Add endo scalar decomposition (#24)

* feat: add endo scalar decomposition

* fix: clippy

* docs: add comments to endo parameters

* remove example glv mul

* cargo fmt

src/arithmetic.rs

@@ -4,6 +4,19 @@
 //! This module is temporary, and the extension traits defined here are expected to be
 //! upstreamed into the `ff` and `group` crates after some refactoring.
 
+use crate::CurveExt;
+
+pub(crate) struct EndoParameters {
+    pub(crate) gamma1: [u64; 4],
+    pub(crate) gamma2: [u64; 4],
+    pub(crate) b1: [u64; 4],
+    pub(crate) b2: [u64; 4],
+}
+
+pub trait CurveEndo: CurveExt {
+    fn decompose_scalar(e: &Self::ScalarExt) -> (u128, bool, u128, bool);
+}
+
 pub trait CurveAffineExt: pasta_curves::arithmetic::CurveAffine {
     fn batch_add<const COMPLETE: bool, const LOAD_POINTS: bool>(
         points: &mut [Self],
@@ -42,3 +55,36 @@ pub(crate) const fn mac(a: u64, b: u64, c: u64, carry: u64) -> (u64, u64) {
     let ret = (a as u128) + ((b as u128) * (c as u128)) + (carry as u128);
     (ret as u64, (ret >> 64) as u64)
 }
+
+/// Compute a + (b * c), returning the result and the new carry over.
+#[inline(always)]
+pub(crate) const fn macx(a: u64, b: u64, c: u64) -> (u64, u64) {
+    let res = (a as u128) + ((b as u128) * (c as u128));
+    (res as u64, (res >> 64) as u64)
+}
+
+/// Compute a * b, returning the result.
+#[inline(always)]
+pub(crate) fn mul_512(a: [u64; 4], b: [u64; 4]) -> [u64; 8] {
+    let (r0, carry) = macx(0, a[0], b[0]);
+    let (r1, carry) = macx(carry, a[0], b[1]);
+    let (r2, carry) = macx(carry, a[0], b[2]);
+    let (r3, carry_out) = macx(carry, a[0], b[3]);
+
+    let (r1, carry) = macx(r1, a[1], b[0]);
+    let (r2, carry) = mac(r2, a[1], b[1], carry);
+    let (r3, carry) = mac(r3, a[1], b[2], carry);
+    let (r4, carry_out) = mac(carry_out, a[1], b[3], carry);
+
+    let (r2, carry) = macx(r2, a[2], b[0]);
+    let (r3, carry) = mac(r3, a[2], b[1], carry);
+    let (r4, carry) = mac(r4, a[2], b[2], carry);
+    let (r5, carry_out) = mac(carry_out, a[2], b[3], carry);
+
+    let (r3, carry) = macx(r3, a[3], b[0]);
+    let (r4, carry) = mac(r4, a[3], b[1], carry);
+    let (r5, carry) = mac(r5, a[3], b[2], carry);
+    let (r6, carry_out) = mac(carry_out, a[3], b[3], carry);
+
+    [r0, r1, r2, r3, r4, r5, r6, carry_out]
+}

src/bn256/curve.rs

@@ -1,6 +1,11 @@
+use crate::arithmetic::mul_512;
+use crate::arithmetic::sbb;
+use crate::arithmetic::CurveEndo;
+use crate::arithmetic::EndoParameters;
 use crate::bn256::Fq;
 use crate::bn256::Fq2;
 use crate::bn256::Fr;
+use crate::endo;
 use crate::ff::WithSmallOrderMulGroup;
 use crate::ff::{Field, PrimeField};
 use crate::group::Curve;
@@ -16,6 +21,7 @@ use core::fmt::Debug;
 use core::iter::Sum;
 use core::ops::{Add, Mul, Neg, Sub};
 use rand::RngCore;
+use std::convert::TryInto;
 use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
 
 #[cfg(feature = "derive_serde")]
@@ -111,6 +117,26 @@ const G2_GENERATOR_Y: Fq2 = Fq2 {
     ]),
 };
 
+// Generated using https://github.com/ConsenSys/gnark-crypto/blob/master/ecc/utils.go
+// with `bn256::Fr::ZETA`
+// See https://github.com/demining/Endomorphism-Secp256k1/blob/main/README.md
+// to have more details about the endomorphism.
+const ENDO_PARAMS: EndoParameters = EndoParameters {
+    // round(b2/n)
+    gamma1: [
+        0x7a7bd9d4391eb18du64,
+        0x4ccef014a773d2cfu64,
+        0x0000000000000002u64,
+        0u64,
+    ],
+    // round(-b1/n)
+    gamma2: [0xd91d232ec7e0b3d7u64, 0x0000000000000002u64, 0u64, 0u64],
+    b1: [0x8211bbeb7d4f1128u64, 0x6f4d8248eeb859fcu64, 0u64, 0u64],
+    b2: [0x89d3256894d213e3u64, 0u64, 0u64, 0u64],
+};
+
+endo!(G1, Fr, ENDO_PARAMS);
+
 impl group::cofactor::CofactorGroup for G1 {
     type Subgroup = G1;
 
@@ -180,9 +206,14 @@ impl CofactorGroup for G2 {
 
 #[cfg(test)]
 mod tests {
+
+    use crate::arithmetic::CurveEndo;
     use crate::bn256::{Fr, G1, G2};
     use crate::CurveExt;
+    use ff::Field;
+    use ff::PrimeField;
     use ff::WithSmallOrderMulGroup;
+    use rand_core::OsRng;
 
     #[test]
     fn test_curve() {
@@ -191,12 +222,24 @@ mod tests {
     }
 
     #[test]
-    fn test_endo_consistency() {
+    fn test_endo() {
         let g = G1::generator();
         assert_eq!(g * Fr::ZETA, g.endo());
-
         let g = G2::generator();
         assert_eq!(g * Fr::ZETA, g.endo());
+        for _ in 0..100000 {
+            let k = Fr::random(OsRng);
+            let (k1, k1_neg, k2, k2_neg) = G1::decompose_scalar(&k);
+            if k1_neg & k2_neg {
+                assert_eq!(k, -Fr::from_u128(k1) + Fr::ZETA * Fr::from_u128(k2))
+            } else if k1_neg {
+                assert_eq!(k, -Fr::from_u128(k1) - Fr::ZETA * Fr::from_u128(k2))
+            } else if k2_neg {
+                assert_eq!(k, Fr::from_u128(k1) + Fr::ZETA * Fr::from_u128(k2))
+            } else {
+                assert_eq!(k, Fr::from_u128(k1) - Fr::ZETA * Fr::from_u128(k2))
+            }
+        }
     }
 
     #[test]

src/derive/curve.rs

@@ -140,6 +140,57 @@ macro_rules! batch_add {
     };
 }
 
+#[macro_export]
+macro_rules! endo {
+    ($name:ident, $field:ident, $params:expr) => {
+        impl CurveEndo for $name {
+            fn decompose_scalar(k: &$field) -> (u128, bool, u128, bool) {
+                let to_limbs = |e: &$field| {
+                    let repr = e.to_repr();
+                    let repr = repr.as_ref();
+                    let tmp0 = u64::from_le_bytes(repr[0..8].try_into().unwrap());
+                    let tmp1 = u64::from_le_bytes(repr[8..16].try_into().unwrap());
+                    let tmp2 = u64::from_le_bytes(repr[16..24].try_into().unwrap());
+                    let tmp3 = u64::from_le_bytes(repr[24..32].try_into().unwrap());
+                    [tmp0, tmp1, tmp2, tmp3]
+                };
+
+                let get_lower_128 = |e: &$field| {
+                    let e = to_limbs(e);
+                    u128::from(e[0]) | (u128::from(e[1]) << 64)
+                };
+
+                let is_neg = |e: &$field| {
+                    let e = to_limbs(e);
+                    let (_, borrow) = sbb(0xffffffffffffffff, e[0], 0);
+                    let (_, borrow) = sbb(0xffffffffffffffff, e[1], borrow);
+                    let (_, borrow) = sbb(0xffffffffffffffff, e[2], borrow);
+                    let (_, borrow) = sbb(0x00, e[3], borrow);
+                    borrow & 1 != 0
+                };
+
+                let input = to_limbs(&k);
+                let c1 = mul_512($params.gamma2, input);
+                let c2 = mul_512($params.gamma1, input);
+                let c1 = [c1[4], c1[5], c1[6], c1[7]];
+                let c2 = [c2[4], c2[5], c2[6], c2[7]];
+                let q1 = mul_512(c1, $params.b1);
+                let q2 = mul_512(c2, $params.b2);
+                let q1 = $field::from_raw([q1[0], q1[1], q1[2], q1[3]]);
+                let q2 = $field::from_raw([q2[0], q2[1], q2[2], q2[3]]);
+                let k2 = q2 - q1;
+                let k1 = k + k2 * $field::ZETA;
+                let k1_neg = is_neg(&k1);
+                let k2_neg = is_neg(&k2);
+                let k1 = if k1_neg { -k1 } else { k1 };
+                let k2 = if k2_neg { -k2 } else { k2 };
+
+                (get_lower_128(&k1), k1_neg, get_lower_128(&k2), k2_neg)
+            }
+        }
+    };
+}
+
 #[macro_export]
 macro_rules! new_curve_impl {
     (($($privacy:tt)*),

src/pasta/mod.rs

@@ -1,4 +1,13 @@
+use crate::arithmetic::mul_512;
+use crate::arithmetic::sbb;
+use crate::{
+    arithmetic::{CurveEndo, EndoParameters},
+    endo,
+};
+use ff::PrimeField;
+use ff::WithSmallOrderMulGroup;
 pub use pasta_curves::{pallas, vesta, Ep, EpAffine, Eq, EqAffine, Fp, Fq};
+use std::convert::TryInto;
 
 impl crate::CurveAffineExt for EpAffine {
     fn batch_add<const COMPLETE: bool, const LOAD_POINTS: bool>(
@@ -25,3 +34,66 @@ impl crate::CurveAffineExt for EqAffine {
         unimplemented!();
     }
 }
+
+// Generated using https://github.com/ConsenSys/gnark-crypto/blob/master/ecc/utils.go
+// with `pasta_curves::Fp::ZETA`
+// See https://github.com/demining/Endomorphism-Secp256k1/blob/main/README.md
+// to have more details about the endomorphism.
+const ENDO_PARAMS_EQ: EndoParameters = EndoParameters {
+    // round(b2/n)
+    gamma1: [0x32c49e4c00000003, 0x279a745902a2654e, 0x1, 0x0],
+    // round(-b1/n)
+    gamma2: [0x31f0256800000002, 0x4f34e8b2066389a4, 0x2, 0x0],
+    b1: [0x8cb1279300000001, 0x49e69d1640a89953, 0x0, 0x0],
+    b2: [0x0c7c095a00000001, 0x93cd3a2c8198e269, 0x0, 0x0],
+};
+
+// Generated using https://github.com/ConsenSys/gnark-crypto/blob/master/ecc/utils.go
+// with `pasta_curves::Fq::ZETA`
+// See https://github.com/demining/Endomorphism-Secp256k1/blob/main/README.md
+// to have more details about the endomorphism.
+const ENDO_PARAMS_EP: EndoParameters = EndoParameters {
+    // round(b2/n)
+    gamma1: [0x32c49e4bffffffff, 0x279a745902a2654e, 0x1, 0x0],
+    // round(-b1/n)
+    gamma2: [0x31f0256800000002, 0x4f34e8b2066389a4, 0x2, 0x0],
+    b1: [0x8cb1279300000000, 0x49e69d1640a89953, 0x0, 0x0],
+    b2: [0x0c7c095a00000001, 0x93cd3a2c8198e269, 0x0, 0x0],
+};
+
+endo!(Eq, Fp, ENDO_PARAMS_EQ);
+endo!(Ep, Fq, ENDO_PARAMS_EP);
+
+#[test]
+fn test_endo() {
+    use ff::Field;
+    use rand_core::OsRng;
+
+    for _ in 0..100000 {
+        let k = Fp::random(OsRng);
+        let (k1, k1_neg, k2, k2_neg) = Eq::decompose_scalar(&k);
+        if k1_neg & k2_neg {
+            assert_eq!(k, -Fp::from_u128(k1) + Fp::ZETA * Fp::from_u128(k2))
+        } else if k1_neg {
+            assert_eq!(k, -Fp::from_u128(k1) - Fp::ZETA * Fp::from_u128(k2))
+        } else if k2_neg {
+            assert_eq!(k, Fp::from_u128(k1) + Fp::ZETA * Fp::from_u128(k2))
+        } else {
+            assert_eq!(k, Fp::from_u128(k1) - Fp::ZETA * Fp::from_u128(k2))
+        }
+    }
+
+    for _ in 0..100000 {
+        let k = Fp::random(OsRng);
+        let (k1, k1_neg, k2, k2_neg) = Eq::decompose_scalar(&k);
+        if k1_neg & k2_neg {
+            assert_eq!(k, -Fp::from_u128(k1) + Fp::ZETA * Fp::from_u128(k2))
+        } else if k1_neg {
+            assert_eq!(k, -Fp::from_u128(k1) - Fp::ZETA * Fp::from_u128(k2))
+        } else if k2_neg {
+            assert_eq!(k, Fp::from_u128(k1) + Fp::ZETA * Fp::from_u128(k2))
+        } else {
+            assert_eq!(k, Fp::from_u128(k1) - Fp::ZETA * Fp::from_u128(k2))
+        }
+    }
+}