Mark the current inv_mod2k() as vartime and add an actual constant-time variant

fjarri · fjarri · commit cf4eb1dfba0d · 2023-08-28T13:50:27.000-07:00
diff --git a/src/ct_choice.rs b/src/ct_choice.rs
@@ -29,6 +29,17 @@ impl CtChoice {
         Self(value.wrapping_neg())
     }
 
+    /// Returns the truthy value if `value != 0`, and the falsy value otherwise.
+    pub(crate) const fn from_usize_being_nonzero(value: usize) -> Self {
+        const HI_BIT: u32 = usize::BITS - 1;
+        Self::from_lsb(((value | value.wrapping_neg()) >> HI_BIT) as Word)
+    }
+
+    /// Returns the truthy value if `x == y`, and the falsy value otherwise.
+    pub(crate) const fn from_usize_equality(x: usize, y: usize) -> Self {
+        Self::from_usize_being_nonzero(x.wrapping_sub(y)).not()
+    }
+
     /// Returns the truthy value if `x < y`, and the falsy value otherwise.
     pub(crate) const fn from_usize_lt(x: usize, y: usize) -> Self {
         let bit = (((!x) & y) | (((!x) | y) & (x.wrapping_sub(y)))) >> (usize::BITS - 1);
diff --git a/src/uint/bits.rs b/src/uint/bits.rs
@@ -69,7 +69,7 @@ impl<const LIMBS: usize> Uint<LIMBS> {
     /// Get the value of the bit at position `index`, as a truthy or falsy `CtChoice`.
     /// Returns the falsy value for indices out of range.
     pub const fn bit(&self, index: usize) -> CtChoice {
-        let limb_num = Limb((index / Limb::BITS) as Word);
+        let limb_num = index / Limb::BITS;
         let index_in_limb = index % Limb::BITS;
         let index_mask = 1 << index_in_limb;
 
@@ -79,18 +79,36 @@ impl<const LIMBS: usize> Uint<LIMBS> {
         let mut i = 0;
         while i < LIMBS {
             let bit = limbs[i] & index_mask;
-            let is_right_limb = Limb::ct_eq(limb_num, Limb(i as Word));
+            let is_right_limb = CtChoice::from_usize_equality(i, limb_num);
             result |= is_right_limb.if_true(bit);
             i += 1;
         }
 
         CtChoice::from_lsb(result >> index_in_limb)
     }
+
+    /// Sets the bit at `index` to 0 or 1 depending on the value of `bit_value`.
+    pub(crate) const fn set_bit(self, index: usize, bit_value: CtChoice) -> Self {
+        let mut result = self;
+        let limb_num = index / Limb::BITS;
+        let index_in_limb = index % Limb::BITS;
+        let index_mask = 1 << index_in_limb;
+
+        let mut i = 0;
+        while i < LIMBS {
+            let is_right_limb = CtChoice::from_usize_equality(i, limb_num);
+            let old_limb = result.limbs[i].0;
+            let new_limb = bit_value.select(old_limb & !index_mask, old_limb | index_mask);
+            result.limbs[i] = Limb(is_right_limb.select(old_limb, new_limb));
+            i += 1;
+        }
+        result
+    }
 }
 
 #[cfg(test)]
 mod tests {
-    use crate::U256;
+    use crate::{CtChoice, U256};
 
     fn uint_with_bits_at(positions: &[usize]) -> U256 {
         let mut result = U256::ZERO;
@@ -159,4 +177,31 @@ mod tests {
         let u = U256::ZERO;
         assert_eq!(u.trailing_zeros() as u32, 256);
     }
+
+    #[test]
+    fn set_bit() {
+        let u = uint_with_bits_at(&[16, 79, 150]);
+        assert_eq!(
+            u.set_bit(127, CtChoice::TRUE),
+            uint_with_bits_at(&[16, 79, 127, 150])
+        );
+
+        let u = uint_with_bits_at(&[16, 79, 150]);
+        assert_eq!(
+            u.set_bit(150, CtChoice::TRUE),
+            uint_with_bits_at(&[16, 79, 150])
+        );
+
+        let u = uint_with_bits_at(&[16, 79, 150]);
+        assert_eq!(
+            u.set_bit(127, CtChoice::FALSE),
+            uint_with_bits_at(&[16, 79, 150])
+        );
+
+        let u = uint_with_bits_at(&[16, 79, 150]);
+        assert_eq!(
+            u.set_bit(150, CtChoice::FALSE),
+            uint_with_bits_at(&[16, 79])
+        );
+    }
 }
diff --git a/src/uint/inv_mod.rs b/src/uint/inv_mod.rs
@@ -1,28 +1,68 @@
 use super::Uint;
-use crate::{CtChoice, Limb};
+use crate::CtChoice;
 
 impl<const LIMBS: usize> Uint<LIMBS> {
-    /// Computes 1/`self` mod 2^k as specified in Algorithm 4 from
-    /// A Secure Algorithm for Inversion Modulo 2k by
-    /// Sadiel de la Fe and Carles Ferrer. See
-    /// <https://www.mdpi.com/2410-387X/2/3/23>.
+    /// Computes 1/`self` mod `2^k`.
+    /// This method is constant-time w.r.t. `self` but not `k`.
     ///
     /// Conditions: `self` < 2^k and `self` must be odd
-    pub const fn inv_mod2k(&self, k: usize) -> Self {
-        let mut x = Self::ZERO;
-        let mut b = Self::ONE;
+    pub const fn inv_mod2k_vartime(&self, k: usize) -> Self {
+        // Using the Algorithm 3 from "A Secure Algorithm for Inversion Modulo 2k"
+        // by Sadiel de la Fe and Carles Ferrer.
+        // See <https://www.mdpi.com/2410-387X/2/3/23>.
+
+        // Note that we are not using Alrgorithm 4, since we have a different approach
+        // of enforcing constant-timeness w.r.t. `self`.
+
+        let mut x = Self::ZERO; // keeps `x` during iterations
+        let mut b = Self::ONE; // keeps `b_i` during iterations
         let mut i = 0;
 
         while i < k {
-            let mut x_i = Self::ZERO;
-            let j = b.limbs[0].0 & 1;
-            x_i.limbs[0] = Limb(j);
-            x = x.bitor(&x_i.shl_vartime(i));
+            // X_i = b_i mod 2
+            let x_i = b.limbs[0].0 & 1;
+            let x_i_choice = CtChoice::from_lsb(x_i);
+            // b_{i+1} = (b_i - a * X_i) / 2
+            b = Self::ct_select(&b, &b.wrapping_sub(self), x_i_choice).shr_vartime(1);
+            // Store the X_i bit in the result (x = x | (1 << X_i))
+            x = x.bitor(&Uint::from_word(x_i).shl_vartime(i));
 
-            let t = b.wrapping_sub(self);
-            b = Self::ct_select(&b, &t, CtChoice::from_lsb(j)).shr_vartime(1);
             i += 1;
         }
+
+        x
+    }
+
+    /// Computes 1/`self` mod `2^k`.
+    ///
+    /// Conditions: `self` < 2^k and `self` must be odd
+    pub const fn inv_mod2k(&self, k: usize) -> Self {
+        // This is the same algorithm as in `inv_mod2k_vartime()`,
+        // but made constant-time w.r.t `k` as well.
+
+        let mut x = Self::ZERO; // keeps `x` during iterations
+        let mut b = Self::ONE; // keeps `b_i` during iterations
+        let mut i = 0;
+
+        while i < Self::BITS {
+            // Only iterations for i = 0..k need to change `x`,
+            // the rest are dummy ones performed for the sake of constant-timeness.
+            let within_range = CtChoice::from_usize_lt(i, k);
+
+            // X_i = b_i mod 2
+            let x_i = b.limbs[0].0 & 1;
+            let x_i_choice = CtChoice::from_lsb(x_i);
+            // b_{i+1} = (b_i - a * X_i) / 2
+            b = Self::ct_select(&b, &b.wrapping_sub(self), x_i_choice).shr_vartime(1);
+
+            // Store the X_i bit in the result (x = x | (1 << X_i))
+            // Don't change the result in dummy iterations.
+            let x_i_choice = x_i_choice.and(within_range);
+            x = x.set_bit(i, x_i_choice);
+
+            i += 1;
+        }
+
         x
     }
 
diff --git a/src/uint/modular/constant_mod/macros.rs b/src/uint/modular/constant_mod/macros.rs
@@ -32,7 +32,7 @@ macro_rules! impl_modulus {
             const MOD_NEG_INV: $crate::Limb = $crate::Limb(
                 $crate::Word::MIN.wrapping_sub(
                     Self::MODULUS
-                        .inv_mod2k($crate::Word::BITS as usize)
+                        .inv_mod2k_vartime($crate::Word::BITS as usize)
                         .as_limbs()[0]
                         .0,
                 ),
diff --git a/src/uint/modular/runtime_mod.rs b/src/uint/modular/runtime_mod.rs
@@ -47,8 +47,9 @@ impl<const LIMBS: usize> DynResidueParams<LIMBS> {
         // Since we are calculating the inverse modulo (Word::MAX+1),
         // we can take the modulo right away and calculate the inverse of the first limb only.
         let modulus_lo = Uint::<1>::from_words([modulus.limbs[0].0]);
-        let mod_neg_inv =
-            Limb(Word::MIN.wrapping_sub(modulus_lo.inv_mod2k(Word::BITS as usize).limbs[0].0));
+        let mod_neg_inv = Limb(
+            Word::MIN.wrapping_sub(modulus_lo.inv_mod2k_vartime(Word::BITS as usize).limbs[0].0),
+        );
 
         let r3 = montgomery_reduction(&r2.square_wide(), modulus, mod_neg_inv);
 
diff --git a/tests/proptests.rs b/tests/proptests.rs
@@ -212,6 +212,27 @@ proptest! {
         }
     }
 
+    #[test]
+    fn inv_mod2k(a in uint(), k in any::<usize>()) {
+        let a = a | U256::ONE; // make odd
+        let k = k % (U256::BITS + 1);
+        let a_bi = to_biguint(&a);
+        let m_bi = BigUint::one() << k;
+
+        let actual = a.inv_mod2k(k);
+        let actual_vartime = a.inv_mod2k_vartime(k);
+        assert_eq!(actual, actual_vartime);
+
+        if k == 0 {
+            assert_eq!(actual, U256::ZERO);
+        }
+        else {
+            let inv_bi = to_biguint(&actual);
+            let res = (inv_bi * a_bi) % m_bi;
+            assert_eq!(res, BigUint::one());
+        }
+    }
+
     #[test]
     fn wrapping_sqrt(a in uint()) {
         let a_bi = to_biguint(&a);
