Implement the Bandersnatch curve (#64)

* impl bandersnatch

* clean up

* update changelog

* Relocate the readme so they show up in the doc

* Delete README.md

* Relocate the changelog entry

* rename & fmt

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

CHANGELOG.md

@@ -19,6 +19,8 @@
 
 ### Features
 
+- [\#64](https://github.com/arkworks-rs/curves/pull/64) Implement the Bandersnatch curve, another twisted Edwards curve for BLS12-381.
+
 ### Improvements
 
 ### Bug fixes

Cargo.toml

@@ -15,6 +15,7 @@ members = [
 
     "bls12_381",
     "ed_on_bls12_381",
+    "ed_on_bls12_381_bandersnatch",
 
     "bn254",
     "ed_on_bn254",

README.md

@@ -5,6 +5,7 @@ This repository contains implementations of some popular elliptic curves. The cu
 ### BLS12-381 and embedded curves
 * [`ark-bls12-381`](bls12_381): Implements the BLS12-381 pairing-friendly curve
 * [`ark-ed-on-bls12-381`](ed_on_bls12_381): Implements a Twisted Edwards curve atop the scalar field of BLS12-381
+* [`ark-ed-on-bls12-381-bandersnatch`](ed_on_bls12_381_bandersnatch): Implements Bandersnatch, another Twisted Edwards curve atop the scalar field of BLS12-381
 
 ### BLS12-377 and related curves
 * [`ark-bls12-377`](bls12_377): Implements the BLS12-377 pairing-friendly curve

ed_on_bls12_381/src/lib.rs

@@ -9,7 +9,7 @@
 #![forbid(unsafe_code)]
 
 //! This library implements a twisted Edwards curve whose base field is the scalar field of the
-//! curve BLS12-377. This allows defining cryptographic primitives that use elliptic curves over
+//! curve BLS12-381. This allows defining cryptographic primitives that use elliptic curves over
 //! the scalar field of the latter curve. This curve was generated by Sean Bowe, and is also known
 //! as [Jubjub](https://github.com/zkcrypto/jubjub).
 //!

ed_on_bls12_381_bandersnatch/Cargo.toml

@@ -0,0 +1,34 @@
+[package]
+name = "ark-ed-on-bls12-381-bandersnatch"
+version = "0.1.0"
+authors = [ "zhenfei zhang", "arkworks contributors" ]
+description = "Bandersnatch: a curve defined over the scalar field of the BLS12-381 curve"
+repository = "https://github.com/zhenfeizhang/bandersnatch-rust"
+keywords = ["cryptography", "finite-fields", "elliptic-curves" ]
+categories = ["cryptography"]
+include = ["Cargo.toml", "src", "README.md", "LICENSE-APACHE", "LICENSE-MIT"]
+license = "MIT/Apache-2.0"
+edition = "2018"
+
+[dependencies]
+ark-ff = { version = "^0.3.0", default-features = false }
+ark-ec = { version = "^0.3.0", default-features = false }
+ark-std = { version = "^0.3.0", default-features = false }
+ark-r1cs-std = { version = "^0.3.0", default-features = false, optional = true }
+ark-bls12-381 = { version = "^0.3.0", default-features = false, features = [ "scalar_field" ] }
+
+[dev-dependencies]
+ark-relations = { version = "^0.3.0", default-features = false }
+ark-serialize = { version = "^0.3.0", default-features = false }
+ark-algebra-test-templates = { version = "^0.3.0", default-features = false }
+ark-curve-constraint-tests = { path = "../curve-constraint-tests", default-features = false }
+
+[features]
+default = []
+std = [ 
+    "ark-std/std", 
+    "ark-ff/std", 
+    "ark-ec/std", 
+    "ark-bls12-381/std" 
+]
+r1cs = ["ark-r1cs-std"]

ed_on_bls12_381_bandersnatch/src/constraints/curves.rs

@@ -0,0 +1,12 @@
+use crate::*;
+use ark_r1cs_std::groups::curves::twisted_edwards::AffineVar;
+
+use crate::constraints::FqVar;
+
+/// A variable that is the R1CS equivalent of `crate::EdwardsAffine`.
+pub type EdwardsVar = AffineVar<EdwardsParameters, FqVar>;
+
+#[test]
+fn test() {
+    ark_curve_constraint_tests::curves::te_test::<_, EdwardsVar>().unwrap();
+}

ed_on_bls12_381_bandersnatch/src/constraints/fields.rs

@@ -0,0 +1,9 @@
+use ark_r1cs_std::fields::fp::FpVar;
+
+/// A variable that is the R1CS equivalent of `crate::Fq`.
+pub type FqVar = FpVar<crate::Fq>;
+
+#[test]
+fn test() {
+    ark_curve_constraint_tests::fields::field_test::<_, _, FqVar>().unwrap();
+}

ed_on_bls12_381_bandersnatch/src/constraints/mod.rs

@@ -0,0 +1,107 @@
+//! This module implements the R1CS equivalent of `ark_bandersnatch`.
+//!
+//! It implements field variables for `crate::Fq`,
+//! and group variables for `crate::GroupProjective`.
+//!
+//! The field underlying these constraints is `crate::Fq`.
+//!
+//! # Examples
+//!
+//! One can perform standard algebraic operations on `FqVar`:
+//!
+//! ```
+//! # fn main() -> Result<(), ark_relations::r1cs::SynthesisError> {
+//! use ark_std::UniformRand;
+//! use ark_relations::r1cs::*;
+//! use ark_r1cs_std::prelude::*;
+//! use ark_ed_on_bls12_381_bandersnatch::{*, constraints::*};
+//!
+//! let cs = ConstraintSystem::<Fq>::new_ref();
+//! // This rng is just for test purposes; do not use it
+//! // in real applications.
+//! let mut rng = ark_std::test_rng();
+//!
+//! // Generate some random `Fq` elements.
+//! let a_native = Fq::rand(&mut rng);
+//! let b_native = Fq::rand(&mut rng);
+//!
+//! // Allocate `a_native` and `b_native` as witness variables in `cs`.
+//! let a = FqVar::new_witness(ark_relations::ns!(cs, "generate_a"), || Ok(a_native))?;
+//! let b = FqVar::new_witness(ark_relations::ns!(cs, "generate_b"), || Ok(b_native))?;
+//!
+//! // Allocate `a_native` and `b_native` as constants in `cs`. This does not add any
+//! // constraints or variables.
+//! let a_const = FqVar::new_constant(ark_relations::ns!(cs, "a_as_constant"), a_native)?;
+//! let b_const = FqVar::new_constant(ark_relations::ns!(cs, "b_as_constant"), b_native)?;
+//!
+//! let one = FqVar::one();
+//! let zero = FqVar::zero();
+//!
+//! // Sanity check one + one = two
+//! let two = &one + &one + &zero;
+//! two.enforce_equal(&one.double()?)?;
+//!
+//! assert!(cs.is_satisfied()?);
+//!
+//! // Check that the value of &a + &b is correct.
+//! assert_eq!((&a + &b).value()?, a_native + &b_native);
+//!
+//! // Check that the value of &a * &b is correct.
+//! assert_eq!((&a * &b).value()?, a_native * &b_native);
+//!
+//! // Check that operations on variables and constants are equivalent.
+//! (&a + &b).enforce_equal(&(&a_const + &b_const))?;
+//! assert!(cs.is_satisfied()?);
+//! # Ok(())
+//! # }
+//! ```
+//!
+//! One can also perform standard algebraic operations on `EdwardsVar`:
+//!
+//! ```
+//! # fn main() -> Result<(), ark_relations::r1cs::SynthesisError> {
+//! # use ark_std::UniformRand;
+//! # use ark_relations::r1cs::*;
+//! # use ark_r1cs_std::prelude::*;
+//! # use ark_ed_on_bls12_381_bandersnatch::{*, constraints::*};
+//!
+//! # let cs = ConstraintSystem::<Fq>::new_ref();
+//! # let mut rng = ark_std::test_rng();
+//!
+//! // Generate some random `Edwards` elements.
+//! let a_native = EdwardsProjective::rand(&mut rng);
+//! let b_native = EdwardsProjective::rand(&mut rng);
+//!
+//! // Allocate `a_native` and `b_native` as witness variables in `cs`.
+//! let a = EdwardsVar::new_witness(ark_relations::ns!(cs, "a"), || Ok(a_native))?;
+//! let b = EdwardsVar::new_witness(ark_relations::ns!(cs, "b"), || Ok(b_native))?;
+//!
+//! // Allocate `a_native` and `b_native` as constants in `cs`. This does not add any
+//! // constraints or variables.
+//! let a_const = EdwardsVar::new_constant(ark_relations::ns!(cs, "a_as_constant"), a_native)?;
+//! let b_const = EdwardsVar::new_constant(ark_relations::ns!(cs, "b_as_constant"), b_native)?;
+//!
+//! // This returns the identity of `Edwards`.
+//! let zero = EdwardsVar::zero();
+//!
+//! // Sanity check one + one = two
+//! let two_a = &a + &a + &zero;
+//! two_a.enforce_equal(&a.double()?)?;
+//!
+//! assert!(cs.is_satisfied()?);
+//!
+//! // Check that the value of &a + &b is correct.
+//! assert_eq!((&a + &b).value()?, a_native + &b_native);
+//!
+//! // Check that operations on variables and constants are equivalent.
+//! (&a + &b).enforce_equal(&(&a_const + &b_const))?;
+//! assert!(cs.is_satisfied()?);
+//! # Ok(())
+//! # }
+//! ```
+
+mod curves;
+mod fields;
+
+pub use curves::*;
+pub use fields::*;

ed_on_bls12_381_bandersnatch/src/curves/mod.rs

@@ -0,0 +1,94 @@
+use crate::{Fq, Fr};
+use ark_ec::{
+    models::{ModelParameters, MontgomeryModelParameters, TEModelParameters},
+    twisted_edwards_extended::{GroupAffine, GroupProjective},
+};
+use ark_ff::{field_new, Field};
+#[cfg(test)]
+mod tests;
+
+pub type EdwardsAffine = GroupAffine<EdwardsParameters>;
+pub type EdwardsProjective = GroupProjective<EdwardsParameters>;
+
+/// `banersnatch` is a twisted Edwards curve. These curves have equations of the
+/// form: ax² + y² = 1 - dx²y².
+/// over some base finite field Fq.
+///
+/// banersnatch's curve equation: -5x² + y² = 1 - dx²y²
+///
+/// q = 52435875175126190479447740508185965837690552500527637822603658699938581184513.
+///
+/// a = 52435875175126190479447740508185965837690552500527637822603658699938581184508.
+/// d = (138827208126141220649022263972958607803/
+/// 171449701953573178309673572579671231137) mod q
+///   = 45022363124591815672509500913686876175488063829319466900776701791074614335719.
+///
+/// Sage script to calculate these:
+///
+/// ```text
+/// q = 52435875175126190479447740508185965837690552500527637822603658699938581184513
+/// Fq = GF(q)
+/// d = (Fq(138827208126141220649022263972958607803)/Fq(171449701953573178309673572579671231137))
+/// ```
+/// These parameters and the sage script obtained from:
+/// <https://github.com/asanso/Bandersnatch/>
+#[derive(Clone, Default, PartialEq, Eq)]
+pub struct EdwardsParameters;
+
+impl ModelParameters for EdwardsParameters {
+    type BaseField = Fq;
+    type ScalarField = Fr;
+}
+
+impl TEModelParameters for EdwardsParameters {
+    /// COEFF_A = -1
+    #[rustfmt::skip]
+    const COEFF_A: Fq = field_new!(Fq, "-5");
+
+    /// COEFF_D = (138827208126141220649022263972958607803/
+    /// 171449701953573178309673572579671231137) mod q
+    #[rustfmt::skip]
+    const COEFF_D: Fq = field_new!(Fq, "45022363124591815672509500913686876175488063829319466900776701791074614335719");
+
+    /// COFACTOR = 4
+    const COFACTOR: &'static [u64] = &[4];
+
+    /// COFACTOR^(-1) mod r =
+    /// 9831726595336160714896451345284868594481866920080427688839802480047265754601
+    #[rustfmt::skip]
+    const COFACTOR_INV: Fr = field_new!(Fr, "9831726595336160714896451345284868594481866920080427688839802480047265754601");
+
+    /// AFFINE_GENERATOR_COEFFS = (GENERATOR_X, GENERATOR_Y)
+    const AFFINE_GENERATOR_COEFFS: (Self::BaseField, Self::BaseField) = (GENERATOR_X, GENERATOR_Y);
+
+    type MontgomeryModelParameters = EdwardsParameters;
+
+    /// Multiplication by `a` is multiply by `-5`.
+    #[inline(always)]
+    fn mul_by_a(elem: &Self::BaseField) -> Self::BaseField {
+        let t = (*elem).double().double();
+        -(t + *elem)
+    }
+}
+
+impl MontgomeryModelParameters for EdwardsParameters {
+    /// COEFF_A = 29978822694968839326280996386011761570173833766074948509196803838190355340952
+    #[rustfmt::skip]
+    const COEFF_A: Fq = field_new!(Fq, "29978822694968839326280996386011761570173833766074948509196803838190355340952");
+    /// COEFF_B = 25465760566081946422412445027709227188579564747101592991722834452325077642517
+    #[rustfmt::skip]
+    const COEFF_B: Fq = field_new!(Fq, "25465760566081946422412445027709227188579564747101592991722834452325077642517");
+
+    type TEModelParameters = EdwardsParameters;
+}
+
+// using the generator from bench.py (in affine form)
+// P = BandersnatchPoint(
+//      13738737789055671334382939318077718462576533426798874551591468520593954805549,
+//      11575885077368931610486103676191793534029821920164915325066801506752632626968,
+//      14458123306641001284399433086015669988340559992755622870694102351476334505845,
+//      C)
+#[rustfmt::skip]
+const GENERATOR_X: Fq = field_new!(Fq, "29627151942733444043031429156003786749302466371339015363120350521834195802525");
+#[rustfmt::skip]
+const GENERATOR_Y: Fq = field_new!(Fq, "27488387519748396681411951718153463804682561779047093991696427532072116857978");

ed_on_bls12_381_bandersnatch/src/curves/tests.rs

@@ -0,0 +1,103 @@
+use crate::*;
+use ark_algebra_test_templates::{curves::*, groups::*};
+use ark_ec::{AffineCurve, ProjectiveCurve};
+use ark_ff::{bytes::FromBytes, Zero};
+use ark_std::{rand::Rng, str::FromStr, test_rng};
+
+#[test]
+fn test_projective_curve() {
+    curve_tests::<EdwardsProjective>();
+
+    edwards_tests::<EdwardsParameters>();
+}
+
+#[test]
+fn test_projective_group() {
+    let mut rng = test_rng();
+    let a = rng.gen();
+    let b = rng.gen();
+    for _i in 0..100 {
+        group_test::<EdwardsProjective>(a, b);
+    }
+}
+
+#[test]
+fn test_affine_group() {
+    let mut rng = test_rng();
+    let a: EdwardsAffine = rng.gen();
+    let b: EdwardsAffine = rng.gen();
+    for _i in 0..100 {
+        group_test::<EdwardsAffine>(a, b);
+    }
+}
+
+#[test]
+fn test_generator() {
+    let generator = EdwardsAffine::prime_subgroup_generator();
+    assert!(generator.is_on_curve());
+    assert!(generator.is_in_correct_subgroup_assuming_on_curve());
+}
+
+#[test]
+fn test_conversion() {
+    let mut rng = test_rng();
+    let a: EdwardsAffine = rng.gen();
+    let b: EdwardsAffine = rng.gen();
+    let a_b = {
+        use ark_ec::group::Group;
+        (a + &b).double().double()
+    };
+    let a_b2 = (a.into_projective() + &b.into_projective())
+        .double()
+        .double();
+    assert_eq!(a_b, a_b2.into_affine());
+    assert_eq!(a_b.into_projective(), a_b2);
+}
+
+#[test]
+fn test_scalar_multiplication() {
+    let f1 = Fr::from_str(
+        "4257185345094557079734489188109952172285839137338142340240392707284963971010",
+    )
+    .unwrap();
+    let f2 = Fr::from_str(
+        "1617998875791656082457755819308421023664764572929977389209373068350490665160",
+    )
+    .unwrap();
+
+    let g = EdwardsAffine::from_str(
+        "(29627151942733444043031429156003786749302466371339015363120350521834195802525, \
+         27488387519748396681411951718153463804682561779047093991696427532072116857978)",
+    )
+    .unwrap();
+    let f1f2g = EdwardsAffine::from_str(
+        "(16530491029447613915334753043669938793793987372416328257719459807614119987301, \
+         42481140308370805476764840229335460092474682686441442216596889726548353970772)",
+    )
+    .unwrap();
+
+    assert!(!g.is_zero());
+    assert!(!f1f2g.is_zero());
+
+    let f1g = g.mul(f1).into_affine();
+    assert_eq!(g.mul(f1 * &f2).into_affine(), f1f2g);
+    assert_eq!(f1g.mul(f2).into_affine(), f1f2g);
+}
+
+#[test]
+fn test_bytes() {
+    let g_from_repr = EdwardsAffine::from_str(
+        "(29627151942733444043031429156003786749302466371339015363120350521834195802525, \
+         27488387519748396681411951718153463804682561779047093991696427532072116857978)",
+    )
+    .unwrap();
+
+    let g_bytes = ark_ff::to_bytes![g_from_repr].unwrap();
+    let g = EdwardsAffine::read(g_bytes.as_slice()).unwrap();
+    assert_eq!(g_from_repr, g);
+}
+
+#[test]
+fn test_montgomery_conversion() {
+    montgomery_conversion_test::<EdwardsParameters>();
+}

ed_on_bls12_381_bandersnatch/src/fields/fq.rs

@@ -0,0 +1 @@
+pub use ark_bls12_381::{Fr as Fq, FrParameters as FqParameters};

ed_on_bls12_381_bandersnatch/src/fields/fr.rs

@@ -0,0 +1,115 @@
+use ark_ff::{
+    biginteger::BigInteger256 as BigInteger,
+    fields::{FftParameters, Fp256, Fp256Parameters, FpParameters},
+};
+
+pub type Fr = Fp256<FrParameters>;
+
+pub struct FrParameters;
+
+impl Fp256Parameters for FrParameters {}
+impl FftParameters for FrParameters {
+    type BigInt = BigInteger;
+
+    /// Let `N` be the size of the multiplicative group defined by the field.
+    /// Then `TWO_ADICITY` is the two-adicity of `N`, i.e. the integer `s`
+    /// such that `N = 2^s * t` for some odd integer `t`.
+    const TWO_ADICITY: u32 = 5;
+
+    /// 2^s root of unity computed by GENERATOR^t
+    /// 4740934665446857387895054948191089665295030226009829406950782728666658007874
+    #[rustfmt::skip]
+    const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([
+        0xa4dcdba087826b42,
+        0x6e4ab162f57f862a,
+        0xabc5492749348d6a,
+        0xa7b462035f8c169,
+    ]);
+}
+impl FpParameters for FrParameters {
+    /// The modulus of the field.
+    /// MODULUS = 13108968793781547619861935127046491459309155893440570251786403306729687672801.
+    #[rustfmt::skip]
+    const MODULUS: BigInteger = BigInteger([
+        0x74fd06b52876e7e1,
+        0xff8f870074190471,
+        0x0cce760202687600,
+        0x1cfb69d4ca675f52,
+    ]);
+
+    /// The number of bits needed to represent the `Self::MODULUS`.
+    const MODULUS_BITS: u32 = 253;
+
+    /// The number of bits that can be reliably stored.
+    /// (Should equal `SELF::MODULUS_BITS - 1`)
+    const CAPACITY: u32 = Self::MODULUS_BITS - 1;
+
+    /// The number of bits that must be shaved from the beginning of
+    /// the representation when randomly sampling.
+    const REPR_SHAVE_BITS: u32 = 4;
+
+    /// Let `M` be the power of 2^64 nearest to `Self::MODULUS_BITS`. Then
+    /// `R = M % Self::MODULUS`.
+    /// R = 10920338887063814464675503992315976178796737518116002025166357554075628257528
+    #[rustfmt::skip]
+    const R: BigInteger = BigInteger([
+        0x5817ca56bc48c0f8,
+        0x0383c7fc5f37dc74,
+        0x998c4fefecbc4ff8,
+        0x1824b159acc5056f,
+    ]);
+
+    /// R2 = R^2 % Self::MODULUS
+    /// R2 = 4932290691328759802879919559207542894238895193980447506221046538067943049163
+    #[rustfmt::skip]
+    const R2: BigInteger = BigInteger([
+        0xdbb4f5d658db47cb,
+        0x40fa7ca27fecb938,
+        0xaa9e6daec0055cea,
+        0xae793ddb14aec7d
+    ]);
+
+    /// INV = -MODULUS^{-1} mod 2^64
+    /// INV = 17410672245482742751
+    const INV: u64 = 0xf19f22295cc063df;
+
+    /// A multiplicative generator of the field.
+    /// `Self::GENERATOR` is an element having multiplicative order
+    /// `Self::MODULUS - 1`.
+    /// n = 9962557815892774795293348142308860067333132192265356416788884706064406244838
+    #[rustfmt::skip]
+    const GENERATOR: BigInteger = BigInteger([
+        0x56b6f3ab7b616de6,
+        0x114f419d6c9083e5,
+        0xbf518d217780c4b9,
+        0x16069b9f45dbce7f,
+    ]);
+
+    /// (Self::MODULUS - 1) / 2
+    /// 6554484396890773809930967563523245729654577946720285125893201653364843836400
+    const MODULUS_MINUS_ONE_DIV_TWO: BigInteger = BigInteger([
+        0xba7e835a943b73f0,
+        0x7fc7c3803a0c8238,
+        0x06673b0101343b00,
+        0xe7db4ea6533afa9,
+    ]);
+
+    /// t for 2^s * t = MODULUS - 1, and t coprime to 2.
+    /// t = 409655274805673363120685472720202858103411121670017820368325103335302739775
+    ///   = (modulus-1)/2^5
+    const T: BigInteger = BigInteger([
+        0x8ba7e835a943b73f,
+        0x07fc7c3803a0c823,
+        0x906673b0101343b0,
+        0xe7db4ea6533afa,
+    ]);
+
+    /// (t - 1) / 2
+    ///  = 204827637402836681560342736360101429051705560835008910184162551667651369887
+    const T_MINUS_ONE_DIV_TWO: BigInteger = BigInteger([
+        0xc5d3f41ad4a1db9f,
+        0x03fe3e1c01d06411,
+        0x483339d80809a1d8,
+        0x73eda753299d7d,
+    ]);
+}

ed_on_bls12_381_bandersnatch/src/fields/mod.rs

@@ -0,0 +1,8 @@
+pub mod fq;
+pub mod fr;
+
+pub use fq::*;
+pub use fr::*;
+
+#[cfg(all(feature = "ed_on_bls12_381_bandersnatch", test))]
+mod tests;

ed_on_bls12_381_bandersnatch/src/fields/tests.rs

@@ -0,0 +1,423 @@
+use crate::{Fq, Fr};
+use ark_algebra_test_templates::fields::*;
+use ark_ff::{
+    biginteger::BigInteger256 as BigInteger,
+    bytes::{FromBytes, ToBytes},
+    fields::{Field, LegendreSymbol::*, SquareRootField},
+    One, Zero,
+};
+use ark_std::{rand::Rng, str::FromStr, test_rng};
+
+#[test]
+fn test_fr() {
+    let mut rng = test_rng();
+    let a: Fr = rng.gen();
+    let b: Fr = rng.gen();
+    field_test(a, b);
+    primefield_test::<Fr>();
+}
+
+#[test]
+fn test_fq() {
+    let mut rng = test_rng();
+    let a: Fq = rng.gen();
+    let b: Fq = rng.gen();
+    field_test(a, b);
+    primefield_test::<Fq>();
+}
+
+#[test]
+fn test_fq_add() {
+    let f1 = Fq::from_str(
+        "18386742314266644595564329008376577163854043021652781768352795308532764650733",
+    )
+    .unwrap();
+    let f2 = Fq::from_str(
+        "39786307610986038981023499868190793548353538256264351797285876981647142458383",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "5737174750126493097140088368381404874517028777389495743035013590241325924603",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f2.is_zero());
+    assert!(!f3.is_zero());
+    assert_eq!(f1 + &f2, f3);
+}
+
+#[test]
+fn test_fq_add_one() {
+    let f1 = Fq::from_str(
+        "4946875394261337176810256604189376311946643975348516311606738923340201185904",
+    )
+    .unwrap();
+    let f2 = Fq::from_str(
+        "4946875394261337176810256604189376311946643975348516311606738923340201185905",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f2.is_zero());
+    assert_eq!(f1 + &Fq::one(), f2);
+}
+
+#[test]
+fn test_fq_mul() {
+    let f1 = Fq::from_str(
+        "24703123148064348394273033316595937198355721297494556079070134653139656190956",
+    )
+    .unwrap();
+    let f2 = Fq::from_str(
+        "38196797080882758914424853878212529985425118523754343117256179679117054302131",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "38057113854472161555556064369220825628027487067886761874351491955834635348140",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f2.is_zero());
+    assert!(!f3.is_zero());
+    assert_eq!(f1 * &f2, f3);
+}
+
+#[test]
+fn test_fq_triple_mul() {
+    let f1 = Fq::from_str(
+        "23834398828139479510988224171342199299644042568628082836691700490363123893905",
+    )
+    .unwrap();
+    let f2 = Fq::from_str(
+        "48343809612844640454129919255697536258606705076971130519928764925719046689317",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "22704845471524346880579660022678666462201713488283356385810726260959369106033",
+    )
+    .unwrap();
+    let f4 = Fq::from_str(
+        "18897508522635316277030308074760673440128491438505204942623624791502972539393",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f2.is_zero());
+    assert!(!f3.is_zero());
+    assert_eq!(f1 * &f2 * &f3, f4);
+}
+
+#[test]
+fn test_fq_div() {
+    let f1 = Fq::from_str(
+        "31892744363926593013886463524057935370302352424137349660481695792871889573091",
+    )
+    .unwrap();
+    let f2 = Fq::from_str(
+        "47695868328933459965610498875668250916462767196500056002116961816137113470902",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "29049672724678710659792141917402891276693777283079976086581207190825261000580",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f2.is_zero());
+    assert!(!f3.is_zero());
+    assert_eq!(f1 / &f2, f3);
+}
+
+#[test]
+fn test_fq_sub() {
+    let f1 = Fq::from_str(
+        "18695869713129401390241150743745601908470616448391638969502807001833388904079",
+    )
+    .unwrap();
+    let f2 = Fq::from_str(
+        "10105476028534616828778879109836101003805485072436929139123765141153277007373",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "8590393684594784561462271633909500904665131375954709830379041860680111896706",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f2.is_zero());
+    assert!(!f3.is_zero());
+    assert_eq!(f1 - &f2, f3);
+}
+
+#[test]
+fn test_fq_double_in_place() {
+    let mut f1 = Fq::from_str(
+        "29729289787452206300641229002276778748586801323231253291984198106063944136114",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "7022704399778222121834717496367591659483050145934868761364737512189307087715",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f3.is_zero());
+    f1.double_in_place();
+    assert_eq!(f1, f3);
+}
+
+#[test]
+fn test_fq_double_in_place_thrice() {
+    let mut f1 = Fq::from_str(
+        "32768907806651393940832831055386272949401004221411141755415956893066040832473",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "52407761752706389608871686410346320244445823769178582752913020344774001921732",
+    )
+    .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f3.is_zero());
+    f1.double_in_place();
+    f1.double_in_place();
+    f1.double_in_place();
+    assert_eq!(f1, f3);
+}
+
+#[test]
+fn test_fq_generate_random_ed_on_bls12_381_point() {
+    let d = Fq::from_str(
+        "19257038036680949359750312669786877991949435402254120286184196891950884077233",
+    )
+    .unwrap();
+    let y = Fq::from_str(
+        "20269054604167148422407276086932743904275456233139568486008667107872965128512",
+    )
+    .unwrap();
+    let x2 = Fq::from_str(
+        "35041048504708632193693740149219726446678304552734087046982753200179718192840",
+    )
+    .unwrap();
+
+    let computed_y2 = y.square();
+    let y2 = Fq::from_str(
+        "22730681238307918419349440108285755984465605552827817317611903495170775437833",
+    )
+    .unwrap();
+    assert_eq!(y2, computed_y2);
+
+    let computed_dy2 = d * &computed_y2;
+    let dy2 = Fq::from_str(
+        "24720347560552809545835752815204882739669031262711919770503096707526812943411",
+    )
+    .unwrap();
+    assert_eq!(dy2, computed_dy2);
+
+    let computed_divisor = computed_dy2 + &Fq::one();
+    let divisor = Fq::from_str(
+        "24720347560552809545835752815204882739669031262711919770503096707526812943412",
+    )
+    .unwrap();
+    assert_eq!(divisor, computed_divisor);
+
+    let computed_x2 = (computed_y2 - &Fq::one()) / &computed_divisor;
+    assert_eq!(x2, computed_x2);
+
+    let x = Fq::from_str(
+        "15337652609730546173818014678723269532482775720866471265774032070871608223361",
+    )
+    .unwrap();
+    let computed_x = computed_x2.sqrt().unwrap();
+    assert_eq!(computed_x.square(), x2);
+    assert_eq!(x, computed_x);
+
+    fn add<'a>(curr: (Fq, Fq), other: &'a (Fq, Fq)) -> (Fq, Fq) {
+        let y1y2 = curr.1 * &other.1;
+        let x1x2 = curr.0 * &other.0;
+        let d = Fq::from_str(
+            "19257038036680949359750312669786877991949435402254120286184196891950884077233",
+        )
+        .unwrap();
+        let dx1x2y1y2 = d * &y1y2 * &x1x2;
+
+        let d1 = Fq::one() + &dx1x2y1y2;
+        let d2 = Fq::one() - &dx1x2y1y2;
+
+        let x1y2 = curr.0 * &other.1;
+        let y1x2 = curr.1 * &other.0;
+
+        let x = (x1y2 + &y1x2) / &d1;
+        let y = (y1y2 + &x1x2) / &d2;
+
+        (x, y)
+    }
+
+    let result = add((x, y), &(x, y));
+    let result = add(result, &result);
+    let result = add(result, &result);
+
+    let point_x = Fq::from_str(
+        "47259664076168047050113154262636619161204477920503059672059915868534495873964",
+    )
+    .unwrap();
+    let point_y = Fq::from_str(
+        "19016409245280491801573912449420132838852726543024859389273314249842195919690",
+    )
+    .unwrap();
+    assert_eq!((point_x, point_y), result);
+}
+
+#[test]
+fn test_fq_square_in_place() {
+    let mut f1 = Fq::from_str(
+        "34864651240005695523200639428464570946052769938774601449735727714436878540682",
+    )
+    .unwrap();
+    let f3 =
+        Fq::from_str("213133100629336594719108316042277780359104840987226496279264105585804377948")
+            .unwrap();
+    assert!(!f1.is_zero());
+    assert!(!f3.is_zero());
+    f1.square_in_place();
+    assert_eq!(f1, f3);
+}
+
+#[test]
+fn test_fq_sqrt() {
+    let f1 = Fq::from_str(
+        "10875927553327821418567659853801220899541454800710193788767706167237535308235",
+    )
+    .unwrap();
+    let f3 = Fq::from_str(
+        "10816221372957505053219354782681292880545918527618367765651802809826238616708",
+    )
+    .unwrap();
+    assert_eq!(f1.sqrt().unwrap(), f3);
+}
+
+#[test]
+fn test_fq_from_str() {
+    let f1_from_repr = Fq::from(BigInteger([
+        0xab8a2535947d1a77,
+        0x9ba74cbfda0bbcda,
+        0xe928b59724d60baf,
+        0x1cccaaeb9bb1680a,
+    ]));
+    let f1 = Fq::from_str(
+        "13026376210409056429264774981357153555336288129100724591327877625017068755575",
+    )
+    .unwrap();
+    let f2_from_repr = Fq::from(BigInteger([
+        0x97e9103775d2f35c,
+        0xbe6756b6c587544b,
+        0x6ee38c3afd88ef4b,
+        0x2bacd150f540c677,
+    ]));
+    let f2 = Fq::from_str(
+        "19754794831832707859764530223239420866832328728734160755396495950822165902172",
+    )
+    .unwrap();
+    assert_eq!(f1_from_repr, f1);
+    assert_eq!(f2_from_repr, f2);
+}
+
+#[test]
+fn test_fq_legendre() {
+    assert_eq!(QuadraticResidue, Fq::one().legendre());
+    assert_eq!(Zero, Fq::zero().legendre());
+
+    let e = BigInteger([
+        0x0dbc5349cd5664da,
+        0x8ac5b6296e3ae29d,
+        0x127cb819feceaa3b,
+        0x3a6b21fb03867191,
+    ]);
+    assert_eq!(QuadraticResidue, Fq::from(e).legendre());
+    let e = BigInteger([
+        0x96341aefd047c045,
+        0x9b5f4254500a4d65,
+        0x1ee08223b68ac240,
+        0x31d9cd545c0ec7c6,
+    ]);
+    assert_eq!(QuadraticNonResidue, Fq::from(e).legendre());
+}
+
+#[test]
+fn test_fq_bytes() {
+    let f1_from_repr = Fq::from(BigInteger([
+        0xab8a2535947d1a77,
+        0x9ba74cbfda0bbcda,
+        0xe928b59724d60baf,
+        0x1cccaaeb9bb1680a,
+    ]));
+
+    let mut f1_bytes = [0u8; 32];
+    f1_from_repr.write(f1_bytes.as_mut()).unwrap();
+
+    let f1 = Fq::read(f1_bytes.as_ref()).unwrap();
+    assert_eq!(f1_from_repr, f1);
+}
+
+#[test]
+fn test_fr_add() {
+    let f1 = Fr::from(BigInteger([
+        0xc81265fb4130fe0c,
+        0xb308836c14e22279,
+        0x699e887f96bff372,
+        0x84ecc7e76c11ad,
+    ]));
+    let f2 = Fr::from(BigInteger([
+        0x71875719b422efb8,
+        0x0043658e68a93612,
+        0x9fa756be2011e833,
+        0xaa2b2cb08dac497,
+    ]));
+    let f3 = Fr::from(BigInteger([
+        0x3999bd14f553edc4,
+        0xb34be8fa7d8b588c,
+        0x0945df3db6d1dba5,
+        0xb279f92f046d645,
+    ]));
+    assert_eq!(f1 + &f2, f3);
+}
+
+#[test]
+fn test_fr_mul() {
+    let f1 = Fr::from(BigInteger([
+        0xc81265fb4130fe0c,
+        0xb308836c14e22279,
+        0x699e887f96bff372,
+        0x84ecc7e76c11ad,
+    ]));
+    let f2 = Fr::from(BigInteger([
+        0x71875719b422efb8,
+        0x43658e68a93612,
+        0x9fa756be2011e833,
+        0xaa2b2cb08dac497,
+    ]));
+    let f3 = Fr::from(BigInteger([
+        0xbe3e50c164fe3381,
+        0x5ac45bc180974585,
+        0x1c234ad6dcdc70c9,
+        0x15a75fba99bc8ad,
+    ]));
+    assert_eq!(f1 * &f2, f3);
+}
+
+#[test]
+fn test_fr_bytes() {
+    let f1_from_repr = Fr::from(BigInteger([
+        0xc81265fb4130fe0c,
+        0xb308836c14e22279,
+        0x699e887f96bff372,
+        0x84ecc7e76c11ad,
+    ]));
+
+    let mut f1_bytes = [0u8; 32];
+    f1_from_repr.write(f1_bytes.as_mut()).unwrap();
+
+    let f1 = Fr::read(f1_bytes.as_ref()).unwrap();
+    assert_eq!(f1_from_repr, f1);
+}
+
+#[test]
+fn test_fr_from_str() {
+    let f100_from_repr = Fr::from(BigInteger([0x64, 0, 0, 0]));
+    let f100 = Fr::from_str("100").unwrap();
+    assert_eq!(f100_from_repr, f100);
+}

ed_on_bls12_381_bandersnatch/src/lib.rs

@@ -0,0 +1,37 @@
+#![cfg_attr(not(feature = "std"), no_std)]
+#![deny(
+    warnings,
+    unused,
+    future_incompatible,
+    nonstandard_style,
+    rust_2018_idioms
+)]
+#![forbid(unsafe_code)]
+
+//! This library implements the Bendersnatch curve, a twisted Edwards curve
+//! whose base field is the scalar field of the curve BLS12-381. This allows
+//! defining cryptographic primitives that use elliptic curves over the scalar
+//! field of the latter curve. This curve was generated by Simon Masson from Anoma,
+//! and Antonio Sanso from Ethereum Foundation, and is also known as [bandersnatch](https://ethresear.ch/t/introducing-bandersnatch-a-fast-elliptic-curve-built-over-the-bls12-381-scalar-field/9957).
+//!
+//! See [here](https://github.com/asanso/Bandersnatch/blob/main/README.md) for the specification of the curve.
+//! There was also a Python implementation [here](https://github.com/asanso/Bandersnatch/).
+//!
+//! Curve information:
+//! * Base field: q =
+//!   52435875175126190479447740508185965837690552500527637822603658699938581184513
+//! * Scalar field: r =
+//!   13108968793781547619861935127046491459309155893440570251786403306729687672801
+//! * Valuation(q - 1, 2) = 32
+//! * Valuation(r - 1, 2) = 5
+//! * Curve equation: ax^2 + y^2 =1 + dx^2y^2, where
+//!    * a = -5
+//!    * d = 45022363124591815672509500913686876175488063829319466900776701791074614335719
+
+#[cfg(feature = "r1cs")]
+pub mod constraints;
+mod curves;
+mod fields;
+
+pub use curves::*;
+pub use fields::*;