Implement Elligator2 hash-to-curve for Twisted Edwards curves

CHANGELOG.md

@@ -7,10 +7,12 @@
 - [\#577](https://github.com/arkworks-rs/algebra/pull/577) (`ark-ff`, `ark-ec`) Add `AdditiveGroup`, a trait for additive groups (equipped with scalar field).
 - [\#593](https://github.com/arkworks-rs/algebra/pull/593) (`ark-ec`) Change `AffineRepr::xy()` to return owned values.
 - [\#633](https://github.com/arkworks-rs/algebra/pull/633) (`ark-ec`) Generic pairing implementation for the curves from the BW6 family.
+- [\#659](https://github.com/arkworks-rs/algebra/pull/659) (`ark-ec`) Move auxiliary `parity` function from `ark_ec::hashing::curve_maps::swu` to `ark_ec::hashing::curve_maps`.
 - [\#748](https://github.com/arkworks-rs/algebra/pull/748) (`ark-ff`) Add `FromStr` for `BigInteger`.
 
 ### Features
 
+- [\#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`.
 - [\#693](https://github.com/arkworks-rs/algebra/pull/693) (`ark-serialize`) Add `serialize_to_vec!` convenience macro.

ec/src/hashing/curve_maps/elligator2.rs

@@ -0,0 +1,295 @@
+use crate::models::twisted_edwards::{MontCurveConfig, TECurveConfig};
+use ark_ff::{Field, One, Zero};
+use core::marker::PhantomData;
+
+use crate::{
+    hashing::{curve_maps::parity, map_to_curve_hasher::MapToCurve, HashToCurveError},
+    models::twisted_edwards::{Affine, Projective},
+};
+
+/// Trait defining the necessary parameters for the Elligator2 hash-to-curve method
+/// for twisted edwards curves form of:
+/// `b * y² = x³ + a * x² + x`
+/// from [\[BHKL13\]], according to [\[HSSWW23\]]
+///
+/// - [\[BHKL13\]] <http://dx.doi.org/10.1145/2508859.2516734>
+/// - [\[HSSWW23\]] <https://datatracker.ietf.org/doc/html/rfc9380>
+pub trait Elligator2Config: TECurveConfig + MontCurveConfig {
+    /// An element of the base field that is not a square root see \[BHKL13, Section 5\].
+    /// When `BaseField` is a prime field, [\[HSSWW23\]] mandates that `Z` is the
+    /// non-square with lowest absolute value in the `BaseField` when its elements
+    /// are represented as [-(q-1)/2, (q-1)/2]
+    const Z: Self::BaseField;
+
+    /// This must be equal to 1/(MontCurveConfig::COEFF_B)^2;
+    const ONE_OVER_COEFF_B_SQUARE: Self::BaseField;
+
+    /// This must be equal to MontCurveConfig::COEFF_A/MontCurveConfig::COEFF_B;
+    const COEFF_A_OVER_COEFF_B: Self::BaseField;
+}
+
+/// Represents the Elligator2 hash-to-curve map defined by `P`.
+pub struct Elligator2Map<P: TECurveConfig>(PhantomData<fn() -> P>);
+
+impl<P: Elligator2Config> MapToCurve<Projective<P>> for Elligator2Map<P> {
+    /// Checks if `P` represents a valid Elligator2 map. Panics otherwise.
+    fn check_parameters() -> Result<(), HashToCurveError> {
+        // We assume that the Montgomery curve is correct and  as such we do
+        // not verify the prerequisite for applicability of Elligator2 map to the TECurveConfing.
+
+        // Verifying that Z is a non-square
+        debug_assert!(
+            !P::Z.legendre().is_qr(),
+            "Z should be a quadratic non-residue for the Elligator2 map"
+        );
+
+        debug_assert_eq!(
+            P::ONE_OVER_COEFF_B_SQUARE,
+            <P as MontCurveConfig>::COEFF_B
+                .square()
+                .inverse()
+                .expect("B coefficient cannot be zero in Montgomery form"),
+            "ONE_OVER_COEFF_B_SQUARE is not equal to 1/COEFF_B^2 in Montgomery form"
+        );
+
+        debug_assert_eq!(
+            P::COEFF_A_OVER_COEFF_B,
+            <P as MontCurveConfig>::COEFF_A / <P as MontCurveConfig>::COEFF_B,
+            "COEFF_A_OVER_COEFF_B is not equal to COEFF_A/COEFF_B in Montgomery form"
+        );
+        Ok(())
+    }
+
+    /// Map an arbitrary base field element `element` to a curve point.
+    fn map_to_curve(element: P::BaseField) -> Result<Affine<P>, HashToCurveError> {
+        // 1. x1 = -(J / K) * inv0(1 + Z * u^2)
+        // 2. If x1 == 0, set x1 = -(J / K)
+        // 3. gx1 = x1^3 + (J / K) * x1^2 + x1 / K^2
+        // 4. x2 = -x1 - (J / K)
+        // 5. gx2 = x2^3 + (J / K) * x2^2 + x2 / K^2
+        // 6. If is_square(gx1), set x = x1, y = sqrt(gx1) with sgn0(y) == 1.
+        // 7. Else set x = x2, y = sqrt(gx2) with sgn0(y) == 0.
+        // 8. s = x * K
+        // 9. t = y * K
+        // 10. return (s, t)
+
+        // ark a is irtf J
+        // ark b is irtf k
+        let k = <P as MontCurveConfig>::COEFF_B;
+        let j_on_k = P::COEFF_A_OVER_COEFF_B;
+        let ksq_inv = P::ONE_OVER_COEFF_B_SQUARE;
+
+        let den_1 = <P::BaseField as One>::one() + P::Z * element.square();
+
+        let x1 = -j_on_k
+            / (if den_1.is_zero() {
+                <P::BaseField as One>::one()
+            } else {
+                den_1
+            });
+        let x1sq = x1.square();
+        let x1cb = x1sq * x1;
+        let gx1 = x1cb + j_on_k * x1sq + x1 * ksq_inv;
+
+        let x2 = -x1 - j_on_k;
+        let x2sq = x2.square();
+        let x2cb = x2sq * x2;
+        let gx2 = x2cb + j_on_k * x2sq + x2 * ksq_inv;
+
+        let (x, mut y, sgn0) = if gx1.legendre().is_qr() {
+            (
+                x1,
+                gx1.sqrt()
+                    .expect("We have checked that gx1 is a quadratic residue. Q.E.D"),
+                true,
+            )
+        } else {
+            (
+                x2,
+                gx2.sqrt()
+                    .expect("gx2 is a quadratic residue because gx1 is not. Q.E.D"),
+                false,
+            )
+        };
+
+        if parity(&y) != sgn0 {
+            y = -y;
+        }
+
+        let s = x * k;
+        let t = y * k;
+
+        // `(s, t)` is an affine point on the Montgomery curve.
+        // Ideally, the TECurve would come with a mapping to its Montgomery curve,
+        // so we could just call that mapping here.
+        // This is currently not supported in arkworks, so
+        // we just implement the rational map here from [\[HSSWW23\]] Appendix D
+
+        let tv1 = s + <P::BaseField as One>::one();
+        let tv2 = tv1 * t;
+        let (v, w) = if tv2.is_zero() {
+            (<P::BaseField as Zero>::zero(), <P::BaseField as One>::one())
+        } else {
+            let tv2_inv = tv2
+                .inverse()
+                .expect("None zero element has inverse. Q.E.D.");
+            let v = tv2_inv * tv1 * s;
+            let w = tv2_inv * t * (s - <P::BaseField as One>::one());
+            (v, w)
+        };
+
+        let point_on_curve = Affine::<P>::new_unchecked(v, w);
+        debug_assert!(
+            point_on_curve.is_on_curve(),
+            "Elligator2 mapped to a point off the curve"
+        );
+        Ok(point_on_curve)
+    }
+}
+
+#[cfg(test)]
+mod test {
+    use crate::{
+        hashing::{map_to_curve_hasher::MapToCurveBasedHasher, HashToCurve},
+        CurveConfig,
+    };
+    use ark_ff::field_hashers::DefaultFieldHasher;
+    use ark_std::vec::Vec;
+
+    use super::*;
+    use ark_ff::{fields::Fp64, MontBackend, MontFp};
+    use hashbrown::HashMap;
+    use sha2::Sha256;
+
+    #[derive(ark_ff::MontConfig)]
+    #[modulus = "101"]
+    #[generator = "2"]
+    pub struct F101Config;
+    pub type F101 = Fp64<MontBackend<F101Config, 1>>;
+
+    #[derive(ark_ff::MontConfig)]
+    #[modulus = "11"]
+    #[generator = "2"]
+    pub struct F11Config;
+    pub type F11 = Fp64<MontBackend<F11Config, 1>>;
+
+    struct TestElligator2MapToCurveConfig;
+
+    impl CurveConfig for TestElligator2MapToCurveConfig {
+        const COFACTOR: &'static [u64] = &[8];
+
+	#[rustfmt::skip]
+        const COFACTOR_INV: F11 = MontFp!("7");
+
+        type BaseField = F101;
+        type ScalarField = F11;
+    }
+
+    /// sage: EnsureValidEdwards(F101,-1,12)
+    /// sage: Curve_EdwardsToMontgomery(F101, -1, 12)
+    /// (76, 23)
+    /// sage: Curve_EdwardsToWeierstrass(F101, -1, 12)
+    /// (11, 5)
+    /// sage: EllipticCurve(F101,[11,5])
+    /// Elliptic Curve defined by y^2 = x^3 + 11*x + 5 over Finite Field of size 101
+    /// sage: EW = EllipticCurve(F101,[11,5])
+    /// sage: EW.order().factor()
+    /// 2^3 * 11
+    /// sage: EW = EdwardsCurve(F101,-1,12)
+    /// sage: EW.gens()[0] * 8
+    /// (5 : 36 : 1)
+    /// Point_WeierstrassToEdwards(F101, 11, 5, F101(5), F101(36), a_given=-1, d_given=12)
+    /// (23, 24)
+    impl TECurveConfig for TestElligator2MapToCurveConfig {
+        /// COEFF_A = -1
+        const COEFF_A: F101 = MontFp!("-1");
+
+        /// COEFF_D = 12
+        const COEFF_D: F101 = MontFp!("12");
+
+        const GENERATOR: Affine<TestElligator2MapToCurveConfig> =
+            Affine::<TestElligator2MapToCurveConfig>::new_unchecked(MontFp!("23"), MontFp!("24"));
+
+        type MontCurveConfig = TestElligator2MapToCurveConfig;
+    }
+
+    impl MontCurveConfig for TestElligator2MapToCurveConfig {
+        /// COEFF_A = 76
+        const COEFF_A: F101 = MontFp!("76");
+
+        /// COEFF_B = 23
+        const COEFF_B: F101 = MontFp!("23");
+
+        type TECurveConfig = TestElligator2MapToCurveConfig;
+    }
+
+    /// sage: find_z_ell2(F101)
+    /// 2
+    /// sage: F101 = FiniteField(101)
+    /// sage: 1/F101("23")^2
+    /// 80
+    /// sage: F101("76")/F101("23")
+    /// 56
+    impl Elligator2Config for TestElligator2MapToCurveConfig {
+        const Z: F101 = MontFp!("2");
+        const ONE_OVER_COEFF_B_SQUARE: F101 = MontFp!("80");
+
+        const COEFF_A_OVER_COEFF_B: F101 = MontFp!("56");
+    }
+
+    /// The point of the test is to get a simple twisted edwards curve and make
+    /// simple hash
+    #[test]
+    fn hash_arbitary_string_to_curve_elligator2() {
+        let test_elligator2_to_curve_hasher = MapToCurveBasedHasher::<
+            Projective<TestElligator2MapToCurveConfig>,
+            DefaultFieldHasher<Sha256, 128>,
+            Elligator2Map<TestElligator2MapToCurveConfig>,
+        >::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"
+        );
+    }
+
+    /// Use a simple twisted edwards curve and map the whole field to it. We observe
+    /// the map behaviour. Specifically, the map should be non-constant, all
+    /// elements should be mapped to curve successfully. everything can be mapped
+    #[test]
+    fn map_field_to_curve_elligator2() {
+        Elligator2Map::<TestElligator2MapToCurveConfig>::check_parameters().unwrap();
+
+        let mut map_range: Vec<Affine<TestElligator2MapToCurveConfig>> = vec![];
+        // We are mapping all elemnts of the field to the curve, verifying that
+        // map is not constant on that set.
+        for current_field_element in 0..101 {
+            map_range.push(
+                Elligator2Map::<TestElligator2MapToCurveConfig>::map_to_curve(F101::from(
+                    current_field_element as u64,
+                ))
+                .unwrap(),
+            );
+        }
+
+        let mut counts = HashMap::new();
+
+        let mode = map_range
+            .iter()
+            .copied()
+            .max_by_key(|&n| {
+                let count = counts.entry(n).or_insert(0);
+                *count += 1;
+                *count
+            })
+            .unwrap();
+
+        assert!(
+            *counts.get(&mode).unwrap() != 101,
+            "a constant hash function is not good."
+        );
+    }
+}

ec/src/hashing/curve_maps/mod.rs

@@ -1,2 +1,14 @@
+use ark_ff::{BigInteger, Field, PrimeField, Zero};
+pub mod elligator2;
 pub mod swu;
 pub mod wb;
+
+//// parity method on the Field elements based on [\[1\]] Section 4.1
+//// which is used by multiple curve maps including Elligator2 and SWU
+/// - [\[1\]] <https://datatracker.ietf.org/doc/html/rfc9380/>
+pub fn parity<F: Field>(element: &F) -> bool {
+    element
+        .to_base_prime_field_elements()
+        .find(|&x| !x.is_zero())
+        .map_or(false, |x| x.into_bigint().is_odd())
+}