Elligator2 hash-to-curve for Bandersnatch (#758)

* Implement Elligator2 hash-to-curve for Bandersnatch curve

* Add relevant entry to `CHANGELOG.md`

* Remove sha3 unused dependancy

* Include the script to compute an efficient z for elligator2 map.

* move Elligator2 for Bandersnatch from Pending to Features

---------

Co-authored-by: Marcin <marcin.gorny.94@protonmail.com>

CHANGELOG.md

@@ -20,6 +20,7 @@
 
 ### Features
 
+- [\#758](https://github.com/arkworks-rs/algebra/pull/758) Implement Elligator2 hash-to-curve parameters for Bandersnatch.
 - [\#659](https://github.com/arkworks-rs/algebra/pull/659) (`ark-ec`) Add Elligator2 hash-to-curve map.
 - [\#689](https://github.com/arkworks-rs/algebra/pull/689) (`ark-serialize`) Add `CanonicalSerialize` and `CanonicalDeserialize` impls for `VecDeque` and `LinkedList`.
 - [\#691](https://github.com/arkworks-rs/algebra/pull/691) (`ark-poly`) Implement `Polynomial` for `SparseMultilinearExtension` and `DenseMultilinearExtension`.

curves/ed_on_bls12_381_bandersnatch/Cargo.toml

@@ -22,6 +22,7 @@ ark-relations = { version = "0.4.0", default-features = false }
 ark-serialize = { version = "0.4.0", default-features = false }
 ark-algebra-test-templates = { version = "0.4.0", default-features = false }
 ark-curve-constraint-tests = { path = "../curve-constraint-tests", default-features = false }
+sha2 = { version = "0.10", default-features = false }
 
 [features]
 default = []

curves/ed_on_bls12_381_bandersnatch/src/curves/mod.rs

@@ -1,4 +1,5 @@
 use ark_ec::{
+    hashing::curve_maps::elligator2::Elligator2Config,
     models::CurveConfig,
     short_weierstrass::{self, SWCurveConfig},
     twisted_edwards::{Affine, MontCurveConfig, Projective, TECurveConfig},
@@ -137,3 +138,55 @@ impl SWCurveConfig for BandersnatchConfig {
     /// generators
     const GENERATOR: SWAffine = SWAffine::new_unchecked(SW_GENERATOR_X, SW_GENERATOR_Y);
 }
+
+// Elligator hash to curve Bandersnatch
+// sage: find_z_ell2(GF(52435875175126190479447740508185965837690552500527637822603658699938581184513))
+// 5
+//
+// sage: Fq = GF(52435875175126190479447740508185965837690552500527637822603658699938581184513)
+// sage: 1/Fq(25465760566081946422412445027709227188579564747101592991722834452325077642517)^2
+// sage: COEFF_A = Fq(29978822694968839326280996386011761570173833766074948509196803838190355340952)
+// sage: COEFF_B = Fq(25465760566081946422412445027709227188579564747101592991722834452325077642517)
+// sage: 1/COEFF_B^2
+// 35484827650731063748396669747216844996598387089274032563585525486049249153249
+// sage: COEFF_A/COEFF_B
+// 22511181562295907836254750456843438087744031914659733450388350895537307167857
+impl Elligator2Config for BandersnatchConfig {
+    const Z: Fq = MontFp!("5");
+
+    /// This must be equal to 1/(MontCurveConfig::COEFF_B)^2;
+    const ONE_OVER_COEFF_B_SQUARE: Fq =
+        MontFp!("35484827650731063748396669747216844996598387089274032563585525486049249153249");
+
+    /// This must be equal to MontCurveConfig::COEFF_A/MontCurveConfig::COEFF_B;
+    const COEFF_A_OVER_COEFF_B: Fq =
+        MontFp!("22511181562295907836254750456843438087744031914659733450388350895537307167857");
+}
+
+#[cfg(test)]
+mod test {
+    use super::*;
+    use ark_ec::hashing::{
+        curve_maps::elligator2::Elligator2Map, map_to_curve_hasher::MapToCurveBasedHasher,
+        HashToCurve,
+    };
+    use ark_ff::field_hashers::DefaultFieldHasher;
+    use sha2::Sha512;
+
+    #[test]
+    fn test_elligtor2_hash2curve_hashes_to_curve() {
+        let test_elligator2_to_curve_hasher = MapToCurveBasedHasher::<
+            Projective<BandersnatchConfig>,
+            DefaultFieldHasher<Sha512, 128>,
+            Elligator2Map<BandersnatchConfig>,
+        >::new(&[1])
+        .unwrap();
+
+        let hash_result = test_elligator2_to_curve_hasher.hash(b"if you stick a Babel fish in your ear you can instantly understand anything said to you in any form of language.").expect("fail to hash the string to curve");
+
+        assert!(
+            hash_result.is_on_curve(),
+            "hash results into a point off the curve"
+        );
+    }
+}

ec/src/hashing/curve_maps/curve_map_parameter_helper.sage

@@ -0,0 +1,39 @@
+# Arguments:
+# - F, a field object, e.g., F = GF(2^521 - 1)
+# - A and B, the coefficients of the curve y^2 = x^3 + A * x + B
+def find_z_sswu(F, A, B):
+    R.<xx> = F[]
+    # Polynomial ring over F
+    g = xx^3 + F(A) * xx + F(B)
+    # y^2 = g(x) = x^3 + A * x + B
+    ctr = F.gen()
+    while True:
+        for Z_cand in (F(ctr), F(-ctr)):
+            # Criterion 1: Z is non-square in F.
+            if is_square(Z_cand):
+                continue
+            # Criterion 2: Z != -1 in F.
+            if Z_cand == F(-1):
+                continue
+            # Criterion 3: g(x) - Z is irreducible over F.
+            if not (g - Z_cand).is_irreducible():
+                continue
+            # Criterion 4: g(B / (Z * A)) is square in F.
+            if is_square(g(B / (Z_cand * A))):
+                return Z_cand
+        ctr += 1
+
+# Finds the smallest z in term of non-zero bit
+# in sage representation for consturcting
+# elligator2 map for a curve defined over field F.
+# Argument:
+# - F, a field object, e.g., F = GF(2^255 - 19)
+def find_z_ell2(F):
+    ctr = F.gen()
+    while True:
+        for Z_cand in (F(ctr), F(-ctr)):
+            # Z must be a non-square in F.
+            if is_square(Z_cand):
+                continue
+            return Z_cand
+        ctr += 1