Add inv_mod() that supports any moduli

Author: fjarri

benches/bench.rs

@@ -153,6 +153,59 @@ fn bench_shifts<M: Measurement>(group: &mut BenchmarkGroup<'_, M>) {
     });
 }
 
+fn bench_inv_mod<M: Measurement>(group: &mut BenchmarkGroup<'_, M>) {
+    group.bench_function("inv_odd_mod, U256", |b| {
+        b.iter_batched(
+            || {
+                let m = U256::random(&mut OsRng) | U256::ONE;
+                loop {
+                    let x = U256::random(&mut OsRng);
+                    let (_, is_some) = x.inv_odd_mod(&m);
+                    if is_some.into() {
+                        break (x, m);
+                    }
+                }
+            },
+            |(x, m)| x.inv_odd_mod(&m),
+            BatchSize::SmallInput,
+        )
+    });
+
+    group.bench_function("inv_mod, U256, odd modulus", |b| {
+        b.iter_batched(
+            || {
+                let m = U256::random(&mut OsRng) | U256::ONE;
+                loop {
+                    let x = U256::random(&mut OsRng);
+                    let (_, is_some) = x.inv_odd_mod(&m);
+                    if is_some.into() {
+                        break (x, m);
+                    }
+                }
+            },
+            |(x, m)| x.inv_mod(&m),
+            BatchSize::SmallInput,
+        )
+    });
+
+    group.bench_function("inv_mod, U256", |b| {
+        b.iter_batched(
+            || {
+                let m = U256::random(&mut OsRng);
+                loop {
+                    let x = U256::random(&mut OsRng);
+                    let (_, is_some) = x.inv_mod(&m);
+                    if is_some.into() {
+                        break (x, m);
+                    }
+                }
+            },
+            |(x, m)| x.inv_mod(&m),
+            BatchSize::SmallInput,
+        )
+    });
+}
+
 fn bench_wrapping_ops(c: &mut Criterion) {
     let mut group = c.benchmark_group("wrapping ops");
     bench_division(&mut group);
@@ -169,6 +222,7 @@ fn bench_montgomery(c: &mut Criterion) {
 fn bench_modular_ops(c: &mut Criterion) {
     let mut group = c.benchmark_group("modular ops");
     bench_shifts(&mut group);
+    bench_inv_mod(&mut group);
     group.finish();
 }
 

src/ct_choice.rs

@@ -50,6 +50,10 @@ impl CtChoice {
         Self(!self.0)
     }
 
+    pub(crate) const fn or(&self, other: Self) -> Self {
+        Self(self.0 | other.0)
+    }
+
     pub(crate) const fn and(&self, other: Self) -> Self {
         Self(self.0 & other.0)
     }

src/uint/inv_mod.rs

@@ -137,10 +137,43 @@ impl<const LIMBS: usize> Uint<LIMBS> {
     }
 
     /// Computes the multiplicative inverse of `self` mod `modulus`, where `modulus` is odd.
-    /// Returns `(inverse, Word::MAX)` if an inverse exists, otherwise `(undefined, Word::ZERO)`.
+    /// Returns `(inverse, CtChoice::TRUE)` if an inverse exists,
+    /// otherwise `(undefined, CtChoice::FALSE)`.
     pub const fn inv_odd_mod(&self, modulus: &Self) -> (Self, CtChoice) {
         self.inv_odd_mod_bounded(modulus, Uint::<LIMBS>::BITS, Uint::<LIMBS>::BITS)
     }
+
+    /// Computes the multiplicative inverse of `self` mod `modulus`.
+    /// Returns `(inverse, CtChoice::TRUE)` if an inverse exists,
+    /// otherwise `(undefined, CtChoice::FALSE)`.
+    pub fn inv_mod(&self, modulus: &Self) -> (Self, CtChoice) {
+        // Decompose `modulus = s * 2^k` where `s` is odd
+        let k = modulus.trailing_zeros();
+        let s = modulus.shr(k);
+
+        // Decompose `self` into RNS with moduli `2^k` and `s` and calculate the inverses.
+        // Using the fact that `(z^{-1} mod (m1 * m2)) mod m1 == z^{-1} mod m1`.
+        let (a, a_is_some) = self.inv_odd_mod(&s);
+        let b = self.inv_mod2k(k);
+        // inverse modulo 2^k exists either if `k` is 0 or if `self` is odd.
+        let b_is_some = CtChoice::from_usize_being_nonzero(k)
+            .not()
+            .or(self.ct_is_odd());
+
+        // Restore from RNS:
+        // self^{-1} = a mod s = b mod 2^k
+        // => self^{-1} = a + s * ((b - a) * s^(-1) mod 2^k)
+        let m_odd_inv = s.inv_mod2k(k); // `s` is odd, so this always exists
+
+        // This part is mod 2^k
+        let mask = (Uint::ONE << k).wrapping_sub(&Uint::ONE);
+        let t = (b.wrapping_sub(&a).wrapping_mul(&m_odd_inv)) & mask;
+
+        // Will not overflow since `a <= s - 1`, `t <= 2^k - 1`,
+        // so `a + s * t <= s * 2^k - 1 == modulus - 1`.
+        let result = a.wrapping_add(&s.wrapping_mul(&t));
+        (result, a_is_some.and(b_is_some))
+    }
 }
 
 #[cfg(test)]
@@ -165,7 +198,7 @@ mod tests {
     }
 
     #[test]
-    fn test_invert() {
+    fn test_invert_odd() {
         let a = U1024::from_be_hex(concat![
             "000225E99153B467A5B451979A3F451DAEF3BF8D6C6521D2FA24BBB17F29544E",
             "347A412B065B75A351EA9719E2430D2477B11CC9CF9C1AD6EDEE26CB15F463F8",
@@ -178,15 +211,45 @@ mod tests {
             "D198D3155E5799DC4EA76652D64983A7E130B5EACEBAC768D28D589C36EC749C",
             "558D0B64E37CD0775C0D0104AE7D98BA23C815185DD43CD8B16292FD94156767"
         ]);
-
-        let (res, is_some) = a.inv_odd_mod(&m);
-
         let expected = U1024::from_be_hex(concat![
             "B03623284B0EBABCABD5C5881893320281460C0A8E7BF4BFDCFFCBCCBF436A55",
             "D364235C8171E46C7D21AAD0680676E57274A8FDA6D12768EF961CACDD2DAE57",
             "88D93DA5EB8EDC391EE3726CDCF4613C539F7D23E8702200CB31B5ED5B06E5CA",
             "3E520968399B4017BF98A864FABA2B647EFC4998B56774D4F2CB026BC024A336"
         ]);
+
+        let (res, is_some) = a.inv_odd_mod(&m);
+        assert!(is_some.is_true_vartime());
+        assert_eq!(res, expected);
+
+        // Even though it is less efficient, it still works
+        let (res, is_some) = a.inv_mod(&m);
+        assert!(is_some.is_true_vartime());
+        assert_eq!(res, expected);
+    }
+
+    #[test]
+    fn test_invert_even() {
+        let a = U1024::from_be_hex(concat![
+            "000225E99153B467A5B451979A3F451DAEF3BF8D6C6521D2FA24BBB17F29544E",
+            "347A412B065B75A351EA9719E2430D2477B11CC9CF9C1AD6EDEE26CB15F463F8",
+            "BCC72EF87EA30288E95A48AA792226CEC959DCB0672D8F9D80A54CBBEA85CAD8",
+            "382EC224DEB2F5784E62D0CC2F81C2E6AD14EBABE646D6764B30C32B87688985"
+        ]);
+        let m = U1024::from_be_hex(concat![
+            "D509E7854ABDC81921F669F1DC6F61359523F3949803E58ED4EA8BC16483DC6F",
+            "37BFE27A9AC9EEA2969B357ABC5C0EE214BE16A7D4C58FC620D5B5A20AFF001A",
+            "D198D3155E5799DC4EA76652D64983A7E130B5EACEBAC768D28D589C36EC749C",
+            "558D0B64E37CD0775C0D0104AE7D98BA23C815185DD43CD8B16292FD94156000"
+        ]);
+        let expected = U1024::from_be_hex(concat![
+            "1EBF391306817E1BC610E213F4453AD70911CCBD59A901B2A468A4FC1D64F357",
+            "DBFC6381EC5635CAA664DF280028AF4651482C77A143DF38D6BFD4D64B6C0225",
+            "FC0E199B15A64966FB26D88A86AD144271F6BDCD3D63193AB2B3CC53B99F21A3",
+            "5B9BFAE5D43C6BC6E7A9856C71C7318C76530E9E5AE35882D5ABB02F1696874D",
+        ]);
+
+        let (res, is_some) = a.inv_mod(&m);
         assert!(is_some.is_true_vartime());
         assert_eq!(res, expected);
     }

tests/proptests.rs

@@ -2,7 +2,7 @@
 
 use crypto_bigint::{
     modular::runtime_mod::{DynResidue, DynResidueParams},
-    Encoding, Limb, NonZero, Word, U256,
+    CtChoice, Encoding, Limb, NonZero, Word, U256,
 };
 use num_bigint::BigUint;
 use num_integer::Integer;
@@ -233,6 +233,23 @@ proptest! {
         }
     }
 
+    #[test]
+    fn inv_mod(a in uint(), b in uint()) {
+        let a_bi = to_biguint(&a);
+        let b_bi = to_biguint(&b);
+
+        let expected_is_some = if a_bi.gcd(&b_bi) == BigUint::one() { CtChoice::TRUE } else { CtChoice::FALSE };
+        let (actual, actual_is_some) = a.inv_mod(&b);
+
+        assert_eq!(bool::from(expected_is_some), bool::from(actual_is_some));
+
+        if actual_is_some.into() {
+            let inv_bi = to_biguint(&actual);
+            let res = (inv_bi * a_bi) % b_bi;
+            assert_eq!(res, BigUint::one());
+        }
+    }
+
     #[test]
     fn wrapping_sqrt(a in uint()) {
         let a_bi = to_biguint(&a);