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>

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
+}