector equality check is equivalent to the following polynomial equality check:
+
+\begin{equation*}
+3 \cdot L([(1, 2), (2, 3)]) + 1 \cdot L([(1, 5), (2, 1)]) = 2 \cdot L([(1, 4), (2, 2)]) + 2 \cdot L([(1, 1), (2, 3)])
+\end{equation*}
+
+where \\(L\\) denotes Lagrange Interpolation. In \\(\mathbb{F}_{11}\\) (field with 11 elements), we have:
+
+\begin{align*}
+L([(1, 2), (2, 3)]) &= x + 1 \\\\
+L([(1, 5), (2, 1)]) &= 7x + 9 \\\\
+L([(1, 4), (2, 2)]) &= 9x + 6 \\\\
+L([(1, 1), (2, 3)]) &= 2x + 10
+\end{align*}
+
+The Schwartz-Zippel Lemma states that we need only one evaluation at a random point to check the equivalence of polynomials with high probability.
+
+However, the homomorphic property for multiplication doesn't hold for Lagrange Interpolation. Thus, we don't have \\(\ell(x) \cdot r(x) = o(x)\\), where \\(\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) are the polynomials resulting from the Lagrange interpolation of \\(L \cdot a\\), \\(R \cdot a\\), and \\(O \cdot a\\) respectively. 
+
+While \\(\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) are of degree at most \\(d-1\\), \\(\ell(x) \cdot r(x)\\) is of degree at most \\(2d-2\\).
+
+To address this discrepancy, we introduce a degree \\(d\\) polynomial \\(t(x) = \prod_{i=1}^{d} (x - i)\\). We can then rewrite the equation as:
+
+\begin{equation}
+\ell(x) \cdot r(x) = o(x) + h(x) \cdot t(x)
+\end{equation}
+
+where \\(h(x) = \frac{\ell(x) \cdot r(x) - o(x)}{t(x)}\\). This formulation allows us to maintain the desired polynomial relationships while accounting for the degree differences.
diff --git a/docs/book/src/zksnark/subsec3.md b/docs/book/src/zksnark/subsec3.md
new file mode 100644
index 0000000..d609a2f
--- /dev/null
+++ b/docs/book/src/zksnark/subsec3.md
@@ -0,0 +1,140 @@
+# Proving Single Polynomial
+
+Before dealing with all of \\(\ell(x)\\), \\(r(x)\\), and \\(o(x)\\) at once, we design a protocol that allows the Prover \\(\mathcal{A}\\) to convince the Verifier \\(\mathcal{B}\\) that \\(\mathcal{A}\\) knows a specific polynomial. Let's denote this polynomial of degree \\(n\\) with coefficients in a finite field as:
+
+\begin{equation}
+    P(x) = c_0 + c_1 x + c_2 x^{2} + \cdots c_n x^{n}
+\end{equation}
+
+Assume \\(P(x)\\) has \\(n\\) roots, \\(a_1, a_2, \ldots, a_n \in \mathbb{F}\\), such that \\(P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)\\). The Verifier \\(\mathcal{B}\\) knows \\(d < n\\) roots of \\(P(x)\\), namely \\(a_1, a_2, \ldots, a_d\\). Let \\(T(x) = (x - a_1)(x - a_2)\cdots(x - a_d)\\). Note that the Prover also knows \\(T(x)\\).
+
+The Prover's objective is to convince the Verifier that \\(\mathcal{A}\\) knows a polynomial \\(H(x) = \frac{P(x)}{T(x)}\\).
+
+### Naive Approach
+
+The simplest approach to prove that \\(\mathcal{A}\\) knows \\(H(x)\\) is as follows:
+
+*Naive Approach*
+
+- *\\(\mathcal{B}\\) sends all possible values in \\(\mathbb{F}\\) to \\(\mathcal{A}\\).*
+- *\\(\mathcal{A}\\) computes and sends all possible outputs of \\(H(x)\\) and \\(P(x)\\).*
+- *\\(\mathcal{B}\\) checks whether \\(H(a)T(a) = P(a)\\) holds for any \\(a\\) in \\(\mathbb{F}\\).*
+
+This protocol is highly inefficient, requiring \\(\mathcal{O}(|\mathbb{F}|)\\) evaluations and communications.
+
+### \\(+\\) Schwartz-Zippel Lemma
+
+Instead of evaluating polynomials at all values in \\(\mathbb{F}\\), we can leverage the Schwartz-Zippel Lemma: if \\(H(s) = \frac{P(s)}{T(s)}\\) or equivalently \\(H(s)T(s) = P(s)\\) for a random element \\(s\\), we can conclude that \\(H(x) = \frac{P(x)}{T(x)}\\) with high probability. Thus, the Prover \\(\mathcal{A}\\) only needs to send evaluations of \\(P(s)\\) and \\(H(s)\\) for a random input \\(s\\) received from \\(\mathcal{B}\\).
+
+*\\(+\\) Schwartz-Zippel Lemma*
+    
+- *\\(\mathcal{B}\\) draws random \\(s\\) from \\(\mathbb{F}\\) and sends it to \\(\mathcal{A}\\).*
+- *\\(\mathcal{A}\\) computes \\(h = H(s)\\) and \\(p = P(s)\\) and send them to \\(\mathcal{B}\\).*
+- *\\(\mathcal{B}\\) checks whether \\(p = t h\\), where \\(t\\) denotes \\(T(s)\\).*
+
+This protocol is efficient, requiring only a constant number of evaluations and communications. However, it is vulnerable to a malicious prover who could send an arbitrary value \\(h'\\) and the corresponding \\(p'\\) such that \\(p' = h't\\).
+
+### \\(+\\) Discrete Logarithm Assumption
+
+To address this vulnerability, the Verifier must hide the randomly chosen input \\(s\\) from the Prover. This can be achieved using the discrete logarithm assumption: it is computationally hard to determine \\(s\\) from \\(\alpha\\), where \\(\alpha = g^s \bmod p\\). Thus, it's safe for the Verifier to send \\(\alpha\\), as the Prover cannot easily derive \\(s\\) from it.
+
+An interesting property of polynomial exponentiation is:
+
+\begin{align}
+    g^{P(x)} &= g^{c_0 + c_1 x + c_2 x^{2} + \cdots c_n x^{n}} = g^{c_0} (g^{x})^{c_1}  (g^{(x^2)})^{c_2} \cdots (g^{(x^n)})^{c_n}
+\end{align}
+
+Instead of sending \\(s\\), the Verifier can send \\(g\\) and \\(\alpha_{i} = g^{(s^i)}\\) for \\(i = 1, \cdots n\\). BE CAREFUL THAT **\\(g^{(s^i)} \neq (g^s)^i\\)**. The Prover can still evaluate \\(g^p = g^{P(s)}\\) using these powers of \\(g\\):
+
+\begin{equation}
+    g^{p} = g^{P(s)} = g^{c_0} \alpha_{1}^{c_1} (\alpha_{2})^{c_2} \cdots (\alpha_{n})^{c_n}
+\end{equation}
+
+Similarly, the Prover can evaluate \\(g^h = g^{H(s)}\\). The Verifier can then check \\(p = ht \iff g^p = (g^h)^t\\). 
+
+*\\(+\\) Discrete Logarithm Assumption*
+
+- *\\(\mathcal{B}\\) randomly draw \\(s\\) from \\(\mathbb{F}\\).*
+- *\\(\mathcal{B}\\) computes and sends \\(\\{\alpha, \alpha_2, ..., \alpha_{n}\\}\\), where \\(\alpha_i= g^{(s^{i})}\\).*
+- *\\(\mathcal{A}\\) computes and sends \\(u = g^{p}\\) and \\(v = g^{h}\\).*
+- *\\(\mathcal{B}\\) checks whether \\(u = v^{t}\\).*
+
+This approach prevents the Prover from obtaining \\(s\\) or \\(t = T(s)\\), making it impossible to send fake \\((h', p')\\) such that \\(p' = h't\\).
+
+However, this protocol still has a flaw. Since the Prover can compute \\(g^t = \alpha_{c_1}(\alpha_2)^{c_2}\cdots(\alpha_d)^{c_d}\\), they could send fake values \\(((g^{t})^{z}, g^{z})\\) instead of \\((g^p, g^h)\\) for an arbitrary value \\(z\\). The verifier's check would still pass, and they could not detect this deception.
+
+### \\(+\\) Knowledge of Exponent Assumption
+
+To address the vulnerability where the verifier \\(\mathcal{B}\\) cannot distinguish if \\(v (= g^h)\\) from the prover is a power of \\(\alpha_i = g^{(s^i)}\\), we can employ the Knowledge of Exponent Assumption. This approach involves the following steps:
+
+- \\(\mathcal{B}\\) sends both \\(\alpha_i\\) and \\(\alpha'_i = \alpha_i^r\\) for a new random value \\(r\\).
+- The prover returns \\(a = (\alpha_i)^{c_i}\\) and \\(a' = (\alpha'\_i)^{c_i}\\) for \\(i = 1, ..., n\\).
+- \\(\mathcal{B}\\) can conclude that \\(a\\) is a power of \\(\alpha_i\\) if \\(a^r = a'\\).
+
+
+Based on this assumption, we can design an improved protocol:
+
+*\\(+\\) Knowledge of Exponent Assumption*
+
+- *\\(\mathcal{B}\\) randomly selects \\(s\\) and \\(r\\) from field \\(\mathbb{F}\\).*
+- *\\(\mathcal{B}\\) computes and sends \\(\{\alpha_1, \alpha_2, ..., \alpha_{n}\}\\) and \\(\{\alpha'\_1, \alpha'\_2, ..., \alpha'\_{n}\}\\), where \\(\alpha_i = g^{(s^i)}\\) and \\(\alpha' = \alpha_{r} = g^{(s^{i})r}\\).*
+- *\\(\mathcal{A}\\) computes and sends \\(u = g^{p}\\), \\(v = g^{h}\\), and \\(w = g^{p'}\\), where \\(g^{p'} = g^{P(sr)}\\).*
+- *\\(\mathcal{B}\\) checks whether \\(u^{r} = w\\).*
+- *\\(\mathcal{B}\\) checks whether \\(u = v^{t}\\).*
+
+
+The prover can compute \\(g^{p'} = g^{P(sr)} = \alpha'^{c_1} (\alpha'^{2})^{c_2} \cdots (\alpha'^{n})^{c_n}\\) using powers of \\(\alpha'\\). This protocol now satisfies the properties of a SNARK: completeness, soundness, and efficiency.
+
+### \\(+\\) Zero Knowledge
+
+To transform the above protocol into a zk-SNARK, we need to ensure that the verifier cannot learn anything about \\(P(x)\\) from the prover's information. This is achieved by having the prover obfuscate all information with a random secret value \\(\delta\\):
+
+*\\(+\\) Zero Knowledge*
+
+- *\\(\mathcal{B}\\) randomly selects \\(s\\) and \\(r\\) from field \\(\mathbb{F}\\).*
+- *\\(\mathcal{B}\\) computes and sends \\(\\{\alpha_1, \alpha_2, ..., \alpha_{n}\\}\\) and \\(\\{\alpha\_1', \alpha'\_2, ..., \alpha'\_{n}\\}\\), where \\(\alpha_i = g^{(s^{i})}\\) and \\(\alpha_i' = \alpha_i^{r} = g^{(s^{i})r}\\).*
+- *\\(\mathcal{A}\\) randomly selects \\(\delta\\) from field \\(\mathbb{F}\\).*
+- *\\(\mathcal{A}\\) computes and sends \\(u' = (g^{p})^{\delta}\\), \\(v' = (g^{h})^{\delta}\\), and \\(w' = (g^{p'})^{\delta}\\).*
+- *\\(\mathcal{B}\\) checks whether \\(u'^{r} = w'\\).*
+- *\\(\mathcal{B}\\) checks whether \\(u' = v'^{t}\\).*
+
+
+
+By introducing the random value \\(\delta\\), the verifier can no longer learn anything about \\(p\\), \\(h\\), or \\(w\\), thus achieving zero knowledge.
+
+### \\(+\\) Non-interactivity
+
+The previously described protocol requires each verifier to generate unique random values, which becomes inefficient when a prover needs to demonstrate knowledge to multiple verifiers. To address this, we aim to eliminate the interaction between the prover and verifier. One effective solution is the use of a trusted setup.
+
+*\\(+\\) Non-Interactive: Trusted Setup*
+
+- ***Secret Seed:** A trusted third party generates the random values \\(s\\) and \\(r\\)*
+- ***Proof Key:** Provided to the prover*
+    - *\\(\\{\alpha_1, \alpha_2, ..., \alpha_{n}\\}\\), where \\(\alpha_{i} = g^{(s^i)}\\)*
+    - *\\(\\{\alpha'\_1, \alpha'\_2, ..., \alpha'\_{n}\\}\\), where \\(\alpha_i' = g^{(s^{i})r}\\)*
+- ***Verification Key:** Distributed to verifiers*
+    - *\\(g^{r}\\)*
+    - *\\(g^{T(s)}\\)* 
+- *After distribution, the original \\(s\\) and \\(r\\) values are securely destroyed.*
+
+
+Then, the non-interactive protocol consists of two main parts: proof generation and verification.
+
+*\\(+\\) Non-Interactive: Proof*
+    
+- *\\(\mathcal{A}\\) receives the proof key*
+- *\\(\mathcal{A}\\) randomly selects \\(\delta\\) from field \\(\mathbb{F}\\).*
+- *\\(\mathcal{A}\\) broadcast the proof \\(\pi = (u' = (g^{p})^{\delta}, v' = (g^{h})^{\delta}, w' = (g^{p'})^{\delta})\\)*
+    
+
+
+Since \\(r\\) is not shared and already destroyed, the verifier \\(\mathcal{B}\\) cannot calculate \\(u'^{r}\\) to check \\(u'^{r} = w'\\). Instead, the verifier can use a paring with bilinear mapping; \\(u'^{r} = w'\\) is equivalent to \\(e(u' = (g^{p})^{\delta}, g^{r}) = e(w'=(g^{p'})^{\delta}, g)\\).
+
+*\\(+\\) Non-Interactive: Verification*
+    
+- *\\(\mathcal{B}\\) receives the verification key.*
+- *\\(\mathcal{B}\\) receives the proof \\(\pi\\).*
+- *\\(\mathcal{B}\\) checks whether \\(e(u', g^{r}) = e(w', g)\\).*
+- *\\(\mathcal{B}\\) checks whether \\(u' = v'^{t}\\).*
+    
+
diff --git a/docs/book/theme/css/custom.css b/docs/book/theme/css/custom.css
new file mode 100644
index 0000000..eb3750b
--- /dev/null
+++ b/docs/book/theme/css/custom.css
@@ -0,0 +1,29 @@
+.navy em {
+    background-color: #1a4d4d;
+    padding: 2px 4px;
+    border-radius: 4px;
+  }
+
+.ayu em {
+    background-color: #2d4a3e;
+    padding: 2px 4px;
+    border-radius: 4px;
+  }
+
+.coal em {
+    background-color: #2d4a3e;
+    padding: 2px 4px;
+    border-radius: 4px;
+  }
+
+.light em {
+    background-color: #f0f7e6;
+    padding: 2px 4px;
+    border-radius: 4px;
+  }
+
+.rust em {
+    background-color: #f7ffe6;
+    padding: 2px 4px;
+    border-radius: 4px;
+  }
\ No newline at end of file
diff --git a/myzkp/Cargo.toml b/myzkp/Cargo.toml
index f577038..a73d822 100644
--- a/myzkp/Cargo.toml
+++ b/myzkp/Cargo.toml
@@ -7,3 +7,5 @@ edition = "2021"
 num-bigint = { version = "0.4", features = ["rand"] }
 num-traits = "0.2"
 rand = "0.8.5"
+lazy_static = "1.4.0"
+paste = "1.0.15"
diff --git a/myzkp/src/modules/bn128.rs b/myzkp/src/modules/bn128.rs
new file mode 100644
index 0000000..248a170
--- /dev/null
+++ b/myzkp/src/modules/bn128.rs
@@ -0,0 +1,399 @@
+use crate::modules::ring::Ring;
+use lazy_static::lazy_static;
+use num_bigint::BigInt;
+use num_bigint::ToBigInt;
+use num_traits::FromPrimitive;
+use num_traits::{One, Zero};
+use paste::paste;
+use std::str::FromStr;
+
+use crate::modules::curve::{miller, EllipticCurve, EllipticCurvePoint};
+use crate::modules::efield::{ExtendedFieldElement, IrreduciblePoly};
+use crate::modules::field::Field;
+use crate::modules::field::{FiniteFieldElement, ModulusValue};
+use crate::modules::polynomial::Polynomial;
+use crate::{define_myzkp_curve_type, define_myzkp_modulus_type};
+
+define_myzkp_modulus_type!(
+    BN128Modulus,
+    "21888242871839275222246405745257275088696311157297823662689037894645226208583"
+);
+define_myzkp_curve_type!(BN128Curve, "0", "3");
+
+type Fq = FiniteFieldElement<BN128Modulus>;
+type G1Point = EllipticCurvePoint<Fq, BN128Curve>;
+
+// Define Fq2 as a quadratic extension field
+#[derive(Debug, Clone, PartialEq, Hash)]
+pub struct Fq2Poly;
+lazy_static! {
+    // x^2 + 1
+    static ref MODULUS_Fq2: Polynomial<Fq> = Polynomial {
+        coef: vec![Fq::one(), Fq::zero(), Fq::one()]
+    };
+}
+
+impl IrreduciblePoly<Fq> for Fq2Poly {
+    fn modulus() -> &'static Polynomial<Fq> {
+        &MODULUS_Fq2
+    }
+}
+type Fq2 = ExtendedFieldElement<BN128Modulus, Fq2Poly>;
+type G2Point = EllipticCurvePoint<Fq2, BN128Curve>;
+
+// Define Fq2 as a quadratic extension field
+#[derive(Debug, Clone, PartialEq, Hash)]
+pub struct Fq12Poly;
+lazy_static! {
+    // x^2 + 1
+    static ref MODULUS_Fq12: Polynomial<Fq> = Polynomial {
+        // x^12 -18x^6 + 82
+        coef: vec![
+            Fq::from_value(82_i32),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::from_value(-18_i32),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::one(),
+        ],
+    };
+}
+impl IrreduciblePoly<Fq> for Fq12Poly {
+    fn modulus() -> &'static Polynomial<Fq> {
+        &MODULUS_Fq12
+    }
+}
+type Fq12 = ExtendedFieldElement<BN128Modulus, Fq12Poly>;
+type G12Point = EllipticCurvePoint<Fq12, BN128Curve>;
+
+pub fn cast_G1_to_G12(g: G1Point) -> G12Point {
+    if g.is_point_at_infinity() {
+        return G12Point::point_at_infinity();
+    }
+
+    G12Point::new(
+        Fq12::new(Polynomial {
+            coef: vec![g.x.unwrap()],
+        }),
+        Fq12::new(Polynomial {
+            coef: vec![g.y.unwrap()],
+        }),
+    )
+}
+
+pub fn twist_G2_to_G12(g: G2Point) -> G12Point {
+    if g.is_point_at_infinity() {
+        return G12Point::point_at_infinity();
+    }
+
+    let w = Fq12::new(Polynomial {
+        coef: vec![Fq::zero(), Fq::one()],
+    });
+
+    let x = g.x.unwrap();
+    let y = g.y.unwrap();
+    let x_coeff = vec![
+        x.poly.coef[0].clone() - x.poly.coef[1].clone() * Fq::from_value(9_i32),
+        x.poly.coef[1].clone(),
+    ];
+    let y_coeff = vec![
+        y.poly.coef[0].clone() - y.poly.coef[1].clone() * Fq::from_value(9_i32),
+        y.poly.coef[1].clone(),
+    ];
+
+    let nx = Fq12::new(Polynomial {
+        coef: vec![
+            x_coeff[0].clone(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            x_coeff[1].clone(),
+        ],
+    });
+    let ny = Fq12::new(Polynomial {
+        coef: vec![
+            y_coeff[0].clone(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            Fq::zero(),
+            y_coeff[1].clone(),
+        ],
+    });
+
+    return G12Point::new(
+        nx * (w.pow(2_i32.to_bigint().unwrap())),
+        ny * (w.pow(3_i32.to_bigint().unwrap())),
+    );
+}
+
+pub fn linefunc<F: Field, E: EllipticCurve>(
+    p: &EllipticCurvePoint<F, E>,
+    q: &EllipticCurvePoint<F, E>,
+    t: &EllipticCurvePoint<F, E>,
+) -> F {
+    let x1 = p.x.as_ref().unwrap();
+    let y1 = p.y.as_ref().unwrap();
+    let x2 = q.x.as_ref().unwrap();
+    let y2 = q.y.as_ref().unwrap();
+    let xt = t.x.as_ref().unwrap();
+    let yt = t.y.as_ref().unwrap();
+
+    if x1 != x2 {
+        let m = (y2.sub_ref(&y1)) / (x2.sub_ref(&x1));
+        return (m.mul_ref(&xt.sub_ref(&x1))).sub_ref(&yt.sub_ref(&y1));
+    } else if y1 == y2 {
+        let m = ((x1.pow(2)).mul_ref(&F::from_value(3))) / (y1.mul_ref(&F::from_value(2)));
+        return (m.mul_ref(&xt.sub_ref(&x1))).sub_ref(&yt.sub_ref(&y1));
+    } else {
+        return xt.sub_ref(x1);
+    }
+}
+
+pub fn vanila_miller<F: Field, E: EllipticCurve>(
+    p: &EllipticCurvePoint<F, E>,
+    q: &EllipticCurvePoint<F, E>,
+    m: &BigInt,
+) -> F {
+    if p == q {
+        return F::one();
+    }
+
+    let mut r = q.clone();
+    let mut f = F::one();
+
+    let ate_loop_count = BigInt::from_str("29793968203157093288").unwrap();
+    let log_ate_loop_count = 63;
+    let field_modulus = BigInt::from_str(
+        "21888242871839275222246405745257275088696311157297823662689037894645226208583",
+    )
+    .unwrap();
+
+    for i in (0..=log_ate_loop_count).rev() {
+        f = f.mul_ref(&f) * linefunc(&r, &r, &p);
+        r = r.double();
+        if ate_loop_count.bit(i) {
+            f = f.mul_ref(&linefunc(&r, &q, &p));
+            r = r.add_ref(&q);
+        }
+    }
+
+    // Assert: r == multiply(&q, &ate_loop_count)
+
+    let q1 = EllipticCurvePoint::<F, E>::new(
+        q.x.clone().unwrap().pow(m.clone()),
+        q.y.clone().unwrap().pow(m.clone()),
+    );
+
+    let nq2 = EllipticCurvePoint::<F, E>::new(
+        q1.x.clone().unwrap().pow(m.clone()),
+        -q1.y.clone().unwrap().pow(m.clone()),
+    );
+
+    f = f.mul_ref(&linefunc(&r, &q1, &p));
+    r = r.add_ref(&q1);
+    f = f.mul_ref(&linefunc(&r, &nq2, &p));
+
+    f
+}
+
+pub fn optimal_ate_pairing(p: G1Point, q: G2Point) -> Fq12 {
+    let p_prime: G12Point = cast_G1_to_G12(p);
+    let q_prime: G12Point = twist_G2_to_G12(q);
+    let m = BN128Modulus::modulus();
+
+    let f = vanila_miller(&p_prime, &q_prime, &m);
+    let exp = (m.pow(12) - BigInt::one()) / (BN128::order());
+    f.pow(exp)
+}
+
+pub struct BN128;
+
+impl BN128 {
+    pub fn generator_g1() -> G1Point {
+        G1Point::new(Fq::from_value(1), Fq::from_value(2))
+    }
+
+    pub fn generator_g2() -> G2Point {
+        G2Point::new(
+            Fq2::new(Polynomial {
+                coef: vec![
+                    Fq::from_value(BigInt::from_str("10857046999023057135944570762232829481370756359578518086990519993285655852781").unwrap()),
+                    Fq::from_value(BigInt::from_str("11559732032986387107991004021392285783925812861821192530917403151452391805634").unwrap()),
+                ],
+            }),
+            Fq2::new(Polynomial {
+                coef: vec![
+                    Fq::from_value(BigInt::from_str("8495653923123431417604973247489272438418190587263600148770280649306958101930").unwrap()),
+                    Fq::from_value(BigInt::from_str("4082367875863433681332203403145435568316851327593401208105741076214120093531").unwrap()),
+                ],
+            }),
+        )
+    }
+
+    pub fn order() -> BigInt {
+        BigInt::from_str(
+            "21888242871839275222246405745257275088548364400416034343698204186575808495617",
+        )
+        .unwrap()
+    }
+
+    pub fn get_b() -> Fq {
+        Fq::from_value(3_i32)
+    }
+
+    pub fn get_b2() -> Fq2 {
+        Fq2::new(Polynomial {
+            coef: vec![Fq::from_value(3_i32)],
+        }) / Fq2::new(Polynomial {
+            coef: vec![Fq::from_value(9_i32), Fq::from_value(1_i32)],
+        })
+    }
+
+    pub fn get_b12() -> Fq12 {
+        Fq12::new(Polynomial {
+            coef: vec![Fq::from_value(3_i32)],
+        })
+    }
+}
+
+// Test the implementation
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn test_fq() {
+        assert_eq!(Fq::from_value(2) * Fq::from_value(2), Fq::from_value(4));
+        assert_eq!(
+            Fq::from_value(2) / Fq::from_value(7) + Fq::from_value(9) / Fq::from_value(7),
+            Fq::from_value(11) / Fq::from_value(7)
+        );
+        assert_eq!(
+            Fq::from_value(2) * Fq::from_value(7) + Fq::from_value(9) * Fq::from_value(7),
+            Fq::from_value(11) * Fq::from_value(7)
+        );
+    }
+
+    #[test]
+    fn test_fq2() {
+        use crate::modules::field::Field;
+
+        let x = Fq2::new(Polynomial {
+            coef: vec![Fq::one(), Fq::zero()],
+        });
+        let f = Fq2::new(Polynomial {
+            coef: vec![Fq::one(), Fq::from_value(2_i32)],
+        });
+        let fpx = Fq2::new(Polynomial {
+            coef: vec![Fq::from_value(2_i32), Fq::from_value(2_i32)],
+        });
+        let fpx2 = Fq2::new(Polynomial {
+            coef: vec![Fq::from_value(2_i32), Fq::from_value(1_i32)],
+        });
+        let one = Fq2::one();
+        assert_eq!(x.add_ref(&f), fpx);
+        assert_eq!(fpx2.div_ref(&fpx2), one);
+        assert_eq!(
+            one.div_ref(&f) + x.div_ref(&f),
+            (one.clone() + x.clone()) / f.clone()
+        );
+        assert_eq!(
+            one.clone() * f.clone() + x.clone() * f.clone(),
+            (one.clone() + x.clone()) * f.clone()
+        );
+        assert_eq!(
+            x.clone()
+                .pow(BN128Modulus::modulus().pow(2) - BigInt::one()),
+            one
+        );
+    }
+
+    #[test]
+    fn test_g1() {
+        let g1 = BN128::generator_g1();
+        assert_eq!(
+            g1.clone().y.unwrap().pow(2) - g1.clone().x.unwrap().pow(3),
+            BN128::get_b()
+        );
+        assert_eq!(
+            g1.clone() * 2 + g1.clone() + g1.clone(),
+            (g1.clone() * 2) * 2
+        );
+        assert_eq!(
+            g1.clone() * 9 + g1.clone() * 5,
+            g1.clone() * 12 + g1.clone() * 2,
+        );
+        assert!((g1.clone() * BN128::order()).is_point_at_infinity());
+    }
+
+    #[test]
+    fn test_g2() {
+        let g2 = BN128::generator_g2();
+        assert_eq!(
+            g2.clone().y.unwrap().pow(2) - g2.clone().x.unwrap().pow(3),
+            BN128::get_b2()
+        );
+        assert_eq!(
+            g2.clone() * 2 + g2.clone() + g2.clone(),
+            (g2.clone() * 2) * 2
+        );
+        assert_eq!(
+            g2.clone() * 9 + g2.clone() * 5,
+            g2.clone() * 12 + g2.clone() * 2,
+        );
+        assert!((g2.clone() * BN128::order()).is_point_at_infinity());
+    }
+
+    #[test]
+    fn test_g12() {
+        let g2 = BN128::generator_g2();
+        let g12 = twist_G2_to_G12(g2);
+        let g12_9 = g12.clone() * 9;
+        assert_eq!(
+            g12_9.clone().y.unwrap().pow(2) - g12_9.clone().x.unwrap().pow(3),
+            BN128::get_b12()
+        );
+        assert_eq!(
+            g12.clone() * 2 + g12.clone() + g12.clone(),
+            (g12.clone() * 2) * 2
+        );
+        assert_eq!(
+            g12.clone() * 9 + g12.clone() * 5,
+            g12.clone() * 12 + g12.clone() * 2,
+        );
+        assert!((g12.clone() * BN128::order()).is_point_at_infinity());
+    }
+
+    #[test]
+    fn test_pairing() {
+        let g1 = BN128::generator_g1();
+        let g2 = BN128::generator_g2();
+        let p1 = optimal_ate_pairing(g1.clone(), g2.clone());
+        let pn1 = optimal_ate_pairing(-g1.clone(), g2.clone());
+        assert_eq!(p1.clone() * pn1.clone(), Fq12::one());
+        let np1 = optimal_ate_pairing(g1.clone(), -g2.clone());
+        assert_eq!(p1.clone() * np1.clone(), Fq12::one());
+        assert_eq!(pn1.clone(), np1.clone());
+        let p2 = optimal_ate_pairing(g1.clone() * 2, g2.clone());
+        assert_eq!(p1.clone() * p1.clone(), p2);
+        assert!(p1 != p2);
+        assert!(p1 != np1);
+        assert!(p2 != np1);
+        let po2 = optimal_ate_pairing(g1.clone(), g2.clone() * 2);
+        assert_eq!(p1.clone() * p1.clone(), po2);
+        let p3 = optimal_ate_pairing(g1.clone() * 37, g2.clone() * 27);
+        let po3 = optimal_ate_pairing(g1.clone() * 999, g2.clone());
+        assert_eq!(p3, po3);
+    }
+}
diff --git a/myzkp/src/modules/curve.rs b/myzkp/src/modules/curve.rs
index 823903c..5460d4c 100644
--- a/myzkp/src/modules/curve.rs
+++ b/myzkp/src/modules/curve.rs
@@ -1,36 +1,41 @@
 use num_bigint::BigInt;
 use num_traits::{One, Zero};
+use std::fmt;
+use std::fmt::Debug;
+use std::marker::PhantomData;
 use std::ops::{Add, Mul, Neg, Sub};
 
 use crate::modules::field::Field;
 
-#[derive(Debug, Clone, PartialEq)]
-pub struct EllipticCurve<F: Field> {
-    pub a: F,
-    pub b: F,
+pub trait EllipticCurve: Debug + Clone + PartialEq {
+    fn get_a() -> BigInt;
+    fn get_b() -> BigInt;
 }
 
 #[derive(Debug, Clone, PartialEq)]
-pub struct EllipticCurvePoint<F: Field> {
+pub struct EllipticCurvePoint<F: Field, E: EllipticCurve> {
     pub x: Option<F>,
     pub y: Option<F>,
-    pub curve: EllipticCurve<F>,
+    _phantom: PhantomData<E>,
 }
 
-impl<F: Field> EllipticCurvePoint<F> {
-    fn new(x: F, y: F, curve: EllipticCurve<F>) -> Self {
+impl<F: Field, E: EllipticCurve> EllipticCurvePoint<F, E> {
+    pub fn new(x: F, y: F) -> Self {
+        // let a = E::get_a();
+        // let b = E::get_b();
+        // assert!(y.pow(2) == x.pow(3) + a * x + b, "Point is not on the curve");
         EllipticCurvePoint {
             x: Some(x),
             y: Some(y),
-            curve: curve,
+            _phantom: PhantomData,
         }
     }
 
-    pub fn point_at_infinity(curve: EllipticCurve<F>) -> Self {
+    pub fn point_at_infinity() -> Self {
         EllipticCurvePoint {
             x: None,
             y: None,
-            curve: curve,
+            _phantom: PhantomData,
         }
     }
 
@@ -43,31 +48,43 @@ impl<F: Field> EllipticCurvePoint<F> {
             return self.clone();
         }
 
-        EllipticCurvePoint::new(
-            self.x.clone().unwrap(),
-            -(self.y.clone().unwrap()),
-            self.curve.clone(),
-        )
+        EllipticCurvePoint::new(self.x.clone().unwrap(), -(self.y.clone().unwrap()))
     }
 
-    pub fn line_slope(&self, other: Self) -> F {
-        let x1 = self.x.clone().unwrap();
-        let y1 = self.y.clone().unwrap();
-        let x2 = other.x.clone().unwrap();
-        let y2 = other.y.clone().unwrap();
+    pub fn line_slope(&self, other: &Self) -> F {
+        let a = F::from_value(E::get_a());
+        // let b = E::get_b();
+
+        let x1 = self.x.as_ref().unwrap();
+        let y1 = self.y.as_ref().unwrap();
+        let x2 = other.x.as_ref().unwrap();
+        let y2 = other.y.as_ref().unwrap();
 
-        if self.x.clone() == other.x.clone() {
-            ((x1.clone() * x1.clone()) * (3_i64) + self.curve.a.clone()) / (y1.clone() * (2_i64))
+        if self.x == other.x {
+            ((x1.mul_ref(&x1)) * &(F::from_value(3_i64)) + &a)
+                / (y1.mul_ref(&(F::from_value(2_i64))))
         } else {
-            (y2.clone() - y1.clone()) / (x2.clone() - x1.clone())
+            (y2.sub_ref(&y1)) / (x2.sub_ref(&x1))
         }
     }
-}
 
-impl<F: Field> Add for EllipticCurvePoint<F> {
-    type Output = Self;
+    pub fn double(&self) -> Self {
+        if self.is_point_at_infinity() {
+            return self.clone();
+        }
 
-    fn add(self, other: Self) -> Self {
+        let slope = self.line_slope(&self);
+        let x = self.x.as_ref().unwrap();
+        let y = self.y.as_ref().unwrap();
+
+        let new_x = slope.mul_ref(&slope).sub_ref(&x).sub_ref(&x);
+        let new_y = -slope.mul_ref(&new_x) + slope * x - y;
+
+        Self::new(new_x, new_y)
+    }
+
+    pub fn add_ref(&self, other: &Self) -> Self {
+        /*
         if self.is_point_at_infinity() {
             return other;
         }
@@ -82,7 +99,7 @@ impl<F: Field> Add for EllipticCurvePoint<F> {
             // Check if they are inverses of each other
             if self.y != other.y {
                 // TODO: check self.y = -other.y
-                return EllipticCurvePoint::point_at_infinity(self.curve.clone());
+                return EllipticCurvePoint::point_at_infinity();
             } else {
                 // P = Q, handle point doubling
                 let x1 = self.x.clone().unwrap();
@@ -94,7 +111,7 @@ impl<F: Field> Add for EllipticCurvePoint<F> {
                 let x3 = m.clone() * m.clone() - x1.clone() - x1.clone();
                 let y3 = m * (x1 - x3.clone()) - y1;
 
-                return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+                return EllipticCurvePoint::new(x3, y3);
             }
         } else {
             // P != Q, handle regular addition
@@ -107,29 +124,81 @@ impl<F: Field> Add for EllipticCurvePoint<F> {
             let x3 = m.clone() * m.clone() - x1.clone() - x2.clone();
             let y3 = m * (x1 - x3.clone()) - y1;
 
-            return EllipticCurvePoint::new(x3, y3, self.curve.clone());
+            return EllipticCurvePoint::new(x3, y3);
+        }*/
+
+        if self.is_point_at_infinity() {
+            return other.clone();
+        }
+        if other.is_point_at_infinity() {
+            return self.clone();
+        }
+
+        if self.x == other.x && self.y == other.y {
+            return self.double();
+        } else if self.x == other.x {
+            return Self::point_at_infinity();
         }
+
+        let slope = self.line_slope(&other);
+        let x1 = self.x.as_ref().unwrap();
+        let y1 = self.y.as_ref().unwrap();
+        let x2 = other.x.as_ref().unwrap();
+        let y2 = other.y.as_ref().unwrap();
+
+        let new_x = slope.mul_ref(&slope).sub_ref(&x1).sub_ref(&x2);
+        let new_y = ((-slope.clone()).mul_ref(&new_x)) + (&slope.mul_ref(&x1).sub_ref(&y1));
+        assert!(new_y == -slope.clone() * &new_x + slope.mul_ref(&x2).sub_ref(&y2));
+
+        Self::new(new_x, new_y)
     }
 }
 
-impl<F: Field> Mul<BigInt> for EllipticCurvePoint<F> {
+impl<F: Field, E: EllipticCurve> Add for EllipticCurvePoint<F, E> {
     type Output = Self;
 
-    fn mul(self, scalar: BigInt) -> Self {
+    fn add(self, other: Self) -> Self {
+        self.add_ref(&other)
+    }
+}
+
+impl<F: Field, E: EllipticCurve> Neg for EllipticCurvePoint<F, E> {
+    type Output = Self;
+    fn neg(self) -> Self {
+        if self.is_point_at_infinity() {
+            return self;
+        } else {
+            EllipticCurvePoint::new(self.x.clone().unwrap(), -self.y.clone().unwrap())
+        }
+    }
+}
+
+impl<F: Field, E: EllipticCurve> Sub for EllipticCurvePoint<F, E> {
+    type Output = Self;
+    fn sub(self, other: Self) -> Self {
+        self + (-other)
+    }
+}
+
+impl<F: Field, E: EllipticCurve, V: Into<BigInt>> Mul<V> for EllipticCurvePoint<F, E> {
+    type Output = Self;
+
+    fn mul(self, scalar_val: V) -> Self {
+        let scalar: BigInt = scalar_val.into();
         if scalar.is_zero() {
             // Return the point at infinity for scalar * 0
-            return EllipticCurvePoint::point_at_infinity(self.curve.clone());
+            return EllipticCurvePoint::point_at_infinity();
         }
 
-        let mut result = EllipticCurvePoint::point_at_infinity(self.curve.clone());
+        let mut result = EllipticCurvePoint::point_at_infinity();
         let mut current = self.clone(); // Start with the current point
         let mut scalar_bits = scalar.clone();
 
         while scalar_bits > BigInt::zero() {
             if &scalar_bits & BigInt::one() == BigInt::one() {
-                result = result + current.clone(); // Add the current point if the bit is 1
+                result = result.add_ref(&current);
             }
-            current = current.clone() + current.clone(); // Double the point
+            current = current.add_ref(&current); // Double the point
             scalar_bits >>= 1; // Move to the next bit
         }
 
@@ -137,121 +206,161 @@ impl<F: Field> Mul<BigInt> for EllipticCurvePoint<F> {
     }
 }
 
-impl<F: Field> Neg for EllipticCurvePoint<F> {
-    type Output = Self;
-    fn neg(self) -> Self {
+impl<F: Field, E: EllipticCurve> fmt::Display for EllipticCurvePoint<F, E> {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
         if self.is_point_at_infinity() {
-            return self;
+            write!(f, "(x=∞, y=∞)")
         } else {
-            EllipticCurvePoint::new(
-                self.x.clone().unwrap(),
-                -self.y.clone().unwrap(),
-                self.curve,
-            )
+            match (self.x.as_ref(), self.y.as_ref()) {
+                (Some(x), Some(y)) => write!(f, "(x={}, y={})", x, y),
+                (Some(x), None) => write!(f, "({}, ∅)", x),
+                (None, Some(y)) => write!(f, "(∅, {})", y),
+                (None, None) => unreachable!(), // This case is handled by is_infinity()
+            }
         }
     }
 }
 
-impl<F: Field> Sub for EllipticCurvePoint<F> {
-    type Output = Self;
-    fn sub(self, other: Self) -> Self {
-        self + (-other)
-    }
-}
-
-pub fn get_lambda<F: Field>(
-    p: EllipticCurvePoint<F>,
-    q: EllipticCurvePoint<F>,
-    r: EllipticCurvePoint<F>,
+pub fn get_lambda<F: Field, E: EllipticCurve>(
+    p: &EllipticCurvePoint<F, E>,
+    q: &EllipticCurvePoint<F, E>,
+    r: &EllipticCurvePoint<F, E>,
 ) -> F {
-    let p_x = p.x.clone().unwrap();
-    let p_y = p.y.clone().unwrap();
-    let q_x = q.x.clone().unwrap();
+    let p_x = p.x.as_ref().unwrap();
+    let p_y = p.y.as_ref().unwrap();
+    let q_x = q.x.as_ref().unwrap();
     // let q_y = q.y.clone().unwrap();
-    let r_x = r.x.clone().unwrap();
-    let r_y = r.y.clone().unwrap();
+    let r_x = r.x.as_ref().unwrap();
+    let r_y = r.y.as_ref().unwrap();
 
-    if (p == q && p_y.clone() == F::zero()) || (p != q && p_x.clone() == q_x.clone()) {
-        return r_x.clone() - p_x.clone();
+    if (p == q && *p_y == F::zero()) || (p != q && *p_x == *q_x) {
+        return r_x.sub_ref(&p_x);
     }
-    let slope = p.line_slope(q.clone());
-    let numerator = r_y.clone() - p_y.clone() - slope.clone() * (r_x.clone() - p_x.clone());
-    let denominator = r_x.clone() + p_x.clone() + q_x.clone() - slope.clone() * slope.clone();
+    let slope = p.line_slope(&q);
+    let numerator = (r_y.sub_ref(&p_y)).sub_ref(&slope.mul_ref(&(r_x.sub_ref(&p_x))));
+    let denominator = r_x
+        .add_ref(&p_x)
+        .add_ref(&q_x)
+        .sub_ref(&slope.mul_ref(&slope));
     return numerator / denominator;
 }
 
-pub fn miller<F: Field>(p: EllipticCurvePoint<F>, q: EllipticCurvePoint<F>, m: BigInt) -> F {
+pub fn miller<F: Field, E: EllipticCurve>(
+    p: &EllipticCurvePoint<F, E>,
+    q: &EllipticCurvePoint<F, E>,
+    m: BigInt,
+) -> F {
     if p == q {
-        F::one();
+        return F::one();
     }
 
     let mut f = F::one();
     let mut t = p.clone();
 
     for i in (1..m.bits()).rev() {
-        f = (f.clone() * f.clone()) * (get_lambda(t.clone(), t.clone(), q.clone()));
-        t = t.clone() + t.clone();
+        f = (f.mul_ref(&f)) * (get_lambda(&t, &t, &q));
+        t = t.add_ref(&t);
         if m.bit(i) {
-            f = f * (get_lambda(t.clone(), p.clone(), q.clone()));
-            t = t.clone() + p.clone();
+            f = f * (get_lambda(&t, &p, &q));
+            t = t.add_ref(&p);
         }
     }
 
     f
 }
 
-pub fn weil_pairing<F: Field>(
-    p: EllipticCurvePoint<F>,
-    q: EllipticCurvePoint<F>,
+pub fn weil_pairing<F: Field, E: EllipticCurve>(
+    p: &EllipticCurvePoint<F, E>,
+    q: &EllipticCurvePoint<F, E>,
     m: BigInt,
-    s: Option<EllipticCurvePoint<F>>,
+    s: Option<&EllipticCurvePoint<F, E>>,
 ) -> F {
     let s_value = s.unwrap();
-    let fp_qs = miller(p.clone(), q.clone() + s_value.clone(), m.clone());
-    let fp_s = miller(p.clone(), s_value.clone(), m.clone());
-    let fq_ps = miller(q.clone(), p.clone() - s_value.clone(), m.clone());
-    let fq_s = miller(q.clone(), -s_value.clone(), m.clone());
+    let fp_qs = miller(&p, &q.add_ref(&s_value), m.clone());
+    let fp_s = miller(&p, &s_value, m.clone());
+    let fq_ps = miller(&q, &p.add_ref(&(s_value.clone().neg())), m.clone());
+    let fq_s = miller(&q, &(-s_value.clone()), m.clone());
 
     return (fp_qs / fp_s) / (fq_ps / fq_s);
 }
 
+pub fn general_tate_pairing<F: Field, E: EllipticCurve>(
+    p: &EllipticCurvePoint<F, E>,
+    q: &EllipticCurvePoint<F, E>,
+    ell: BigInt,
+    modulus: BigInt,
+    s: Option<&EllipticCurvePoint<F, E>>,
+) -> F {
+    let s_value = s.unwrap();
+    let fp_qs = miller(&p, &q.add_ref(&s_value), ell.clone());
+    let fp_s = miller(&p, &s_value, ell.clone());
+    let f = fp_qs.div_ref(&fp_s);
+
+    return f.pow((modulus - BigInt::one()) / ell);
+}
+
+pub fn tate_pairing<F: Field, E: EllipticCurve>(
+    p: EllipticCurvePoint<F, E>,
+    q: EllipticCurvePoint<F, E>,
+    ell: BigInt,
+    modulus: BigInt,
+) -> F {
+    let fp_q = miller(&p, &q, ell.clone());
+
+    return fp_q.pow((modulus - BigInt::one()) / ell);
+}
+
+#[macro_export]
+macro_rules! define_myzkp_curve_type {
+    ($name:ident, $a:expr, $b:expr) => {
+        #[derive(Debug, Clone, PartialEq)]
+        pub struct $name;
+
+        impl EllipticCurve for $name {
+            fn get_a() -> BigInt {
+                BigInt::from_str($a).unwrap()
+            }
+            fn get_b() -> BigInt {
+                BigInt::from_str($b).unwrap()
+            }
+        }
+    };
+}
+
 #[cfg(test)]
 mod tests {
     use super::*;
+    use crate::modules::ring::Ring;
     use crate::{
         define_myzkp_modulus_type,
         modules::field::{FiniteFieldElement, ModulusValue},
     };
+    use lazy_static::lazy_static;
     use num_bigint::{BigInt, ToBigInt};
+    use paste::paste;
     use std::str::FromStr;
 
+    define_myzkp_modulus_type!(Mod631, "631");
+    define_myzkp_curve_type!(CurveA30B34, "30", "34");
+
     #[test]
     fn test_weil_pairing() {
-        define_myzkp_modulus_type!(Mod631, "631");
-
-        let a = FiniteFieldElement::<Mod631>::from_value(30_i64);
-        let b = FiniteFieldElement::<Mod631>::from_value(34_i64);
-        let curve = EllipticCurve { a, b };
-
-        let p = EllipticCurvePoint::new(
+        let p = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
             FiniteFieldElement::<Mod631>::from_value(36_i64),
             FiniteFieldElement::<Mod631>::from_value(60_i64),
-            curve.clone(),
         );
-        let q = EllipticCurvePoint::new(
+        let q = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
             FiniteFieldElement::<Mod631>::from_value(121_i64),
             FiniteFieldElement::<Mod631>::from_value(387_i64),
-            curve.clone(),
         );
-        let s = EllipticCurvePoint::new(
+        let s = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
             FiniteFieldElement::<Mod631>::from_value(0_i64),
             FiniteFieldElement::<Mod631>::from_value(36_i64),
-            curve.clone(),
         );
         let order = 5.to_bigint().unwrap();
 
-        let fp_qs = miller(p.clone(), q.clone() + s.clone(), order.clone());
-        let fp_s = miller(p.clone(), s.clone(), order.clone());
+        let fp_qs = miller(&p, &(q.add_ref(&s)), order.clone());
+        let fp_s = miller(&p, &s, order.clone());
         assert_eq!(fp_qs.sanitize().value, 103_i32.to_bigint().unwrap());
         assert_eq!(fp_s.sanitize().value, 219_i32.to_bigint().unwrap());
         assert_eq!(
@@ -259,39 +368,112 @@ mod tests {
             473_i32.to_bigint().unwrap()
         );
 
-        let fq_ps = miller(q.clone(), p.clone() - s.clone(), order.clone());
-        let fq_s = miller(q.clone(), -s.clone(), order.clone());
+        let fq_ps = miller(&q, &p.add_ref(&s.clone().neg()), order.clone());
+        let fq_s = miller(&q, &(-s.clone()), order.clone());
         assert_eq!(fq_ps.sanitize().value, 284_i32.to_bigint().unwrap());
         assert_eq!(fq_s.sanitize().value, 204_i32.to_bigint().unwrap());
         assert_eq!((fq_ps / fq_s).sanitize().value, 88_i32.to_bigint().unwrap());
 
-        let w = weil_pairing(p.clone(), q.clone(), order.clone(), Some(s.clone()));
+        let w = weil_pairing(&p, &q, order.clone(), Some(&s));
         assert_eq!(w.sanitize().value, 242.to_bigint().unwrap());
 
-        let p_prime = EllipticCurvePoint::new(
+        let p_prime = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
             FiniteFieldElement::<Mod631>::from_value(617_i64),
             FiniteFieldElement::<Mod631>::from_value(5_i64),
-            curve.clone(),
         );
-        let q_prime = EllipticCurvePoint::new(
+        let q_prime = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
             FiniteFieldElement::<Mod631>::from_value(121_i64),
             FiniteFieldElement::<Mod631>::from_value(244_i64),
-            curve.clone(),
         );
 
-        let w_prime = weil_pairing(
-            p_prime.clone(),
-            q_prime.clone(),
+        let w_prime = weil_pairing(&p_prime, &q_prime, order.clone(), Some(&s));
+        assert_eq!(w_prime.sanitize().value, 512_i32.to_bigint().unwrap());
+
+        let y_prime = weil_pairing(&p_prime, &p_prime, order.clone(), Some(&s));
+        assert_eq!(y_prime.sanitize().value, 1_i32.to_bigint().unwrap());
+
+        let y_prime2 = weil_pairing(
+            &(p_prime.clone() * 3_i32.to_bigint().unwrap()),
+            &(p_prime.clone() * 2_i32.to_bigint().unwrap()),
             order.clone(),
-            Some(s.clone()),
+            Some(&s),
         );
-        assert_eq!(w_prime.sanitize().value, 512_i32.to_bigint().unwrap());
+        assert_eq!(y_prime2.sanitize().value, 1_i32.to_bigint().unwrap());
+
+        let y_prime2 = weil_pairing(
+            &(p_prime.clone() * 3647912234556789907_i64.to_bigint().unwrap()),
+            &(p_prime.clone() * 289_i32.to_bigint().unwrap()),
+            order.clone(),
+            Some(&s),
+        );
+        assert_eq!(y_prime2.sanitize().value, 1_i32.to_bigint().unwrap());
 
         assert_eq!(p.clone() * 3_i32.to_bigint().unwrap(), p_prime.clone());
         assert_eq!(q.clone() * 4_i32.to_bigint().unwrap(), q_prime.clone());
-        assert_eq!(
-            w.pow(12_i32.to_bigint().unwrap()).sanitize(),
-            w_prime.sanitize()
+        assert_eq!(w.pow(12_i32).sanitize(), w_prime.sanitize());
+    }
+
+    #[test]
+    fn test_general_tate_pairing() {
+        let p = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(36_i64),
+            FiniteFieldElement::<Mod631>::from_value(60_i64),
+        );
+        let q = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(121_i64),
+            FiniteFieldElement::<Mod631>::from_value(387_i64),
+        );
+        let p_prime = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(617_i64),
+            FiniteFieldElement::<Mod631>::from_value(5_i64),
         );
+        let q_prime = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(121_i64),
+            FiniteFieldElement::<Mod631>::from_value(244_i64),
+        );
+        let s = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(0_i64),
+            FiniteFieldElement::<Mod631>::from_value(36_i64),
+        );
+        let order = 5.to_bigint().unwrap();
+
+        let tate = general_tate_pairing(&p, &q, order.clone(), BigInt::from(631), Some(&s));
+
+        let tate_prime = general_tate_pairing(
+            &p_prime,
+            &q_prime,
+            order.clone(),
+            BigInt::from(631),
+            Some(&s),
+        );
+
+        assert_eq!(tate.pow(12_i32).sanitize(), tate_prime.sanitize());
+    }
+
+    #[test]
+    fn test_tate_pairing() {
+        let p = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(36_i64),
+            FiniteFieldElement::<Mod631>::from_value(60_i64),
+        );
+        let q = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(121_i64),
+            FiniteFieldElement::<Mod631>::from_value(387_i64),
+        );
+        let p_prime = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(617_i64),
+            FiniteFieldElement::<Mod631>::from_value(5_i64),
+        );
+        let q_prime = EllipticCurvePoint::<FiniteFieldElement<Mod631>, CurveA30B34>::new(
+            FiniteFieldElement::<Mod631>::from_value(121_i64),
+            FiniteFieldElement::<Mod631>::from_value(244_i64),
+        );
+        let order = 5.to_bigint().unwrap();
+
+        let tate = tate_pairing(p, q, order.clone(), BigInt::from(631));
+
+        let tate_prime = tate_pairing(p_prime, q_prime, order.clone(), BigInt::from(631));
+
+        assert_eq!(tate.pow(12_i32).sanitize(), tate_prime.sanitize());
     }
 }
diff --git a/myzkp/src/modules/educational_protocols/dl.rs b/myzkp/src/modules/educational_protocols/dl.rs
index 33be609..9baeeb4 100644
--- a/myzkp/src/modules/educational_protocols/dl.rs
+++ b/myzkp/src/modules/educational_protocols/dl.rs
@@ -1,9 +1,10 @@
-use crate::modules::field::Field;
-use crate::modules::polynomial::Polynomial;
 use num_bigint::{BigInt, RandBigInt, ToBigInt};
 use num_traits::Zero;
 use std::str::FromStr;
 
+use crate::modules::field::Field;
+use crate::modules::polynomial::Polynomial;
+
 pub struct Prover<F: Field> {
     pub p: Polynomial<F>,
     pub t: Polynomial<F>,
@@ -107,8 +108,10 @@ pub fn malicious_discrete_log_protocol<F: Field>(
 #[cfg(test)]
 mod tests {
     use super::*;
+
     use crate::modules::field::{FiniteFieldElement, ModEIP197};
     use crate::modules::polynomial::Polynomial;
+    use crate::modules::ring::Ring;
 
     #[test]
     fn test_dl_protocol() {
diff --git a/myzkp/src/modules/educational_protocols/kea.rs b/myzkp/src/modules/educational_protocols/kea.rs
index 683c514..ae0fbdb 100644
--- a/myzkp/src/modules/educational_protocols/kea.rs
+++ b/myzkp/src/modules/educational_protocols/kea.rs
@@ -1,9 +1,10 @@
-use crate::modules::field::Field;
-use crate::modules::polynomial::Polynomial;
 use num_bigint::{BigInt, RandBigInt, ToBigInt};
 use num_traits::Zero;
 use std::str::FromStr;
 
+use crate::modules::field::Field;
+use crate::modules::polynomial::Polynomial;
+
 pub struct Prover<F: Field> {
     pub p: Polynomial<F>,
     pub t: Polynomial<F>,
@@ -93,8 +94,10 @@ pub fn knowledge_of_exponent_protocol<F: Field>(
 #[cfg(test)]
 mod tests {
     use super::*;
+
     use crate::modules::field::{FiniteFieldElement, ModEIP197};
     use crate::modules::polynomial::Polynomial;
+    use crate::modules::ring::Ring;
 
     #[test]
     fn test_kea_protocol() {
diff --git a/myzkp/src/modules/educational_protocols/mod.rs b/myzkp/src/modules/educational_protocols/mod.rs
index 7cf8605..bdc2eec 100644
--- a/myzkp/src/modules/educational_protocols/mod.rs
+++ b/myzkp/src/modules/educational_protocols/mod.rs
@@ -1,4 +1,6 @@
 pub mod dl;
 pub mod kea;
 pub mod naive;
+//pub mod nizk;
 pub mod sz;
+pub mod zk;
diff --git a/myzkp/src/modules/educational_protocols/naive.rs b/myzkp/src/modules/educational_protocols/naive.rs
index 3e35616..0a0bb5d 100644
--- a/myzkp/src/modules/educational_protocols/naive.rs
+++ b/myzkp/src/modules/educational_protocols/naive.rs
@@ -65,13 +65,17 @@ pub fn naive_protocol<F: Field>(prover: &Prover<F>, verifier: &Verifier<F>, modu
 #[cfg(test)]
 mod tests {
     use super::*;
-    use crate::modules::polynomial::Polynomial;
+    use num_bigint::BigInt;
+    use std::str::FromStr;
+
+    use crate::modules::ring::Ring;
     use crate::{
         define_myzkp_modulus_type,
         modules::field::{FiniteFieldElement, ModulusValue},
+        modules::polynomial::Polynomial,
     };
-    use num_bigint::BigInt;
-    use std::str::FromStr;
+    use lazy_static::lazy_static;
+    use paste::paste;
 
     #[test]
     fn test_naive_protocol() {
diff --git a/myzkp/src/modules/educational_protocols/nizk.rs b/myzkp/src/modules/educational_protocols/nizk.rs
new file mode 100644
index 0000000..7ae4599
--- /dev/null
+++ b/myzkp/src/modules/educational_protocols/nizk.rs
@@ -0,0 +1,231 @@
+use crate::modules::curve::{weil_pairing, EllipticCurve, EllipticCurvePoint};
+use crate::modules::field::Field;
+use crate::modules::polynomial::Polynomial;
+use num_bigint::BigInt;
+
+pub struct ProofKey<F: Field, E: EllipticCurve> {
+    alpha: Vec<EllipticCurvePoint<F, E>>,
+    alpha_prime: Vec<EllipticCurvePoint<F, E>>,
+}
+
+pub struct VerificationKey<F: Field, E: EllipticCurve> {
+    g_r: EllipticCurvePoint<F, E>,
+    g_t_s: EllipticCurvePoint<F, E>,
+}
+
+pub struct Proof<F: Field, E: EllipticCurve> {
+    u_prime: EllipticCurvePoint<F, E>,
+    v_prime: EllipticCurvePoint<F, E>,
+    w_prime: EllipticCurvePoint<F, E>,
+}
+
+pub struct TrustedSetup<F: Field, E: EllipticCurve> {
+    proof_key: ProofKey<F, E>,
+    verification_key: VerificationKey<F, E>,
+}
+
+impl<F: Field, E: EllipticCurve> TrustedSetup<F, E> {
+    pub fn generate(
+        g: &EllipticCurvePoint<F, E>,
+        t: &Polynomial<F>,
+        n: usize,
+        s: &F,
+        r: &F,
+    ) -> Self {
+        let mut alpha = Vec::with_capacity(n);
+        let mut alpha_prime = Vec::with_capacity(n);
+
+        let mut s_power = F::one();
+        for _ in 0..1 + n {
+            alpha.push(g.clone() * s_power.clone().get_value());
+            alpha_prime.push(g.clone() * (s_power.clone() * r.clone()).get_value());
+            s_power = s_power * s.clone();
+        }
+
+        let g_r = g.clone() * r.clone().get_value();
+        let g_t_s = g.clone() * t.eval(s).get_value();
+
+        TrustedSetup {
+            proof_key: ProofKey { alpha, alpha_prime },
+            verification_key: VerificationKey { g_r, g_t_s },
+        }
+    }
+}
+
+pub struct Prover<F: Field, E: EllipticCurve> {
+    pub p: Polynomial<F>,
+    pub h: Polynomial<F>,
+    pub g: EllipticCurvePoint<F, E>,
+}
+
+impl<F: Field, E: EllipticCurve> Prover<F, E> {
+    pub fn new(p: Polynomial<F>, t: Polynomial<F>, g: EllipticCurvePoint<F, E>) -> Self {
+        let h = p.clone() / t;
+        Prover { p, h, g }
+    }
+
+    pub fn generate_proof(&self, proof_key: &ProofKey<F, E>, delta: &F) -> Proof<F, E> {
+        let g_p = self.p.eval_with_powers_on_curve(&proof_key.alpha);
+        let g_h = self.h.eval_with_powers_on_curve(&proof_key.alpha);
+        let g_p_prime = self.p.eval_with_powers_on_curve(&proof_key.alpha_prime);
+
+        Proof {
+            u_prime: g_p * delta.get_value(),
+            v_prime: g_h * delta.get_value(),
+            w_prime: g_p_prime * delta.get_value(),
+        }
+    }
+}
+
+pub struct Verifier<F: Field, E: EllipticCurve> {
+    g: EllipticCurvePoint<F, E>,
+}
+
+impl<F: Field, E: EllipticCurve> Verifier<F, E> {
+    pub fn new(g: EllipticCurvePoint<F, E>) -> Self {
+        Verifier { g }
+    }
+
+    pub fn verify(
+        &self,
+        proof: &Proof<F, E>,
+        vk: &VerificationKey<F, E>,
+        curve_order: BigInt,
+        s: EllipticCurvePoint<F, E>,
+    ) -> bool {
+        // Check e(u', g^r) = e(w', g)
+        let pairing1 = weil_pairing(
+            proof.u_prime.clone(),
+            vk.g_r.clone(),
+            BigInt::from(curve_order.clone()),
+            Some(self.g.clone()),
+        );
+        let pairing2 = weil_pairing(
+            proof.w_prime.clone(),
+            self.g.clone(),
+            BigInt::from(curve_order.clone()),
+            Some(self.g.clone()),
+        );
+        let check1 = pairing1 == pairing2;
+
+        println!("^^^^ {} {} {}", pairing1, pairing2, check1);
+
+        println!(
+            "{}",
+            proof.u_prime.clone() == self.g.clone() * BigInt::from(24)
+        );
+
+        println!(
+            "{}",
+            proof.v_prime.clone() == self.g.clone() * BigInt::from(4)
+        );
+
+        println!("{}", vk.g_t_s.clone() == self.g.clone() * BigInt::from(6));
+
+        // Check e(u', g^t) = e(v', g)
+        let pairing3 = weil_pairing(
+            proof.u_prime.clone(),
+            self.g.clone(),
+            BigInt::from(curve_order.clone()),
+            Some(s.clone()),
+        );
+        let pairing4 = weil_pairing(
+            proof.v_prime.clone(),
+            vk.g_t_s.clone(),
+            BigInt::from(curve_order.clone()),
+            Some(s.clone()),
+        );
+        let check2 = pairing3 == pairing4;
+
+        println!("^^^^ {} {} {}", pairing3, pairing4, check2);
+
+        check1 && check2
+    }
+}
+
+pub fn non_interactive_zkp_protocol<F: Field, E: EllipticCurve>(
+    prover: &Prover<F, E>,
+    verifier: &Verifier<F, E>,
+    setup: &TrustedSetup<F, E>,
+    delta: &F,
+    curve_order: BigInt,
+    ms: EllipticCurvePoint<F, E>,
+) -> bool {
+    let proof = prover.generate_proof(&setup.proof_key, delta);
+    verifier.verify(&proof, &setup.verification_key, curve_order, ms)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use std::str::FromStr;
+
+    use crate::modules::field::ModulusValue;
+    use crate::modules::ring::Ring;
+    use crate::{
+        define_myzkp_curve_type, define_myzkp_modulus_type,
+        modules::field::{FiniteFieldElement, ModEIP197},
+    };
+
+    #[test]
+    fn test_non_interactive_zkp() {
+        define_myzkp_modulus_type!(Mod631, "631");
+        define_myzkp_curve_type!(BLS12to381, "1", "4");
+        type F = FiniteFieldElement<Mod631>;
+        type E = CurveA30B34;
+
+        // Set up the generator point
+        let g = EllipticCurvePoint::<F, E>::new(F::from_value(36), F::from_value(60));
+
+        // Set up the polynomials
+        let p =
+            Polynomial::from_monomials(&[F::from_value(-1), F::from_value(-2), F::from_value(-3)]);
+        let t = Polynomial::from_monomials(&[F::from_value(-1), F::from_value(-2)]);
+
+        // Generate trusted setup
+        let s = F::from_value(2);
+        let r = F::from_value(1); //F::from_value(456);
+        let setup = TrustedSetup::generate(&g, &t, 3, &s, &r);
+
+        // Create prover and verifier
+        let prover = Prover::new(p, t.clone(), g.clone());
+        let verifier = Verifier::new(g.clone());
+
+        let ms = EllipticCurvePoint::<F, E>::new(F::from_value(0_i64), F::from_value(36_i64));
+
+        // Run the protocol
+        let delta = F::from_value(23);
+        let order = BigInt::from_str("5").unwrap();
+        let result = non_interactive_zkp_protocol(&prover, &verifier, &setup, &delta, order, ms);
+
+        let proof = prover.generate_proof(&setup.proof_key, &delta);
+        /*
+        println!(
+            "v_prime={}, {}",
+            proof.v_prime.clone(),
+            proof.v_prime == g.clone() * prover.h.eval(&s).get_value()
+        );
+
+        println!(
+            "p(s)={} h(s)={} t(s)={}",
+            prover.p.eval(&s).get_value(),
+            prover.h.eval(&s).get_value(),
+            t.eval(&s).get_value()
+        );
+        */
+        println!(
+            "@@  {}",
+            //proof.u_prime.clone() * r.get_value(),
+            //proof.w_prime.clone(),
+            proof.u_prime.clone() * r.get_value() == proof.w_prime
+        );
+        println!(
+            "@@  {}",
+            //proof.u_prime.clone(),
+            //proof.v_prime.clone() * t.eval(&s).get_value(),
+            proof.u_prime.clone() == proof.v_prime * t.eval(&s).get_value()
+        );
+
+        assert!(result);
+    }
+}
diff --git a/myzkp/src/modules/educational_protocols/sz.rs b/myzkp/src/modules/educational_protocols/sz.rs
index bfd5417..44f9f52 100644
--- a/myzkp/src/modules/educational_protocols/sz.rs
+++ b/myzkp/src/modules/educational_protocols/sz.rs
@@ -85,8 +85,10 @@ pub fn malicious_schwartz_zippel_protocol<F: Field>(
 #[cfg(test)]
 mod tests {
     use super::*;
+
     use crate::modules::field::{FiniteFieldElement, ModEIP197};
     use crate::modules::polynomial::Polynomial;
+    use crate::modules::ring::Ring;
 
     #[test]
     fn test_sz_protocol() {
diff --git a/myzkp/src/modules/educational_protocols/zk.rs b/myzkp/src/modules/educational_protocols/zk.rs
index 6b7e4b7..3577f36 100644
--- a/myzkp/src/modules/educational_protocols/zk.rs
+++ b/myzkp/src/modules/educational_protocols/zk.rs
@@ -1,16 +1,17 @@
-use crate::modules::field::Field;
-use crate::modules::polynomial::Polynomial;
 use num_bigint::{BigInt, RandBigInt, ToBigInt};
 use num_traits::Zero;
 use std::str::FromStr;
 
-struct Prover<F: Field> {
-    p: Polynomial<F>,
-    t: Polynomial<F>,
-    h: Polynomial<F>,
+use crate::modules::field::Field;
+use crate::modules::polynomial::Polynomial;
+
+pub struct Prover<F: Field> {
+    pub p: Polynomial<F>,
+    pub t: Polynomial<F>,
+    pub h: Polynomial<F>,
 }
 
-struct Verifier<F: Field> {
+pub struct Verifier<F: Field> {
     t: Polynomial<F>,
     s: F,
     r: F,
@@ -18,88 +19,109 @@ struct Verifier<F: Field> {
 }
 
 impl<F: Field> Prover<F> {
-    fn new(p: Polynomial<F>, t: Polynomial<F>) -> Self {
+    pub fn new(p: Polynomial<F>, t: Polynomial<F>) -> Self {
         let h = p.clone() / t.clone();
         Prover { p, t, h }
     }
 
-    fn compute_values(&self, g: &F, alpha_powers: &[F], alpha_prime_powers: &[F]) -> (F, F, F) {
+    pub fn compute_values(&self, alpha_powers: &[F], alpha_prime_powers: &[F]) -> (F, F, F) {
         let mut rng = rand::thread_rng();
-        let delta = FiniteFieldElement::<MODULUS>::from(rng.gen_range(0..MODULUS));
+        let delta = F::from_value(rng.gen_bigint_range(
+            &BigInt::zero(),
+            &BigInt::from_str("4835703278458516698824704").unwrap(), // 2^82
+        ));
 
-        let g_p = self.p.eval_with_powers(g, alpha_powers);
-        let g_h = self.h.eval_with_powers(g, alpha_powers);
-        let g_p_prime = self.p.eval_with_powers(g, alpha_prime_powers);
+        let g_p = self.p.eval_with_powers(alpha_powers);
+        let g_h = self.h.eval_with_powers(alpha_powers);
+        let g_p_prime = self.p.eval_with_powers(alpha_prime_powers);
 
-        let u_prime = g_p.pow(delta.value);
-        let v_prime = g_h.pow(delta.value);
-        let w_prime = g_p_prime.pow(delta.value);
+        let u_prime = g_p.pow(delta.get_value());
+        let v_prime = g_h.pow(delta.get_value());
+        let w_prime = g_p_prime.pow(delta.get_value());
 
         (u_prime, v_prime, w_prime)
     }
 }
 
 impl<F: Field> Verifier<F> {
-    fn new(t: Polynomial<F>) -> Self {
+    pub fn new(t: Polynomial<F>, generator: i128) -> Self {
         let mut rng = rand::thread_rng();
-        let s = FiniteFieldElement::<MODULUS>::from(rng.gen_range(0..MODULUS));
-        let r = FiniteFieldElement::<MODULUS>::from(rng.gen_range(0..MODULUS));
-        let g = FiniteFieldElement::<MODULUS>::from(GENERATOR);
+        let s = F::from_value(rng.gen_bigint_range(
+            &BigInt::zero(),
+            &BigInt::from_str("4835703278458516698824704").unwrap(), // 2^82
+        ));
+        let r = F::from_value(rng.gen_bigint_range(
+            &BigInt::zero(),
+            &BigInt::from_str("4835703278458516698824704").unwrap(), // 2^82
+        ));
+        let g = F::from_value(generator);
         Verifier { t, s, r, g }
     }
 
-    fn generate_challenge(&self, max_degree: usize) -> (F, Vec<F>, Vec<F>) {
-        let alpha = self.g.pow(self.s.clone().value);
-        let alpha_prime = alpha.pow(self.r.clone().value);
+    pub fn generate_challenge(&self, max_degree: usize) -> (Vec<F>, Vec<F>) {
+        let mut alpha_powers = vec![];
+        let mut alpha_prime_powers = vec![];
 
-        let mut alpha_powers = vec![alpha.clone()];
-        let mut alpha_prime_powers = vec![alpha_prime.clone()];
-
-        for _ in 1..max_degree {
-            alpha_powers.push(alpha_powers.last().unwrap() * alpha.clone());
-            alpha_prime_powers.push(alpha_prime_powers.last().unwrap() * alpha_prime.clone());
+        for i in 0..(max_degree + 1) {
+            alpha_powers.push(
+                self.g
+                    .pow(self.s.clone().pow(i.to_bigint().unwrap()).get_value()),
+            );
+            alpha_prime_powers.push(alpha_powers.last().unwrap().pow(self.r.get_value()));
         }
 
-        (self.g.clone(), alpha_powers, alpha_prime_powers)
+        (alpha_powers, alpha_prime_powers)
     }
 
-    fn verify(&self, u_prime: &F, v_prime: &F, w_prime: &F) -> bool {
+    pub fn verify(&self, u_prime: &F, v_prime: &F, w_prime: &F) -> bool {
         let t_s = self.t.eval(&self.s);
-        let u_prime_r = u_prime.pow(self.r.clone().value);
+        let u_prime_r = u_prime.pow(self.r.clone().get_value());
 
         // Check 1: u'^r = w'
         let check1 = u_prime_r == *w_prime;
 
         // Check 2: u' = v'^t
-        let check2 = *u_prime == v_prime.pow(t_s.value);
+        let check2 = *u_prime == v_prime.pow(t_s.get_value());
 
         check1 && check2
     }
 }
 
-fn zk_snark_protocol<F: Field>(prover: &Prover<F>, verifier: &Verifier<F>) -> bool {
+pub fn zk_protocol<F: Field>(prover: &Prover<F>, verifier: &Verifier<F>) -> bool {
     // Step 1 & 2: Verifier generates a challenge
     let max_degree = std::cmp::max(prover.p.degree(), prover.h.degree()) as usize;
-    let (g, alpha_powers, alpha_prime_powers) = verifier.generate_challenge(max_degree + 1);
+    let (alpha_powers, alpha_prime_powers) = verifier.generate_challenge(max_degree + 1);
 
     // Step 3 & 4: Prover computes and sends u' = (g^p)^δ, v' = (g^h)^δ, and w' = (g^p')^δ
-    let (u_prime, v_prime, w_prime) = prover.compute_values(&g, &alpha_powers, &alpha_prime_powers);
+    let (u_prime, v_prime, w_prime) = prover.compute_values(&alpha_powers, &alpha_prime_powers);
 
     // Step 5 & 6: Verifier checks whether u'^r = w' and u' = v'^t
     verifier.verify(&u_prime, &v_prime, &w_prime)
 }
 
-fn main() {
-    type F = FiniteFieldElement<MODULUS>;
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    use crate::modules::field::{FiniteFieldElement, ModEIP197};
+    use crate::modules::polynomial::Polynomial;
+    use crate::modules::ring::Ring;
+
+    #[test]
+    fn test_kea_protocol() {
+        const GENERATOR: i128 = 5; // A primitive root modulo MODULUS
 
-    // Create polynomials P(x) and T(x)
-    let p = Polynomial::from_monomials(&[F::from(1), F::from(2), F::from(3)]);
-    let t = Polynomial::from_monomials(&[F::from(1), F::from(2)]);
+        type F = FiniteFieldElement<ModEIP197>;
 
-    // ZK-SNARK protocol
-    let prover = Prover::new(p.clone(), t.clone());
-    let verifier = Verifier::new(t.clone());
+        // Create polynomials P(x) and T(x)
+        let p = Polynomial::from_monomials(&[F::from_value(1), F::from_value(2), F::from_value(3)]);
+        let t = Polynomial::from_monomials(&[F::from_value(1), F::from_value(2)]);
 
-    let result = zk_snark_protocol(&prover, &verifier);
-    println!("ZK-SNARK protocol result: {}", result);
+        // ZK protocol
+        let prover = Prover::new(p.clone(), t.clone());
+        let verifier = Verifier::new(t.clone(), GENERATOR);
+
+        let result = zk_protocol(&prover, &verifier);
+        println!("ZK protocol result: {}", result);
+    }
 }
diff --git a/myzkp/src/modules/efield.rs b/myzkp/src/modules/efield.rs
new file mode 100644
index 0000000..a48d045
--- /dev/null
+++ b/myzkp/src/modules/efield.rs
@@ -0,0 +1,399 @@
+use num_bigint::BigInt;
+use num_traits::One;
+use num_traits::Zero;
+use std::fmt;
+use std::fmt::Debug;
+use std::hash::Hash;
+use std::hash::Hasher;
+use std::marker::PhantomData;
+use std::ops::{Add, Div, Mul, Neg, Sub};
+
+use crate::modules::field::{Field, FiniteFieldElement, ModulusValue};
+use crate::modules::polynomial::Polynomial;
+use crate::modules::ring::Ring;
+use crate::modules::utils::extended_euclidean;
+
+pub trait IrreduciblePoly<F: Field>: Debug + Clone + Hash {
+    fn modulus() -> &'static Polynomial<F>;
+}
+
+#[derive(Clone, Debug)]
+pub struct ExtendedFieldElement<M: ModulusValue, P: IrreduciblePoly<FiniteFieldElement<M>>> {
+    pub poly: Polynomial<FiniteFieldElement<M>>,
+    _phantom: PhantomData<P>,
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>>
+    ExtendedFieldElement<M, P>
+{
+    pub fn new(poly: Polynomial<FiniteFieldElement<M>>) -> Self {
+        let result = Self {
+            poly: poly,
+            _phantom: PhantomData,
+        };
+        result.reduce()
+    }
+
+    fn reduce(&self) -> Self {
+        Self {
+            poly: &self.poly.reduce() % P::modulus(),
+            _phantom: PhantomData,
+        }
+    }
+
+    pub fn degree(&self) -> isize {
+        P::modulus().degree()
+    }
+
+    pub fn from_base_field(value: FiniteFieldElement<M>) -> Self {
+        Self::new((Polynomial { coef: vec![value] }).reduce()).reduce()
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Field
+    for ExtendedFieldElement<M, P>
+{
+    fn inverse(&self) -> Self {
+        //if self.poly.is_zero() {
+        //    return None; // Zero has no inverse
+        //}
+
+        let mut lm = Polynomial::<FiniteFieldElement<M>>::one();
+        let mut hm = Polynomial::zero();
+        let mut low = self.poly.clone();
+        let mut high = P::modulus().clone();
+
+        while !low.is_zero() {
+            let q = &high / &low;
+            let r = &high % &low;
+            let nm = hm - (&lm * &q);
+            high = low;
+            hm = lm;
+            low = r;
+            lm = nm;
+        }
+
+        //if high.degree() != 0 {
+        //    return None; // Not invertible
+        //}
+
+        Self::new(hm * high.coef[0].inverse())
+    }
+
+    fn div_ref(&self, other: &Self) -> Self {
+        self.mul_ref(&other.inverse())
+    }
+}
+
+impl<M: ModulusValue, P: IrreduciblePoly<FiniteFieldElement<M>>> Hash
+    for ExtendedFieldElement<M, P>
+{
+    fn hash<H: Hasher>(&self, state: &mut H) {
+        for v in &self.poly.coef {
+            v.hash(state);
+        }
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> PartialEq
+    for ExtendedFieldElement<M, P>
+{
+    fn eq(&self, other: &Self) -> bool {
+        self.poly == other.poly
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Eq
+    for ExtendedFieldElement<M, P>
+{
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Zero
+    for ExtendedFieldElement<M, P>
+{
+    fn zero() -> Self {
+        ExtendedFieldElement::<M, P>::new(Polynomial::<FiniteFieldElement<M>>::zero())
+    }
+
+    fn is_zero(&self) -> bool {
+        self.poly.is_zero()
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> One
+    for ExtendedFieldElement<M, P>
+{
+    fn one() -> Self {
+        ExtendedFieldElement::<M, P>::new(Polynomial::<FiniteFieldElement<M>>::one())
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Add
+    for ExtendedFieldElement<M, P>
+{
+    type Output = Self;
+
+    fn add(self, other: Self) -> Self {
+        self.add_ref(&other)
+    }
+}
+
+impl<'a, M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>>
+    Add<&'a ExtendedFieldElement<M, P>> for ExtendedFieldElement<M, P>
+{
+    type Output = ExtendedFieldElement<M, P>;
+
+    fn add(self, other: &'a ExtendedFieldElement<M, P>) -> ExtendedFieldElement<M, P> {
+        self.add_ref(other)
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Sub
+    for ExtendedFieldElement<M, P>
+{
+    type Output = Self;
+
+    fn sub(self, other: Self) -> Self {
+        self.sub_ref(&other)
+    }
+}
+
+impl<'a, M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>>
+    Sub<&'a ExtendedFieldElement<M, P>> for ExtendedFieldElement<M, P>
+{
+    type Output = ExtendedFieldElement<M, P>;
+
+    fn sub(self, other: &'a ExtendedFieldElement<M, P>) -> ExtendedFieldElement<M, P> {
+        self.sub_ref(other)
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Mul
+    for ExtendedFieldElement<M, P>
+{
+    type Output = Self;
+
+    fn mul(self, other: Self) -> Self {
+        self.mul_ref(&other)
+    }
+}
+
+impl<'a, M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>>
+    Mul<&'a ExtendedFieldElement<M, P>> for ExtendedFieldElement<M, P>
+{
+    type Output = ExtendedFieldElement<M, P>;
+
+    fn mul(self, other: &'a ExtendedFieldElement<M, P>) -> ExtendedFieldElement<M, P> {
+        self.mul_ref(other)
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Div
+    for ExtendedFieldElement<M, P>
+{
+    type Output = Self;
+
+    fn div(self, other: Self) -> Self {
+        self.div_ref(&other)
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Neg
+    for ExtendedFieldElement<M, P>
+{
+    type Output = Self;
+
+    fn neg(self) -> Self {
+        Self::new(-self.poly)
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> Ring
+    for ExtendedFieldElement<M, P>
+{
+    fn add_ref(&self, other: &Self) -> Self {
+        Self::new(&self.poly + &other.poly)
+    }
+
+    fn mul_ref(&self, other: &Self) -> Self {
+        Self::new(&self.poly * &other.poly)
+    }
+
+    fn sub_ref(&self, other: &Self) -> Self {
+        Self::new(&self.poly - &other.poly)
+    }
+
+    fn pow<V: Into<BigInt>>(&self, n: V) -> Self {
+        let mut base = self.clone();
+        let mut exponent: BigInt = n.into();
+
+        let mut result = Self::one();
+        while exponent > BigInt::zero() {
+            if &exponent % BigInt::from(2) == BigInt::one() {
+                result = result.mul_ref(&base);
+            }
+            exponent /= 2;
+            base = base.mul_ref(&base);
+        }
+
+        result
+    }
+
+    fn get_value(&self) -> BigInt {
+        unimplemented!("Not applicable for extended field elements")
+    }
+
+    fn from_value<V: Into<BigInt>>(value: V) -> Self {
+        ExtendedFieldElement::<M, P>::new(
+            Polynomial::<FiniteFieldElement<M>>::one() * FiniteFieldElement::<M>::from_value(value),
+        )
+    }
+
+    fn random_element(_exclude_elements: &[Self]) -> Self {
+        unimplemented!("Random element generation for extended field not implemented")
+    }
+}
+
+impl<M: ModulusValue + 'static, P: IrreduciblePoly<FiniteFieldElement<M>>> fmt::Display
+    for ExtendedFieldElement<M, P>
+{
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        write!(f, "{}", self.poly)
+    }
+}
+
+#[macro_export]
+macro_rules! define_extension_field {
+    ($name:ident, $base_field:ty, $modulus:expr) => {
+        paste! {#[derive(Debug, Clone, PartialEq, Hash)]
+            pub struct $name;
+
+            lazy_static! {
+                static ref [<MODULUS_ $name>]: Polynomial<$base_field> = $modulus;
+            }
+
+            impl IrreduciblePoly<$base_field> for $name {
+                fn modulus() -> &'static Polynomial<$base_field> {
+                    &[<MODULUS_ $name>]
+                }
+            }
+        }
+    };
+}
+
+// Test the implementation
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use lazy_static::lazy_static;
+    use paste::paste;
+    use std::str::FromStr;
+
+    use crate::define_myzkp_modulus_type;
+
+    define_myzkp_modulus_type!(Mod2, "2");
+    define_myzkp_modulus_type!(Mod7, "7");
+
+    define_extension_field!(
+        Ip2,
+        FiniteFieldElement<Mod2>,
+        Polynomial {
+            coef: vec![
+                FiniteFieldElement::<Mod2>::from_value(1),
+                FiniteFieldElement::<Mod2>::from_value(1),
+                FiniteFieldElement::<Mod2>::from_value(1),
+            ],
+        }
+    );
+
+    define_extension_field!(
+        Ip7,
+        FiniteFieldElement<Mod7>,
+        Polynomial {
+            coef: vec![
+                FiniteFieldElement::<Mod7>::from_value(1),
+                FiniteFieldElement::<Mod7>::from_value(0),
+                FiniteFieldElement::<Mod7>::from_value(1),
+            ],
+        }
+    );
+
+    #[test]
+    fn test_extended_field_operations_mod2() {
+        // x + 1
+        let a = ExtendedFieldElement::<Mod2, Ip2>::new(Polynomial {
+            coef: vec![
+                FiniteFieldElement::from_value(1),
+                FiniteFieldElement::from_value(1),
+            ],
+        });
+
+        // x
+        let b = ExtendedFieldElement::<Mod2, Ip2>::new(Polynomial {
+            coef: vec![
+                FiniteFieldElement::from_value(0),
+                FiniteFieldElement::from_value(1),
+            ],
+        });
+
+        // Addition
+        let sum = a.add_ref(&b);
+        assert_eq!(
+            sum,
+            ExtendedFieldElement::<Mod2, Ip2>::new(Polynomial {
+                coef: vec![FiniteFieldElement::from_value(1)],
+            })
+        );
+
+        // Multiplication
+        let product = a.mul_ref(&b);
+        assert_eq!(
+            product,
+            ExtendedFieldElement::<Mod2, Ip2>::new(Polynomial {
+                coef: vec![FiniteFieldElement::from_value(1)],
+            },)
+        );
+
+        // Inverse
+        let inv_a = a.inverse();
+        let product = a.mul_ref(&inv_a);
+        assert_eq!(
+            product,
+            ExtendedFieldElement::<Mod2, Ip2>::from_base_field(FiniteFieldElement::one())
+        );
+
+        assert_eq!(
+            a.div_ref(&a),
+            ExtendedFieldElement::<Mod2, Ip2>::from_base_field(FiniteFieldElement::one())
+        );
+    }
+
+    #[test]
+    fn test_extended_field_operations_mod7() {
+        // x + 2
+        let a = ExtendedFieldElement::<Mod7, Ip7>::new(Polynomial {
+            coef: vec![
+                FiniteFieldElement::from_value(2),
+                FiniteFieldElement::from_value(1),
+            ],
+        });
+
+        // 4x + 6
+        let b = ExtendedFieldElement::<Mod7, Ip7>::new(Polynomial {
+            coef: vec![
+                FiniteFieldElement::from_value(6),
+                FiniteFieldElement::from_value(4),
+            ],
+        });
+
+        // Inverse
+        let inv_a = a.inverse();
+        assert_eq!(inv_a, b);
+
+        let product = a.mul_ref(&inv_a);
+        assert_eq!(
+            product,
+            ExtendedFieldElement::<Mod7, Ip7>::from_base_field(FiniteFieldElement::one())
+        );
+    }
+}
diff --git a/myzkp/src/modules/field.rs b/myzkp/src/modules/field.rs
index 15bc061..8ce4ef2 100644
--- a/myzkp/src/modules/field.rs
+++ b/myzkp/src/modules/field.rs
@@ -1,5 +1,7 @@
-use num_bigint::{BigInt, RandBigInt, ToBigInt};
+use lazy_static::lazy_static;
+use num_bigint::{BigInt, RandBigInt};
 use num_traits::{One, Signed, Zero};
+use paste::paste;
 use std::fmt;
 use std::fmt::Debug;
 use std::hash::Hash;
@@ -9,17 +11,13 @@ use std::ops::{Add, Div, Mul, Neg, Sub};
 use std::str::FromStr;
 
 use crate::modules::ring::Ring;
+use crate::modules::utils::extended_euclidean;
 
 pub trait Field: Ring + Div<Output = Self> + PartialEq + Eq + Hash {
     // A commutative ring with a multiplicative identity element
     // where every non-zero element has a multiplicative inverse is called a field.
     fn inverse(&self) -> Self;
-
-    // Utility functions
-    fn get_value(&self) -> BigInt;
-    fn pow(&self, n: BigInt) -> Self;
-    fn from_value<M: Into<BigInt>>(value: M) -> Self;
-    fn random_element(exclude_elements: &[Self]) -> Self;
+    fn div_ref(&self, other: &Self) -> Self;
 }
 
 #[derive(Debug, Clone)]
@@ -29,18 +27,14 @@ pub struct FiniteFieldElement<M> {
 }
 
 pub trait ModulusValue: Debug + Clone + Hash {
-    fn modulus() -> BigInt;
+    fn modulus() -> &'static BigInt;
 }
 
 impl<M: ModulusValue> FiniteFieldElement<M> {
     // Constructor with optional modulus and generator_value.
     pub fn new(value: BigInt) -> Self {
         let modulus = M::modulus();
-        let value_sanitized = value % &modulus;
-
-        //if value_sanitized < 0_i32.to_bigint().unwrap() {
-        //    value_sanitized += modulus.clone();
-        //}
+        let value_sanitized = value % modulus;
 
         FiniteFieldElement {
             value: value_sanitized,
@@ -50,9 +44,9 @@ impl<M: ModulusValue> FiniteFieldElement<M> {
 
     pub fn sanitize(&self) -> Self {
         let modulus = M::modulus();
-        let mut value_sanitized = self.value.clone() % &modulus;
-        if value_sanitized < 0_i32.to_bigint().unwrap() {
-            value_sanitized += modulus.clone();
+        let mut value_sanitized = &self.value % modulus;
+        if value_sanitized < BigInt::zero() {
+            value_sanitized += modulus;
         }
         FiniteFieldElement {
             value: value_sanitized,
@@ -66,12 +60,12 @@ impl<M: ModulusValue> FiniteFieldElement<M> {
         let identity = FiniteFieldElement::one();
         let mut h = identity.clone();
         for _ in 1..n {
-            h = h * self.clone();
+            h = h * self;
             if h == identity {
                 return false;
             }
         }
-        h * self.clone() == identity
+        h * self == identity
     }
 
     /*
@@ -81,11 +75,18 @@ impl<M: ModulusValue> FiniteFieldElement<M> {
     }
     */
 }
-impl<M: ModulusValue> Ring for FiniteFieldElement<M> {
+
+impl<M: ModulusValue> Zero for FiniteFieldElement<M> {
     fn zero() -> Self {
         FiniteFieldElement::<M>::new(BigInt::zero())
     }
 
+    fn is_zero(&self) -> bool {
+        self.value.is_zero()
+    }
+}
+
+impl<M: ModulusValue> One for FiniteFieldElement<M> {
     fn one() -> Self {
         FiniteFieldElement::<M>::new(BigInt::one())
     }
@@ -98,52 +99,58 @@ impl<M: ModulusValue> Hash for FiniteFieldElement<M> {
 }
 
 impl<M: ModulusValue> Field for FiniteFieldElement<M> {
-    fn get_value(&self) -> BigInt {
-        self.value.clone()
-    }
     fn inverse(&self) -> Self {
         let modulus = M::modulus();
-        let mut t = BigInt::zero();
-        let mut new_t = BigInt::one();
         let mut r = modulus.clone();
         let mut new_r = self.value.clone();
 
         // Ensure r and new_r are non-negative
         if r.is_negative() {
-            r += &modulus;
+            r += modulus;
         }
         if new_r.is_negative() {
-            new_r += &modulus;
+            new_r += modulus;
         }
 
-        while !new_r.is_zero() {
-            let quotient = &r / &new_r;
-
-            // Update (t, new_t) = (new_t, t - quotient * new_t)
-            let temp_t = new_t.clone();
-            new_t = &t - &quotient * &new_t;
-            t = temp_t;
-
-            // Update (r, new_r) = (new_r, r - quotient * new_r)
-            let temp_r = new_r.clone();
-            new_r = &r - &quotient * &new_r;
-            r = temp_r;
-        }
+        let (_, _, mut t) = extended_euclidean(r, new_r);
 
         // At this point, r should be 1 if the inverse exists
-        assert_eq!(r, BigInt::one(), "Inverse does not exist");
-
+        //if final_r == BigInt::one() {
         // Ensure t is within the correct range
-        t %= &modulus;
+        t %= modulus;
         if t.is_negative() {
-            t += &modulus;
+            t += modulus;
         }
 
         FiniteFieldElement::<M>::new(t)
+        //} else {
+        //    panic!("r={}: Inverse does not exist", r)
+        //}
+    }
+
+    fn div_ref(&self, other: &Self) -> Self {
+        self.mul_ref(&other.inverse())
+    }
+}
+
+impl<M: ModulusValue> Ring for FiniteFieldElement<M> {
+    fn add_ref(&self, other: &Self) -> Self {
+        let modulus = M::modulus();
+        FiniteFieldElement::<M>::new((&self.value + &other.value) % modulus)
+    }
+
+    fn mul_ref(&self, other: &Self) -> Self {
+        let modulus = M::modulus();
+        FiniteFieldElement::<M>::new((&self.value * &other.value) % modulus)
     }
 
-    fn pow(&self, n: BigInt) -> Self {
-        let mut exponent = n;
+    fn sub_ref(&self, other: &Self) -> Self {
+        let modulus = M::modulus();
+        FiniteFieldElement::<M>::new((&self.value - &other.value) % modulus)
+    }
+
+    fn pow<V: Into<BigInt>>(&self, n: V) -> Self {
+        let mut exponent: BigInt = n.into();
         let mut base = self.clone();
 
         if exponent.is_negative() {
@@ -154,11 +161,11 @@ impl<M: ModulusValue> Field for FiniteFieldElement<M> {
 
         let mut result = FiniteFieldElement::one();
         while exponent.is_positive() && !exponent.is_zero() {
-            if exponent.clone() % 2 != BigInt::zero() {
-                result = result * base.clone();
+            if &exponent % 2 != BigInt::zero() {
+                result = result * &base;
             }
             exponent /= 2;
-            base = base.clone() * base;
+            base = base.mul_ref(&base);
         }
         result
     }
@@ -167,13 +174,17 @@ impl<M: ModulusValue> Field for FiniteFieldElement<M> {
         FiniteFieldElement::<M>::new(value.into())
     }
 
+    fn get_value(&self) -> BigInt {
+        self.value.clone()
+    }
+
     // Random element excluding a set of elements.
     fn random_element(exclude_elements: &[FiniteFieldElement<M>]) -> Self {
         let modulus = M::modulus();
         let mut rng = rand::thread_rng();
-        let mut fe = FiniteFieldElement::<M>::new(rng.gen_bigint_range(&BigInt::zero(), &modulus));
+        let mut fe = FiniteFieldElement::<M>::new(rng.gen_bigint_range(&BigInt::zero(), modulus));
         while exclude_elements.contains(&fe) {
-            fe = FiniteFieldElement::<M>::new(rng.gen_bigint_range(&BigInt::zero(), &modulus));
+            fe = FiniteFieldElement::<M>::new(rng.gen_bigint_range(&BigInt::zero(), modulus));
         }
         fe
     }
@@ -183,7 +194,7 @@ impl<M: ModulusValue> Field for FiniteFieldElement<M> {
 impl<M: ModulusValue> fmt::Display for FiniteFieldElement<M> {
     fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
         //let modulus = M::modulus();
-        //let repr_value = (&self.value + &modulus / 2) % &modulus - &modulus / 2;
+        //let repr_value = (&self.value + modulus / 2) % modulus - modulus / 2;
         write!(f, "{}", &self.value)
     }
 }
@@ -201,8 +212,15 @@ impl<M: ModulusValue> Add for FiniteFieldElement<M> {
     type Output = Self;
 
     fn add(self, other: Self) -> Self {
-        let modulus = M::modulus();
-        FiniteFieldElement::<M>::new((&self.value + &other.value) % &modulus)
+        self.add_ref(&other)
+    }
+}
+
+impl<'a, M: ModulusValue> Add<&'a FiniteFieldElement<M>> for FiniteFieldElement<M> {
+    type Output = FiniteFieldElement<M>;
+
+    fn add(self, other: &'a FiniteFieldElement<M>) -> FiniteFieldElement<M> {
+        self.add_ref(other)
     }
 }
 
@@ -210,8 +228,15 @@ impl<M: ModulusValue> Sub for FiniteFieldElement<M> {
     type Output = Self;
 
     fn sub(self, other: Self) -> Self {
-        let modulus = M::modulus();
-        FiniteFieldElement::<M>::new((&self.value - &other.value) % &modulus)
+        self.sub_ref(&other)
+    }
+}
+
+impl<'a, M: ModulusValue> Sub<&'a FiniteFieldElement<M>> for FiniteFieldElement<M> {
+    type Output = Self;
+
+    fn sub(self, other: &'a FiniteFieldElement<M>) -> FiniteFieldElement<M> {
+        self.sub_ref(&other)
     }
 }
 
@@ -219,17 +244,15 @@ impl<M: ModulusValue> Mul<FiniteFieldElement<M>> for FiniteFieldElement<M> {
     type Output = Self;
 
     fn mul(self, other: Self) -> Self {
-        let modulus = M::modulus();
-        FiniteFieldElement::<M>::new((&self.value * &other.value) % &modulus)
+        self.mul_ref(&other)
     }
 }
 
-impl<M: ModulusValue> Mul<i64> for FiniteFieldElement<M> {
-    type Output = Self;
+impl<'a, M: ModulusValue> Mul<&'a FiniteFieldElement<M>> for FiniteFieldElement<M> {
+    type Output = FiniteFieldElement<M>;
 
-    fn mul(self, n: i64) -> Self {
-        let modulus = M::modulus();
-        FiniteFieldElement::<M>::new((&self.value * n.to_bigint().unwrap()) % &modulus)
+    fn mul(self, other: &'a FiniteFieldElement<M>) -> FiniteFieldElement<M> {
+        self.mul_ref(other)
     }
 }
 
@@ -245,20 +268,33 @@ impl<M: ModulusValue> Div for FiniteFieldElement<M> {
     type Output = Self;
 
     fn div(self, other: Self) -> Self {
-        self * other.inverse()
+        self.div_ref(&other)
     }
 }
 
 #[macro_export]
 macro_rules! define_myzkp_modulus_type {
     ($name:ident, $modulus:expr) => {
-        #[derive(Debug, Hash, Clone)]
-        pub struct $name;
+        paste! {#[derive(Debug, Hash, Clone)]
+            pub struct $name;
+
+            lazy_static! {
+                static ref [<MODULUS_ $name>]: BigInt = BigInt::from_str($modulus).unwrap();
+            }
+
+            impl ModulusValue for $name {
+                fn modulus() -> &'static BigInt {
+                    &[<MODULUS_ $name>]
+                }
+            }
 
-        impl ModulusValue for $name {
-            fn modulus() -> BigInt {
-                BigInt::from_str($modulus).unwrap()
+            /*
+            impl ModulusValue for $name {
+                fn modulus() -> BigInt {
+                    BigInt::from_str($modulus).unwrap()
+                }
             }
+            */
         }
     };
 }
@@ -271,6 +307,7 @@ define_myzkp_modulus_type!(
 #[cfg(test)]
 mod tests {
     use super::*;
+    use crate::modules::ring::Ring;
     use num_bigint::ToBigInt;
 
     define_myzkp_modulus_type!(Mod7, "7");
@@ -344,7 +381,7 @@ mod tests {
     fn test_division() {
         let fe1 = FiniteFieldElement::<Mod17>::from_value(12);
         let fe2 = FiniteFieldElement::<Mod17>::from_value(3);
-        let fe_div = fe1 / fe2.clone();
+        let fe_div = fe1.div_ref(&fe2);
         assert_eq!(
             fe_div.value,
             (12.to_bigint().unwrap() * fe2.inverse().value) % 17
@@ -354,7 +391,7 @@ mod tests {
     #[test]
     fn test_exponentiation() {
         let fe1 = FiniteFieldElement::<Mod17>::from_value(3);
-        let fe_exp = fe1.pow(4.to_bigint().unwrap());
+        let fe_exp = fe1.pow(4);
         assert_eq!(
             fe_exp.value,
             3.to_bigint().unwrap().pow(4) % 17.to_bigint().unwrap()
diff --git a/myzkp/src/modules/mod.rs b/myzkp/src/modules/mod.rs
index 1823d31..c9e6315 100644
--- a/myzkp/src/modules/mod.rs
+++ b/myzkp/src/modules/mod.rs
@@ -1,6 +1,9 @@
+pub mod bn128;
 pub mod curve;
 pub mod educational_protocols;
+pub mod efield;
 pub mod field;
 pub mod polynomial;
 pub mod ring;
+pub mod utils;
 // pub mod snark;
diff --git a/myzkp/src/modules/polynomial.rs b/myzkp/src/modules/polynomial.rs
index 7c2d1be..6ed72a2 100644
--- a/myzkp/src/modules/polynomial.rs
+++ b/myzkp/src/modules/polynomial.rs
@@ -1,37 +1,42 @@
+use num_traits::{One, Zero};
 use std::fmt;
 use std::ops::{Add, Div, Mul, Neg, Rem, Sub};
 
-// Assuming FiniteFieldElement is already implemented with necessary traits like Add, Sub, Mul, Div.
+use crate::modules::curve::{EllipticCurve, EllipticCurvePoint};
 use crate::modules::field::Field;
 
 /// Polynomial struct representing a polynomial over Field.
 #[derive(Debug, Clone, PartialEq)]
 pub struct Polynomial<F: Field> {
-    pub poly: Vec<F>,
-    pub var: String, // Variable for representation, default to 'x'
+    pub coef: Vec<F>,
 }
 
 impl<F: Field> Polynomial<F> {
     /// Creates a polynomial representing the variable `x`.
     pub fn x() -> Self {
         Polynomial {
-            poly: vec![F::zero(), F::one()],
-            var: "x".to_string(),
+            coef: vec![F::zero(), F::one()],
         }
     }
 
     /// Removes trailing zeroes from a polynomial's coefficients.
-    fn trim_trailing_zeros(poly: Vec<F>) -> Vec<F> {
-        let mut trimmed = poly;
+    fn trim_trailing_zeros(coef: Vec<F>) -> Vec<F> {
+        let mut trimmed = coef;
         while trimmed.last() == Some(&F::zero()) {
             trimmed.pop();
         }
         trimmed
     }
 
+    pub fn reduce(&self) -> Self {
+        Polynomial {
+            coef: Self::trim_trailing_zeros(self.coef.clone()),
+        }
+    }
+
     /// Returns the degree of the polynomial.
     pub fn degree(&self) -> isize {
-        let trimmed = Self::trim_trailing_zeros(self.poly.clone());
+        let trimmed = Self::trim_trailing_zeros(self.coef.clone());
         if trimmed.is_empty() {
             -1
         } else {
@@ -44,27 +49,38 @@ impl<F: Field> Polynomial<F> {
         if n > self.degree() as usize {
             F::zero()
         } else {
-            self.poly[n].clone()
+            self.coef[n].clone()
         }
     }
 
     /// Evaluate the polynomial at a given point.
     pub fn eval(&self, point: &F) -> F {
         let mut result = F::zero();
-        for coef in self.poly.iter().rev() {
-            result = result * point.clone() + coef.clone();
+        for coef in self.coef.iter().rev() {
+            result = result.mul_ref(&point) + coef;
         }
         result
     }
 
     pub fn eval_with_powers(&self, powers: &[F]) -> F {
         let mut result = F::one();
-        for (i, coef) in self.poly.iter().enumerate() {
+        for (i, coef) in self.coef.iter().enumerate() {
             result = result * powers[i].pow(coef.clone().get_value());
         }
         result
     }
 
+    pub fn eval_with_powers_on_curve<E: EllipticCurve>(
+        &self,
+        powers: &[EllipticCurvePoint<F, E>],
+    ) -> EllipticCurvePoint<F, E> {
+        let mut result = EllipticCurvePoint::point_at_infinity();
+        for (i, coef) in self.coef.iter().enumerate() {
+            result = result + powers[i].clone() * coef.clone().get_value();
+        }
+        result
+    }
+
     /// Lagrange interpolation to compute polynomials.
     pub fn interpolate(x_values: &[F], y_values: &[F]) -> Polynomial<F> {
         let mut lagrange_polys = vec![];
@@ -74,7 +90,7 @@ impl<F: Field> Polynomial<F> {
             let mut denominator = F::one();
             for i in 0..x_values.len() {
                 if i != j {
-                    denominator = denominator * (x_values[j].clone() - x_values[i].clone());
+                    denominator = denominator * (x_values[j].sub_ref(&x_values[i]));
                 }
             }
             let cur_poly = numerators
@@ -83,10 +99,7 @@ impl<F: Field> Polynomial<F> {
             lagrange_polys.push(cur_poly);
         }
 
-        let mut result = Polynomial {
-            poly: vec![],
-            var: "x".to_string(),
-        };
+        let mut result = Polynomial { coef: vec![] };
         for (j, lagrange_poly) in lagrange_polys.iter().enumerate() {
             result = result + lagrange_poly.clone() * y_values[j].clone();
         }
@@ -96,24 +109,87 @@ impl<F: Field> Polynomial<F> {
     /// Helper to create polynomial from a single monomial.
     pub fn from_monomials(x_values: &[F]) -> Polynomial<F> {
         let mut poly = Polynomial {
-            poly: vec![F::one()],
-            var: "x".to_string(),
+            coef: vec![F::one()],
         };
         for x in x_values {
             poly = poly.mul(Polynomial {
-                poly: vec![F::zero() - x.clone(), F::one()],
-                var: "x".to_string(),
+                coef: vec![F::zero() - x, F::one()],
             });
         }
         poly
     }
+
+    fn add_ref<'b>(&self, other: &'b Polynomial<F>) -> Polynomial<F> {
+        let max_len = std::cmp::max(self.coef.len(), other.coef.len());
+        let mut result = Vec::with_capacity(max_len);
+
+        let zero = F::zero();
+
+        for i in 0..max_len {
+            let a = self.coef.get(i).unwrap_or(&zero);
+            let b = other.coef.get(i).unwrap_or(&zero);
+            result.push(a.add_ref(b));
+        }
+        Polynomial {
+            coef: Self::trim_trailing_zeros(result),
+        }
+    }
+
+    fn mul_ref<'b>(&self, other: &'b Polynomial<F>) -> Polynomial<F> {
+        if self.is_zero() || other.is_zero() {
+            return Polynomial::<F>::zero();
+        }
+        let mut result = vec![F::zero(); (self.degree() + other.degree() + 1) as usize];
+
+        for (i, a) in self.coef.iter().enumerate() {
+            for (j, b) in other.coef.iter().enumerate() {
+                result[i + j] = result[i + j].add_ref(&a.mul_ref(b));
+            }
+        }
+        Polynomial {
+            coef: Polynomial::<F>::trim_trailing_zeros(result),
+        }
+    }
+
+    fn div_rem_ref<'b>(&self, other: &'b Polynomial<F>) -> (Polynomial<F>, Polynomial<F>) {
+        if self.degree() < other.degree() {
+            return (Polynomial::zero(), self.clone());
+        }
+
+        let mut remainder_coeffs = Self::trim_trailing_zeros(self.coef.clone());
+        let divisor_coeffs = Self::trim_trailing_zeros(other.coef.clone());
+        let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse();
+
+        let mut quotient = vec![F::zero(); self.degree() as usize - other.degree() as usize + 1];
+
+        while remainder_coeffs.len() >= divisor_coeffs.len() {
+            let lead_term = remainder_coeffs.last().unwrap().mul_ref(&divisor_lead_inv);
+            let deg_diff = remainder_coeffs.len() - divisor_coeffs.len();
+            quotient[deg_diff] = lead_term.clone();
+
+            for i in 0..divisor_coeffs.len() {
+                remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i]
+                    .sub_ref(&(lead_term.mul_ref(&divisor_coeffs[i])));
+            }
+            remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs);
+        }
+
+        (
+            Polynomial {
+                coef: Self::trim_trailing_zeros(quotient),
+            },
+            Polynomial {
+                coef: remainder_coeffs,
+            },
+        )
+    }
 }
 
 impl<F: Field> fmt::Display for Polynomial<F> {
     fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
         let mut terms = Vec::new();
 
-        for (i, coeff) in self.poly.iter().enumerate() {
+        for (i, coeff) in self.coef.iter().enumerate() {
             if coeff != &F::zero() {
                 let term = if i == 0 {
                     // Constant term
@@ -121,16 +197,16 @@ impl<F: Field> fmt::Display for Polynomial<F> {
                 } else if i == 1 {
                     // Linear term (e.g., 3x)
                     if coeff == &F::one() {
-                        format!("{}", self.var)
+                        format!("{}", "x")
                     } else {
-                        format!("{}{}", coeff, self.var)
+                        format!("{}{}", coeff, "x")
                     }
                 } else {
                     // Higher degree terms (e.g., 3x^2)
                     if coeff == &F::one() {
-                        format!("{}^{}", self.var, i)
+                        format!("{}^{}", "x", i)
                     } else {
-                        format!("{}{}^{}", coeff, self.var, i)
+                        format!("{}{}^{}", coeff, "x", i)
                     }
                 };
                 terms.push(term);
@@ -147,25 +223,50 @@ impl<F: Field> fmt::Display for Polynomial<F> {
     }
 }
 
+impl<F: Field> Zero for Polynomial<F> {
+    fn zero() -> Self {
+        Polynomial {
+            coef: vec![F::zero()],
+        }
+    }
+
+    fn is_zero(&self) -> bool {
+        self.degree() == -1
+    }
+}
+
+impl<F: Field> One for Polynomial<F> {
+    fn one() -> Self {
+        Polynomial {
+            coef: vec![F::one()],
+        }
+    }
+}
+
 // Arithmetic operations implementation for Polynomial.
+impl<F: Field> Neg for Polynomial<F> {
+    type Output = Self;
+
+    fn neg(self) -> Polynomial<F> {
+        Polynomial {
+            coef: self.coef.iter().map(|x| -x.clone()).collect(),
+        }
+    }
+}
+
 impl<F: Field> Add for Polynomial<F> {
     type Output = Self;
 
     fn add(self, other: Self) -> Polynomial<F> {
-        let max_len = std::cmp::max(self.poly.len(), other.poly.len());
-        let mut result = Vec::with_capacity(max_len);
+        self.add_ref(&other)
+    }
+}
 
-        let zero = F::zero();
+impl<'a, 'b, F: Field> Add<&'b Polynomial<F>> for &'a Polynomial<F> {
+    type Output = Polynomial<F>;
 
-        for i in 0..max_len {
-            let a = self.poly.get(i).unwrap_or(&zero);
-            let b = other.poly.get(i).unwrap_or(&zero);
-            result.push(a.clone() + b.clone());
-        }
-        Polynomial {
-            poly: Self::trim_trailing_zeros(result),
-            var: self.var.clone(),
-        }
+    fn add(self, other: &'b Polynomial<F>) -> Polynomial<F> {
+        self.add_ref(other)
     }
 }
 
@@ -173,51 +274,31 @@ impl<F: Field> Sub for Polynomial<F> {
     type Output = Self;
 
     fn sub(self, other: Self) -> Polynomial<F> {
-        let max_len = std::cmp::max(self.poly.len(), other.poly.len());
-        let mut result = Vec::with_capacity(max_len);
-
-        let zero = F::zero();
-
-        for i in 0..max_len {
-            let a = self.poly.get(i).unwrap_or(&zero);
-            let b = other.poly.get(i).unwrap_or(&zero);
-            result.push(a.clone() - b.clone());
-        }
-        Polynomial {
-            poly: Self::trim_trailing_zeros(result),
-            var: self.var.clone(),
-        }
+        self.add_ref(&-other)
     }
 }
 
-impl<F: Field> Neg for Polynomial<F> {
-    type Output = Self;
+impl<'a, 'b, F: Field> Sub<&'b Polynomial<F>> for &'a Polynomial<F> {
+    type Output = Polynomial<F>;
 
-    /// Negation of a polynomial.
-    fn neg(self) -> Polynomial<F> {
-        Polynomial {
-            poly: self.poly.iter().map(|x| -x.clone()).collect(),
-            var: self.var.clone(),
-        }
+    fn sub(self, other: &'b Polynomial<F>) -> Polynomial<F> {
+        self.add_ref(&-other.clone())
     }
 }
 
 impl<F: Field> Mul<Polynomial<F>> for Polynomial<F> {
     type Output = Self;
 
-    /// Multiplication of two polynomials.
     fn mul(self, other: Polynomial<F>) -> Polynomial<F> {
-        let mut result = vec![F::zero(); self.degree() as usize + other.degree() as usize + 1];
+        self.mul_ref(&other)
+    }
+}
 
-        for (i, a) in self.poly.iter().enumerate() {
-            for (j, b) in other.poly.iter().enumerate() {
-                result[i + j] = result[i + j].clone() + (a.clone() * b.clone());
-            }
-        }
-        Polynomial {
-            poly: Self::trim_trailing_zeros(result),
-            var: self.var.clone(),
-        }
+impl<'a, 'b, F: Field> Mul<&'b Polynomial<F>> for &'a Polynomial<F> {
+    type Output = Polynomial<F>;
+
+    fn mul(self, other: &'b Polynomial<F>) -> Polynomial<F> {
+        self.mul_ref(other)
     }
 }
 
@@ -226,12 +307,7 @@ impl<F: Field> Mul<F> for Polynomial<F> {
 
     fn mul(self, scalar: F) -> Polynomial<F> {
         Polynomial {
-            poly: self
-                .poly
-                .iter()
-                .map(|x| x.clone() * scalar.clone())
-                .collect(),
-            var: self.var.clone(),
+            coef: self.coef.iter().map(|x| x.mul_ref(&scalar)).collect(),
         }
     }
 }
@@ -239,90 +315,61 @@ impl<F: Field> Mul<F> for Polynomial<F> {
 impl<F: Field> Div for Polynomial<F> {
     type Output = Self;
 
-    /// Division of two polynomials, returns quotient.
     fn div(self, other: Polynomial<F>) -> Polynomial<F> {
-        let mut remainder_coeffs = Self::trim_trailing_zeros(self.poly.clone());
-        let divisor_coeffs = Self::trim_trailing_zeros(other.poly.clone());
-        let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse();
-
-        let mut quotient = vec![F::zero(); self.degree() as usize - other.degree() as usize + 1];
-
-        while remainder_coeffs.len() >= divisor_coeffs.len() {
-            let lead_term = remainder_coeffs.last().unwrap().clone() * divisor_lead_inv.clone();
-            let deg_diff = remainder_coeffs.len() - divisor_coeffs.len();
-            quotient[deg_diff] = lead_term.clone();
+        self.div_rem_ref(&other).0
+    }
+}
 
-            for i in 0..divisor_coeffs.len() {
-                remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i].clone()
-                    - (lead_term.clone() * divisor_coeffs[i].clone());
-            }
-            remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs);
-        }
+impl<'a, 'b, F: Field> Div<&'b Polynomial<F>> for &'a Polynomial<F> {
+    type Output = Polynomial<F>;
 
-        Polynomial {
-            poly: Self::trim_trailing_zeros(quotient),
-            var: self.var.clone(),
-        }
+    fn div(self, other: &'b Polynomial<F>) -> Polynomial<F> {
+        self.div_rem_ref(other).0
     }
 }
 
 impl<F: Field> Rem for Polynomial<F> {
     type Output = Self;
 
-    /// Division of two polynomials, returns quotient.
     fn rem(self, other: Polynomial<F>) -> Polynomial<F> {
-        let mut remainder_coeffs = Self::trim_trailing_zeros(self.poly.clone());
-        let divisor_coeffs = Self::trim_trailing_zeros(other.poly.clone());
-        let divisor_lead_inv = divisor_coeffs.last().unwrap().inverse();
-
-        let mut quotient = vec![F::zero(); self.degree() as usize - other.degree() as usize + 1];
+        self.div_rem_ref(&other).1
+    }
+}
 
-        while remainder_coeffs.len() >= divisor_coeffs.len() {
-            let lead_term = remainder_coeffs.last().unwrap().clone() * divisor_lead_inv.clone();
-            let deg_diff = remainder_coeffs.len() - divisor_coeffs.len();
-            quotient[deg_diff] = lead_term.clone();
+impl<'a, 'b, F: Field> Rem<&'b Polynomial<F>> for &'a Polynomial<F> {
+    type Output = Polynomial<F>;
 
-            for i in 0..divisor_coeffs.len() {
-                remainder_coeffs[deg_diff + i] = remainder_coeffs[deg_diff + i].clone()
-                    - (lead_term.clone() * divisor_coeffs[i].clone());
-            }
-            remainder_coeffs = Self::trim_trailing_zeros(remainder_coeffs);
-        }
-
-        Polynomial {
-            poly: remainder_coeffs,
-            var: self.var.clone(),
-        }
+    fn rem(self, other: &'b Polynomial<F>) -> Polynomial<F> {
+        self.div_rem_ref(other).1
     }
 }
 
 #[cfg(test)]
 mod tests {
     use super::*;
+
     use crate::modules::field::{FiniteFieldElement, ModEIP197};
+    use crate::modules::ring::Ring;
 
     #[test]
     fn test_polynomial_addition() {
         let poly1 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
             ],
-            var: "x".to_string(),
         };
         let poly2 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
             ],
-            var: "x".to_string(),
         };
         let expected = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(5_i32),
             ],
-            var: "x".to_string(),
         };
         assert_eq!(poly1 + poly2, expected);
     }
@@ -330,25 +377,22 @@ mod tests {
     #[test]
     fn test_polynomial_subtraction() {
         let poly1 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(4_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(5_i32),
             ],
-            var: "x".to_string(),
         };
         let poly2 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
             ],
-            var: "x".to_string(),
         };
         let expected = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
             ],
-            var: "x".to_string(),
         };
         assert_eq!(poly1 - poly2, expected);
     }
@@ -356,18 +400,16 @@ mod tests {
     #[test]
     fn test_polynomial_negation() {
         let poly = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(4_i32),
             ],
-            var: "x".to_string(),
         };
         let expected = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(-3_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(-4_i32),
             ],
-            var: "x".to_string(),
         };
         assert_eq!(-poly, expected);
     }
@@ -375,26 +417,23 @@ mod tests {
     #[test]
     fn test_polynomial_multiplication() {
         let poly1 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
             ], // 1 + 2x
-            var: "x".to_string(),
         };
         let poly2 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
             ], // 2 + 3x
-            var: "x".to_string(),
         };
         let expected = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32), // constant term
                 FiniteFieldElement::<ModEIP197>::from_value(7_i32), // x term
                 FiniteFieldElement::<ModEIP197>::from_value(6_i32), // x^2 term
             ],
-            var: "x".to_string(),
         };
         assert_eq!(poly1 * poly2, expected);
     }
@@ -402,19 +441,17 @@ mod tests {
     #[test]
     fn test_polynomial_scalar_multiplication() {
         let poly = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
             ], // 1 + 2x
-            var: "x".to_string(),
         };
         let scalar = FiniteFieldElement::<ModEIP197>::from_value(3_i32);
         let expected = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(6_i32),
             ], // 3 + 6x
-            var: "x".to_string(),
         };
         assert_eq!(poly * scalar, expected);
     }
@@ -422,32 +459,28 @@ mod tests {
     #[test]
     fn test_polynomial_division() {
         let poly1 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
             ], // 3 + 3x + x^2
-            var: "x".to_string(),
         };
         let poly2 = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
             ], // 1 + x
-            var: "x".to_string(),
         };
-        let quotient = poly1.clone() / poly2.clone();
+        let quotient = &poly1 / &poly2;
         let expected_quotient = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
             ], // 2 + x
-            var: "x".to_string(),
         };
-        let remainder = poly1.clone() % poly2.clone();
+        let remainder = &poly1 % &poly2;
         let expected_remainder = Polynomial {
-            poly: vec![FiniteFieldElement::<ModEIP197>::from_value(1_i32)], // remainder is 1
-            var: "x".to_string(),
+            coef: vec![FiniteFieldElement::<ModEIP197>::from_value(1_i32)], // remainder is 1
         };
         assert_eq!(quotient, expected_quotient);
         assert_eq!(remainder, expected_remainder);
@@ -456,11 +489,10 @@ mod tests {
     #[test]
     fn test_polynomial_evaluation() {
         let poly = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(2_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
             ], // 2 + 3x
-            var: "x".to_string(),
         };
         let point = FiniteFieldElement::<ModEIP197>::from_value(2_i32);
         let expected = FiniteFieldElement::<ModEIP197>::from_value(8_i32); // 2 + 3 * 2 = 8
@@ -470,12 +502,11 @@ mod tests {
     #[test]
     fn test_polynomial_degree() {
         let poly = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(0_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(3_i32),
             ], // 1 + 0x + 3x^2
-            var: "x".to_string(),
         };
         assert_eq!(poly.degree(), 2); // The degree should be 2
     }
@@ -494,12 +525,11 @@ mod tests {
         ];
         let result = Polynomial::interpolate(&x_values, &y_values);
         let expected = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(-1_i32).sanitize(),
                 FiniteFieldElement::<ModEIP197>::from_value(0_i32),
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32),
             ], // x^2 - 1
-            var: "x".to_string(),
         };
         assert_eq!(result, expected);
     }
@@ -513,12 +543,11 @@ mod tests {
         let result = Polynomial::from_monomials(&points);
         // (x - 2) * (x - 3) = x^2 - 5x + 6
         let expected = Polynomial {
-            poly: vec![
+            coef: vec![
                 FiniteFieldElement::<ModEIP197>::from_value(6_i32), // constant term
                 FiniteFieldElement::<ModEIP197>::from_value(-5_i32), // x term
                 FiniteFieldElement::<ModEIP197>::from_value(1_i32), // x^2 term
             ],
-            var: "x".to_string(),
         };
         assert_eq!(result, expected);
     }
diff --git a/myzkp/src/modules/ring.rs b/myzkp/src/modules/ring.rs
index 19f2e5e..3a2c0b9 100644
--- a/myzkp/src/modules/ring.rs
+++ b/myzkp/src/modules/ring.rs
@@ -1,3 +1,5 @@
+use num_bigint::BigInt;
+use num_traits::{One, Zero};
 use std::fmt;
 use std::ops::{Add, Mul, Neg, Sub};
 
@@ -7,12 +9,23 @@ pub trait Ring:
     + PartialEq
     + fmt::Display
     + Add<Output = Self>
+    + for<'a> Add<&'a Self, Output = Self>
     + Sub<Output = Self>
+    + for<'a> Sub<&'a Self, Output = Self>
     + Mul<Output = Self>
-    + Mul<i64, Output = Self>
+    + for<'a> Mul<&'a Self, Output = Self>
     + Neg<Output = Self>
+    + One
+    + Zero
 {
     // A ring is an algebraic structure with addition and multiplication
-    fn zero() -> Self;
-    fn one() -> Self;
+    fn add_ref(&self, rhs: &Self) -> Self;
+    fn sub_ref(&self, rhs: &Self) -> Self;
+    fn mul_ref(&self, rhs: &Self) -> Self;
+
+    // Utility functions
+    fn pow<M: Into<BigInt>>(&self, n: M) -> Self;
+    fn get_value(&self) -> BigInt;
+    fn from_value<M: Into<BigInt>>(value: M) -> Self;
+    fn random_element(exclude_elements: &[Self]) -> Self;
 }
diff --git a/myzkp/src/modules/snark.rs b/myzkp/src/modules/snark.rs
index c0d791f..e69de29 100644
--- a/myzkp/src/modules/snark.rs
+++ b/myzkp/src/modules/snark.rs
@@ -1,92 +0,0 @@
-use crate::modules::field::FiniteFieldElement;
-use crate::modules::polynomial::Polynomial;
-
-pub struct SNARK {
-    g: FiniteFieldElement,
-    t: Polynomial,
-    n: i32,
-}
-
-pub struct ProofKey {
-    alpha: FiniteFieldElement,
-    alpha_seq: Vec<FiniteFieldElement>,
-    alpha_prime: FiniteFieldElement,
-    alpha_prime_seq: Vec<FiniteFieldElement>,
-}
-
-pub struct VerificationKey {
-    gr: FiniteFieldElement,
-    gts: FiniteFieldElement,
-}
-
-pub struct Proof {
-    u_prime: FiniteFieldElement,
-    v_prime: FiniteFieldElement,
-    w_prime: FiniteFieldElement,
-}
-
-/// Function simulating bilinear pairing on two group elements.
-//fn simulate_bilinear_pairing(g1: FiniteFieldElement, g2: FiniteFieldElement) -> FiniteFieldElement {
-//    // Pairing function e(g1, g2) = g1.element^g2.element mod modulus
-//    g1.element.pow(g2.element.value.to_u64_digits()[0]) // Simplified pairing logic
-//}
-
-impl SNARK {
-    pub fn trusted_setup(&self) -> (ProofKey, VerificationKey) {
-        let excluded_elements: Vec<FiniteFieldElement> = vec![];
-        let s = FiniteFieldElement::random_element(&excluded_elements);
-        let r = FiniteFieldElement::random_element(&excluded_elements);
-
-        let alpha = self.g.pow(s.clone());
-        let alpha_prime = self.g.pow(s.clone() * r.clone()); // g^(sr)
-
-        let proof_key = ProofKey {
-            alpha,
-            alpha_seq: (1..self.n)
-                .map(|i| self.g.pow(s.mul_scalar(i.into())))
-                .collect(),
-            alpha_prime,
-            alpha_prime_seq: (1..self.n)
-                .map(|i| self.g.pow((s.clone() * r.clone()).mul_scalar(i.into())))
-                .collect(),
-        };
-
-        let verifier_key = VerificationKey {
-            gr: self.g.pow(r),                // g^r
-            gts: self.g.pow(self.t.eval(&s)), // g^T(s)
-        };
-
-        (proof_key, verifier_key)
-    }
-
-    pub fn prove(&self, proof_key: &ProofKey, p: &Polynomial, h: &Polynomial) -> Proof {
-        let excluded_elements: Vec<FiniteFieldElement> = vec![];
-        let delta = FiniteFieldElement::random_element(&excluded_elements);
-
-        let mut gp = self.g.pow(p.poly[0].clone());
-        for i in 1..p.poly.len() {
-            gp = gp * (proof_key.alpha_seq[i - 1]).pow(p.poly[i].clone());
-        }
-        let u_prime = gp.pow(delta.clone());
-
-        let mut gh = self.g.pow(h.poly[0].clone());
-        for i in 1..h.poly.len() {
-            gh = gh * (proof_key.alpha_seq[i - 1]).pow(h.poly[i].clone());
-        }
-        let v_prime = gh.pow(delta.clone());
-
-        let mut gp_prime = self.g.pow(p.poly[0].clone());
-        for i in 1..p.poly.len() {
-            gp_prime = gp_prime * (proof_key.alpha_prime_seq[i - 1]).pow(p.poly[i].clone());
-        }
-        let w_prime = gp_prime.pow(delta.clone());
-
-        Proof {
-            u_prime: u_prime,
-            v_prime: v_prime,
-            w_prime: w_prime,
-        }
-    }
-
-    //pub fn verify(&self, verification_key: &VerificationKey, proof: Proof) -> bool {}
-}
diff --git a/myzkp/src/modules/utils.rs b/myzkp/src/modules/utils.rs
new file mode 100644
index 0000000..8dd0648
--- /dev/null
+++ b/myzkp/src/modules/utils.rs
@@ -0,0 +1,29 @@
+use num_traits::{One, Zero};
+use std::ops::{Div, Rem, Sub};
+
+pub fn extended_euclidean<F>(a: F, b: F) -> (F, F, F)
+where
+    F: Clone + PartialEq + Sub<Output = F> + Div<Output = F> + Rem<Output = F> + Zero + One,
+{
+    let mut r0 = a;
+    let mut r1 = b;
+    let mut s0 = F::one();
+    let mut s1 = F::zero();
+    let mut t0 = F::zero();
+    let mut t1 = F::one();
+
+    while !r1.is_zero() {
+        let q = r0.clone() / r1.clone();
+        let r = r0.clone() % r1.clone();
+        r0 = r1;
+        r1 = r;
+        let new_s = s0.clone() - q.clone() * s1.clone();
+        s0 = s1;
+        s1 = new_s;
+        let new_t = t0.clone() - q * t1.clone();
+        t0 = t1;
+        t1 = new_t;
+    }
+
+    (r0, s0, t0)
+}