Use Scott's subgroup membership tests for `G1` and `G2` of BLS12-381. (#74)

* implementation of the fast subgroup check for bls12_381

* add a bench

* subgroup check for g1

* subgroup check modifications

* remove useless test

* fmt

* need the last version of arkworks/algebra

* remove Parameters0

* using projective points is more efficient

* use of projective coordinates in G2

* fmt

* documentation on the constants and the psi function

* references for algorithms of eprint 2021/1130

* fmt

* sed ^ **

* minor improvement

* fmt

* fix Cargo toml

* nits

* some cleanup for g1

* add the beta test back

* fmt

* g2

* changelog

* add a  note on the Cargo.toml

* nits

* avoid variable name conflicts

* add the early-out optimization

Co-authored-by: weikeng <w.k@berkeley.edu>

CHANGELOG.md

@@ -8,6 +8,8 @@
 
 ### Improvements
 
+- [\#74](https://github.com/arkworks-rs/curves/pull/74) Use Scott's subgroup membership tests for `G1` and `G2` of BLS12-381.
+
 ### Bug fixes
 
 ## v0.3.0 

Cargo.toml

@@ -56,3 +56,9 @@ lto = "thin"
 incremental = true
 debug-assertions = true
 debug = true
+
+# To be removed in the new release.
+[patch.crates-io]
+ark-ec = { git = "https://github.com/arkworks-rs/algebra" }
+ark-ff = { git = "https://github.com/arkworks-rs/algebra" }
+ark-serialize = { git = "https://github.com/arkworks-rs/algebra" }

bls12_381/src/curves/g1.rs

@@ -1,9 +1,13 @@
 use crate::*;
 use ark_ec::{
     bls12,
+    bls12::Bls12Parameters,
     models::{ModelParameters, SWModelParameters},
+    short_weierstrass_jacobian::GroupAffine,
+    AffineCurve, ProjectiveCurve,
 };
-use ark_ff::{field_new, Zero};
+use ark_ff::{biginteger::BigInteger256, field_new, Zero};
+use ark_std::ops::Neg;
 
 pub type G1Affine = bls12::G1Affine<crate::Parameters>;
 pub type G1Projective = bls12::G1Projective<crate::Parameters>;
@@ -40,6 +44,25 @@ impl SWModelParameters for Parameters {
     fn mul_by_a(_: &Self::BaseField) -> Self::BaseField {
         Self::BaseField::zero()
     }
+
+    fn is_in_correct_subgroup_assuming_on_curve(p: &GroupAffine<Parameters>) -> bool {
+        // Algorithm from Section 6 of https://eprint.iacr.org/2021/1130.
+        //
+        // Check that endomorphism_p(P) == -[X^2]P
+
+        let x = BigInteger256::new([crate::Parameters::X[0], 0, 0, 0]);
+
+        // An early-out optimization described in Section 6.
+        // If uP == P but P != point of infinity, then the point is not in the right subgroup.
+        let x_times_p = p.mul(x);
+        if x_times_p.eq(p) && !p.infinity {
+            return false;
+        }
+
+        let minus_x_squared_times_p = x_times_p.mul(x).neg();
+        let endomorphism_p = endomorphism(p);
+        minus_x_squared_times_p.eq(&endomorphism_p)
+    }
 }
 
 /// G1_GENERATOR_X =
@@ -51,3 +74,14 @@ pub const G1_GENERATOR_X: Fq = field_new!(Fq, "368541675371338701678108831518307
 /// 1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569
 #[rustfmt::skip]
 pub const G1_GENERATOR_Y: Fq = field_new!(Fq, "1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569");
+
+/// BETA is a non-trivial cubic root of unity in Fq.
+pub const BETA: Fq = field_new!(Fq, "793479390729215512621379701633421447060886740281060493010456487427281649075476305620758731620350");
+
+pub fn endomorphism(p: &GroupAffine<Parameters>) -> GroupAffine<Parameters> {
+    // Endomorphism of the points on the curve.
+    // endomorphism_p(x,y) = (BETA * x, y) where BETA is a non-trivial cubic root of unity in Fq.
+    let mut res = (*p).clone();
+    res.x *= BETA;
+    res
+}

bls12_381/src/curves/g2.rs

@@ -1,9 +1,12 @@
 use crate::*;
+use ark_ec::bls12::Bls12Parameters;
 use ark_ec::{
     bls12,
     models::{ModelParameters, SWModelParameters},
+    short_weierstrass_jacobian::GroupAffine,
+    AffineCurve,
 };
-use ark_ff::{field_new, Zero};
+use ark_ff::{biginteger::BigInteger256, field_new, Field, Zero};
 
 pub type G2Affine = bls12::G2Affine<crate::Parameters>;
 pub type G2Projective = bls12::G2Projective<crate::Parameters>;
@@ -51,6 +54,21 @@ impl SWModelParameters for Parameters {
     fn mul_by_a(_: &Self::BaseField) -> Self::BaseField {
         Self::BaseField::zero()
     }
+
+    fn is_in_correct_subgroup_assuming_on_curve(point: &GroupAffine<Parameters>) -> bool {
+        // Algorithm from Section 4 of https://eprint.iacr.org/2021/1130.
+        //
+        // Checks that [p]P = [X]P
+
+        let mut x_times_point = point.mul(BigInteger256([crate::Parameters::X[0], 0, 0, 0]));
+        if crate::Parameters::X_IS_NEGATIVE {
+            x_times_point = -x_times_point;
+        }
+
+        let p_times_point = p_power_endomorphism(point);
+
+        x_times_point.eq(&p_times_point)
+    }
 }
 
 pub const G2_GENERATOR_X: Fq2 = field_new!(Fq2, G2_GENERATOR_X_C0, G2_GENERATOR_X_C1);
@@ -75,3 +93,52 @@ pub const G2_GENERATOR_Y_C0: Fq = field_new!(Fq, "198515060228729193556805452117
 /// 927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582
 #[rustfmt::skip]
 pub const G2_GENERATOR_Y_C1: Fq = field_new!(Fq, "927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582");
+
+// psi(x,y) = (x**p * PSI_X, y**p * PSI_Y) is the Frobenius composed
+// with the quadratic twist and its inverse
+
+// PSI_X = 1/(u+1)^((p-1)/3)
+pub const P_POWER_ENDOMORPHISM_COEFF_0 : Fq2 = field_new!(
+    Fq2,
+    FQ_ZERO,
+    field_new!(
+       Fq,
+       "4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939437"
+    )
+);
+
+// PSI_Y = 1/(u+1)^((p-1)/2)
+pub const P_POWER_ENDOMORPHISM_COEFF_1: Fq2 = field_new!(
+    Fq2,
+    field_new!(
+       Fq,
+       "2973677408986561043442465346520108879172042883009249989176415018091420807192182638567116318576472649347015917690530"),
+    field_new!(
+       Fq,
+       "1028732146235106349975324479215795277384839936929757896155643118032610843298655225875571310552543014690878354869257")
+);
+
+pub fn p_power_endomorphism(p: &GroupAffine<Parameters>) -> GroupAffine<Parameters> {
+    // The p-power endomorphism for G2 is defined as follows:
+    // 1. Note that G2 is defined on curve E': y^2 = x^3 + 4(u+1). To map a point (x, y) in E' to (s, t) in E,
+    //    one set s = x / ((u+1) ^ (1/3)), t = y / ((u+1) ^ (1/2)), because E: y^2 = x^3 + 4.
+    // 2. Apply the Frobenius endomorphism (s, t) => (s', t'), another point on curve E,
+    //    where s' = s^p, t' = t^p.
+    // 3. Map the point from E back to E'; that is,
+    //    one set x' = s' * ((u+1) ^ (1/3)), y' = t' * ((u+1) ^ (1/2)).
+    //
+    // To sum up, it maps
+    // (x,y) -> (x^p / ((u+1)^((p-1)/3)), y^p / ((u+1)^((p-1)/2)))
+    // as implemented in the code as follows.
+
+    let mut res = *p;
+    res.x.frobenius_map(1);
+    res.y.frobenius_map(1);
+
+    let tmp_x = res.x.clone();
+    res.x.c0 = -P_POWER_ENDOMORPHISM_COEFF_0.c1 * &tmp_x.c1;
+    res.x.c1 = P_POWER_ENDOMORPHISM_COEFF_0.c1 * &tmp_x.c0;
+    res.y *= P_POWER_ENDOMORPHISM_COEFF_1;
+
+    res
+}

bls12_381/src/curves/tests.rs

@@ -1,8 +1,12 @@
 #![allow(unused_imports)]
-use ark_ec::{models::SWModelParameters, AffineCurve, PairingEngine, ProjectiveCurve};
+use ark_ec::{
+    models::SWModelParameters,
+    short_weierstrass_jacobian::{GroupAffine, GroupProjective},
+    AffineCurve, PairingEngine, ProjectiveCurve,
+};
 use ark_ff::{
     fields::{Field, FpParameters, PrimeField, SquareRootField},
-    One, Zero,
+    BitIteratorBE, One, UniformRand, Zero,
 };
 use ark_serialize::CanonicalSerialize;
 use ark_std::rand::Rng;
@@ -11,6 +15,7 @@ use core::ops::{AddAssign, MulAssign};
 
 use crate::{g1, g2, Bls12_381, Fq, Fq12, Fq2, Fr, G1Affine, G1Projective, G2Affine, G2Projective};
 use ark_algebra_test_templates::{curves::*, groups::*};
+use ark_ec::group::Group;
 
 #[test]
 fn test_g1_projective_curve() {
@@ -115,3 +120,54 @@ fn test_g1_generator_raw() {
         x.add_assign(&Fq::one());
     }
 }
+
+#[test]
+fn test_g1_endomorphism_beta() {
+    assert!(g1::BETA.pow(&[3u64]).is_one());
+}
+
+#[test]
+fn test_g1_subgroup_membership_via_endomorphism() {
+    let mut rng = test_rng();
+    let generator = G1Projective::rand(&mut rng).into_affine();
+    assert!(generator.is_in_correct_subgroup_assuming_on_curve());
+}
+
+#[test]
+fn test_g1_subgroup_non_membership_via_endomorphism() {
+    let mut rng = test_rng();
+    loop {
+        let x = Fq::rand(&mut rng);
+        let greatest = rng.gen();
+
+        if let Some(p) = G1Affine::get_point_from_x(x, greatest) {
+            if !p.into_projective().mul(Fr::characteristic()).is_zero() {
+                assert!(!p.is_in_correct_subgroup_assuming_on_curve());
+                return;
+            }
+        }
+    }
+}
+
+#[test]
+fn test_g2_subgroup_membership_via_endomorphism() {
+    let mut rng = test_rng();
+    let generator = G2Projective::rand(&mut rng).into_affine();
+    assert!(generator.is_in_correct_subgroup_assuming_on_curve());
+}
+
+#[test]
+fn test_g2_subgroup_non_membership_via_endomorphism() {
+    let mut rng = test_rng();
+    loop {
+        let x = Fq2::rand(&mut rng);
+        let greatest = rng.gen();
+
+        if let Some(p) = G2Affine::get_point_from_x(x, greatest) {
+            if !p.into_projective().mul(Fr::characteristic()).is_zero() {
+                assert!(!p.is_in_correct_subgroup_assuming_on_curve());
+                return;
+            }
+        }
+    }
+}

curve-benches/src/macros/ec.rs

@@ -196,6 +196,32 @@ macro_rules! ec_bench {
             });
         }
 
+        fn deser_uncompressed(b: &mut $crate::bencher::Bencher) {
+            use ark_ec::ProjectiveCurve;
+            use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
+            const SAMPLES: usize = 1000;
+
+            let mut rng = ark_std::test_rng();
+
+            let mut num_bytes = 0;
+            let tmp = <$projective>::rand(&mut rng).into_affine();
+            let v: Vec<_> = (0..SAMPLES)
+                .flat_map(|_| {
+                    let mut bytes = Vec::with_capacity(1000);
+                    tmp.serialize_uncompressed(&mut bytes).unwrap();
+                    num_bytes = bytes.len();
+                    bytes
+                })
+                .collect();
+
+            let mut count = 0;
+            b.iter(|| {
+                count = (count + 1) % SAMPLES;
+                let index = count * num_bytes;
+                <$affine>::deserialize_uncompressed(&v[index..(index + num_bytes)]).unwrap()
+            });
+        }
+
         fn msm_131072(b: &mut $crate::bencher::Bencher) {
             use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
             const SAMPLES: usize = 131072;
@@ -224,6 +250,7 @@ macro_rules! ec_bench {
             deser,
             ser_unchecked,
             deser_unchecked,
+            deser_uncompressed,
             msm_131072,
         );
     };