Add Pallas and Vesta curves (#21)

Co-authored-by: Ying Tong Lai <yingtong@electriccoin.co>
Co-authored-by: Daira Hopwood <daira@jacaranda.org>
Co-authored-by: Pratyush Mishra <pratyushmishra@berkeley.edu>
Co-authored-by: therealyingtong <yingtong@z.cash>

Cargo.toml

@@ -27,6 +27,9 @@ members = [
     "mnt4_753",
     "mnt6_753",
     "ed_on_mnt4_298",
+
+    "pallas",
+    "vesta",
 ]
 
 [profile.release]

README.md

@@ -29,3 +29,7 @@ This repository contains implementations of some popular elliptic curves. The cu
 * [`ark-mnt4-753`](mnt4_753): Implements the MNT4-753 pairing-friendly curve. This curve forms a pairing-friendly cycle with MNT6-753
 * [`ark-mnt6-753`](mnt6_753): Implements the MNT6-753 pairing-friendly curve. This curve forms a pairing-friendly cycle with MNT4-753
 * [`ark-ed-on-mnt4-753`](ed_on_mnt4_753): Implements a Twisted Edwards curve atop the scalar field of MNT4-753
+
+### [Pasta](https://electriccoin.co/blog/the-pasta-curves-for-halo-2-and-beyond/) cycle of curves
+* [`ark-pallas`](pallas): Implements Pallas, a prime-order curve that forms an amicable pair with Vesta
+* [`ark-vesta`](vesta): Implements Vesta, a prime-order curve that forms an amicable pair with Pallas

curve-tests/src/fields.rs

@@ -417,20 +417,14 @@ pub fn field_serialization_test(buf_size: usize) {
         #[derive(Default, Clone, Copy, Debug)]
         struct DummyFlags;
         impl Flags for DummyFlags {
+            const BIT_SIZE: usize = 200;
+
             fn u8_bitmask(&self) -> u8 {
                 0
             }
 
-            fn from_u8(_value: u8) -> Self {
-                DummyFlags
-            }
-
-            fn from_u8_remove_flags(_value: &mut u8) -> Self {
-                DummyFlags
-            }
-
-            fn len() -> usize {
-                200
+            fn from_u8(_value: u8) -> Option<Self> {
+                Some(DummyFlags)
             }
         }
 

pallas/Cargo.toml

@@ -0,0 +1,36 @@
+[package]
+name = "ark-pallas"
+version = "0.1.0"
+authors = [ "Ying Tong Lai", "Daira Hopwood", "O(1) Labs", "arkworks contributors" ]
+description = "The Pallas prime-order elliptic curve"
+homepage = "https://arkworks.rs"
+repository = "https://github.com/arkworks-rs/curves"
+documentation = "https://docs.rs/ark-pallas/"
+keywords = ["cryptography", "finite fields", "elliptic curves" ]
+categories = ["cryptography"]
+include = ["Cargo.toml", "src"]
+license = "MIT/Apache-2.0"
+edition = "2018"
+
+[dependencies]
+ark-ff = { git = "https://github.com/arkworks-rs/algebra", default-features = false }
+ark-ec = { git = "https://github.com/arkworks-rs/algebra", default-features = false }
+ark-r1cs-std = {  git = "https://github.com/arkworks-rs/r1cs-std", default-features = false, optional = true }
+ark-std = { git = "https://github.com/arkworks-rs/utils", default-features = false }
+
+[dev-dependencies]
+ark-relations = { git = "https://github.com/arkworks-rs/snark", default-features = false }
+ark-serialize = { git = "https://github.com/arkworks-rs/algebra", default-features = false }
+ark-curve-tests = { path = "../curve-tests", default-features = false }
+ark-curve-constraint-tests = { path = "../curve-constraint-tests", default-features = false }
+rand = { version = "0.7", default-features = false }
+rand_xorshift = "0.2"
+
+[features]
+default = [ "curve" ]
+std = [ "ark-std/std", "ark-ff/std", "ark-ec/std" ]
+
+curve = [ "scalar_field", "base_field" ]
+scalar_field = []
+base_field = []
+r1cs = [ "base_field", "ark-r1cs-std" ]

pallas/src/constraints/curves.rs

@@ -0,0 +1,12 @@
+use crate::*;
+use ark_r1cs_std::groups::curves::short_weierstrass::ProjectiveVar;
+
+use crate::constraints::FBaseVar;
+
+/// A group element in the Pallas prime-order group.
+pub type GVar = ProjectiveVar<PallasParameters, FBaseVar>;
+
+#[test]
+fn test() {
+    ark_curve_constraint_tests::curves::sw_test::<PallasParameters, GVar>().unwrap();
+}

pallas/src/constraints/fields.rs

@@ -0,0 +1,10 @@
+use crate::fq::Fq;
+use ark_r1cs_std::fields::fp::FpVar;
+
+/// A variable that is the R1CS equivalent of `crate::Fq`.
+pub type FBaseVar = FpVar<Fq>;
+
+#[test]
+fn test() {
+    ark_curve_constraint_tests::fields::field_test::<_, _, FBaseVar>().unwrap();
+}

pallas/src/constraints/mod.rs

@@ -0,0 +1,107 @@
+//! This module implements the R1CS equivalent of `ark_pallas`.
+//!
+//! 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 `FBaseVar`:
+//!
+//! ```
+//! # fn main() -> Result<(), ark_relations::r1cs::SynthesisError> {
+//! use ark_std::UniformRand;
+//! use ark_relations::r1cs::*;
+//! use ark_r1cs_std::prelude::*;
+//! use ark_pallas::{*, 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 = FBaseVar::new_witness(ark_relations::ns!(cs, "generate_a"), || Ok(a_native))?;
+//! let b = FBaseVar::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 = FBaseVar::new_constant(ark_relations::ns!(cs, "a_as_constant"), a_native)?;
+//! let b_const = FBaseVar::new_constant(ark_relations::ns!(cs, "b_as_constant"), b_native)?;
+//!
+//! let one = FBaseVar::one();
+//! let zero = FBaseVar::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 `GVar`:
+//!
+//! ```
+//! # fn main() -> Result<(), ark_relations::r1cs::SynthesisError> {
+//! # use ark_std::UniformRand;
+//! # use ark_relations::r1cs::*;
+//! # use ark_r1cs_std::prelude::*;
+//! # use ark_pallas::{*, constraints::*};
+//!
+//! # let cs = ConstraintSystem::<Fq>::new_ref();
+//! # let mut rng = ark_std::test_rng();
+//!
+//! // Generate some random `Projective` elements.
+//! let a_native = Projective::rand(&mut rng);
+//! let b_native = Projective::rand(&mut rng);
+//!
+//! // Allocate `a_native` and `b_native` as witness variables in `cs`.
+//! let a = GVar::new_witness(ark_relations::ns!(cs, "a"), || Ok(a_native))?;
+//! let b = GVar::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 = GVar::new_constant(ark_relations::ns!(cs, "a_as_constant"), a_native)?;
+//! let b_const = GVar::new_constant(ark_relations::ns!(cs, "b_as_constant"), b_native)?;
+//!
+//! // This returns the identity.
+//! let zero = GVar::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::*;

pallas/src/curves/mod.rs

@@ -0,0 +1,49 @@
+use crate::{fq::Fq, fr::Fr};
+use ark_ec::{
+    models::{ModelParameters, SWModelParameters},
+    short_weierstrass_jacobian::{GroupAffine, GroupProjective},
+};
+use ark_ff::{field_new, Zero};
+
+#[cfg(test)]
+mod tests;
+
+#[derive(Copy, Clone, Default, PartialEq, Eq)]
+pub struct PallasParameters;
+
+impl ModelParameters for PallasParameters {
+    type BaseField = Fq;
+    type ScalarField = Fr;
+}
+
+pub type Affine = GroupAffine<PallasParameters>;
+pub type Projective = GroupProjective<PallasParameters>;
+
+impl SWModelParameters for PallasParameters {
+    /// COEFF_A = 0
+    const COEFF_A: Fq = field_new!(Fq, "0");
+
+    /// COEFF_B = 5
+    const COEFF_B: Fq = field_new!(Fq, "5");
+
+    /// COFACTOR = 1
+    const COFACTOR: &'static [u64] = &[0x1];
+
+    /// COFACTOR_INV = 1
+    const COFACTOR_INV: Fr = field_new!(Fr, "1");
+
+    /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
+    const AFFINE_GENERATOR_COEFFS: (Self::BaseField, Self::BaseField) =
+        (G_GENERATOR_X, G_GENERATOR_Y);
+
+    #[inline(always)]
+    fn mul_by_a(_: &Self::BaseField) -> Self::BaseField {
+        Self::BaseField::zero()
+    }
+}
+
+/// G_GENERATOR_X = -1
+pub const G_GENERATOR_X: Fq = field_new!(Fq, "-1");
+
+/// G_GENERATOR_Y = 2
+pub const G_GENERATOR_Y: Fq = field_new!(Fq, "2");

pallas/src/curves/tests.rs

@@ -0,0 +1,39 @@
+#![allow(unused_imports)]
+use ark_ff::{
+    fields::{Field, FpParameters, PrimeField, SquareRootField},
+    One, Zero,
+};
+use ark_serialize::CanonicalSerialize;
+use ark_std::test_rng;
+
+use ark_ec::{models::SWModelParameters, AffineCurve, PairingEngine, ProjectiveCurve};
+use core::ops::{AddAssign, MulAssign};
+use rand::Rng;
+
+use crate::{Affine, PallasParameters, Projective};
+
+use ark_curve_tests::{
+    curves::{curve_tests, sw_tests},
+    groups::group_test,
+};
+
+#[test]
+fn test_projective_curve() {
+    curve_tests::<Projective>();
+    sw_tests::<PallasParameters>();
+}
+
+#[test]
+fn test_projective_group() {
+    let mut rng = test_rng();
+    let a: Projective = rng.gen();
+    let b: Projective = rng.gen();
+    group_test(a, b);
+}
+
+#[test]
+fn test_generator() {
+    let generator = Affine::prime_subgroup_generator();
+    assert!(generator.is_on_curve());
+    assert!(generator.is_in_correct_subgroup_assuming_on_curve());
+}

pallas/src/fields/fq.rs

@@ -0,0 +1,90 @@
+use ark_ff::{
+    biginteger::BigInteger256 as BigInteger,
+    fields::{FftParameters, Fp256, Fp256Parameters},
+};
+
+pub type Fq = Fp256<FqParameters>;
+
+pub struct FqParameters;
+
+impl Fp256Parameters for FqParameters {}
+impl FftParameters for FqParameters {
+    type BigInt = BigInteger;
+
+    const TWO_ADICITY: u32 = 32;
+
+    // TWO_ADIC_ROOT_OF_UNITY = GENERATOR^T
+    // Encoded in Montgomery form, so the value here is (5^T)R mod p.
+    const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([
+        0xa28db849bad6dbf0,
+        0x9083cd03d3b539df,
+        0xfba6b9ca9dc8448e,
+        0x3ec928747b89c6da,
+    ]);
+}
+
+impl ark_ff::fields::FpParameters for FqParameters {
+    // 28948022309329048855892746252171976963363056481941560715954676764349967630337
+    const MODULUS: BigInteger = BigInteger([
+        0x992d30ed00000001,
+        0x224698fc094cf91b,
+        0x0000000000000000,
+        0x4000000000000000,
+    ]);
+
+    // R = 2^256 mod p
+    const R: BigInteger = BigInteger([
+        0x34786d38fffffffd,
+        0x992c350be41914ad,
+        0xffffffffffffffff,
+        0x3fffffffffffffff,
+    ]);
+
+    // R2 = (2^256)^2 mod p
+    const R2: BigInteger = BigInteger([
+        0x8c78ecb30000000f,
+        0xd7d30dbd8b0de0e7,
+        0x7797a99bc3c95d18,
+        0x096d41af7b9cb714,
+    ]);
+
+    const MODULUS_MINUS_ONE_DIV_TWO: BigInteger = BigInteger([
+        0xcc96987680000000,
+        0x11234c7e04a67c8d,
+        0x0000000000000000,
+        0x2000000000000000,
+    ]);
+
+    // T and T_MINUS_ONE_DIV_TWO, where MODULUS - 1 = 2^S * T
+    const T: BigInteger = BigInteger([
+        0x094cf91b992d30ed,
+        0x00000000224698fc,
+        0x0000000000000000,
+        0x0000000040000000,
+    ]);
+
+    const T_MINUS_ONE_DIV_TWO: BigInteger = BigInteger([
+        0x04a67c8dcc969876,
+        0x0000000011234c7e,
+        0x0000000000000000,
+        0x0000000020000000,
+    ]);
+
+    // GENERATOR = 5
+    // Encoded in Montgomery form, so the value here is 5R mod p.
+    const GENERATOR: BigInteger = BigInteger([
+        0xa1a55e68ffffffed,
+        0x74c2a54b4f4982f3,
+        0xfffffffffffffffd,
+        0x3fffffffffffffff,
+    ]);
+
+    const MODULUS_BITS: u32 = 255;
+
+    const CAPACITY: u32 = Self::MODULUS_BITS - 1;
+
+    const REPR_SHAVE_BITS: u32 = 1;
+
+    // INV = -p^{-1} (mod 2^64)
+    const INV: u64 = 11037532056220336127;
+}

pallas/src/fields/fr.rs

@@ -0,0 +1,91 @@
+use ark_ff::{
+    biginteger::BigInteger256 as BigInteger,
+    fields::{FftParameters, Fp256, Fp256Parameters, FpParameters},
+};
+
+pub struct FrParameters;
+
+pub type Fr = Fp256<FrParameters>;
+
+impl Fp256Parameters for FrParameters {}
+impl FftParameters for FrParameters {
+    type BigInt = BigInteger;
+
+    const TWO_ADICITY: u32 = 32;
+
+    // TWO_ADIC_ROOT_OF_UNITY = GENERATOR^T
+    // Encoded in Montgomery form, so the value here is (5^T)R mod q.
+    const TWO_ADIC_ROOT_OF_UNITY: BigInteger = BigInteger([
+        0x218077428c9942de,
+        0xcc49578921b60494,
+        0xac2e5d27b2efbee2,
+        0x0b79fa897f2db056,
+    ]);
+}
+
+impl FpParameters for FrParameters {
+    // 28948022309329048855892746252171976963363056481941647379679742748393362948097
+    const MODULUS: BigInteger = BigInteger([
+        0x8c46eb2100000001,
+        0x224698fc0994a8dd,
+        0x0000000000000000,
+        0x4000000000000000,
+    ]);
+
+    // R = 2^256 mod q
+    const R: BigInteger = BigInteger([
+        0x5b2b3e9cfffffffd,
+        0x992c350be3420567,
+        0xffffffffffffffff,
+        0x3fffffffffffffff,
+    ]);
+
+    // R2 = (2^256)^2 mod q
+    const R2: BigInteger = BigInteger([
+        0xfc9678ff0000000f,
+        0x67bb433d891a16e3,
+        0x7fae231004ccf590,
+        0x096d41af7ccfdaa9,
+    ]);
+
+    const MODULUS_MINUS_ONE_DIV_TWO: BigInteger = BigInteger([
+        0xc623759080000000,
+        0x11234c7e04ca546e,
+        0x0000000000000000,
+        0x2000000000000000,
+    ]);
+
+    // T and T_MINUS_ONE_DIV_TWO, where MODULUS - 1 = 2^S * T
+
+    const T: BigInteger = BigInteger([
+        0x0994a8dd8c46eb21,
+        0x00000000224698fc,
+        0x0000000000000000,
+        0x0000000040000000,
+    ]);
+
+    const T_MINUS_ONE_DIV_TWO: BigInteger = BigInteger([
+        0x04ca546ec6237590,
+        0x0000000011234c7e,
+        0x0000000000000000,
+        0x0000000020000000,
+    ]);
+
+    // GENERATOR = 5
+    // Encoded in Montgomery form, so the value here is 5R mod q.
+    const GENERATOR: BigInteger = BigInteger([
+        0x96bc8c8cffffffed,
+        0x74c2a54b49f7778e,
+        0xfffffffffffffffd,
+        0x3fffffffffffffff,
+    ]);
+
+    const MODULUS_BITS: u32 = 255;
+
+    const CAPACITY: u32 = Self::MODULUS_BITS - 1;
+
+    const REPR_SHAVE_BITS: u32 = 1;
+
+    // INV = -q^{-1} (mod 2^64)
+    const INV: u64 = 10108024940646105087;
+}

pallas/src/fields/mod.rs

@@ -0,0 +1,12 @@
+#[cfg(feature = "base_field")]
+pub mod fq;
+#[cfg(feature = "base_field")]
+pub use self::fq::*;
+
+#[cfg(feature = "scalar_field")]
+pub mod fr;
+#[cfg(feature = "scalar_field")]
+pub use self::fr::*;
+
+#[cfg(all(feature = "curve", test))]
+mod tests;

pallas/src/fields/tests.rs

@@ -0,0 +1,26 @@
+use ark_std::test_rng;
+use rand::Rng;
+
+use crate::*;
+
+use ark_curve_tests::fields::*;
+
+#[test]
+fn test_fr() {
+    let mut rng = test_rng();
+    let a: Fr = rng.gen();
+    let b: Fr = rng.gen();
+    field_test(a, b);
+    sqrt_field_test(a);
+    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);
+    sqrt_field_test(a);
+    primefield_test::<Fq>();
+}

pallas/src/lib.rs

@@ -0,0 +1,36 @@
+#![cfg_attr(not(feature = "std"), no_std)]
+#![deny(
+    warnings,
+    unused,
+    future_incompatible,
+    nonstandard_style,
+    rust_2018_idioms
+)]
+#![forbid(unsafe_code)]
+
+//! This library implements the prime-order curve Pallas, generated by
+//! [Daira Hopwood](https://github.com/zcash/pasta). The main feature of this
+//! curve is that it forms a cycle with Vesta, i.e. its scalar field and base
+//! field respectively are the base field and scalar field of Vesta.
+//!
+//!
+//! Curve information:
+//! * Base field: q =
+//!   28948022309329048855892746252171976963363056481941560715954676764349967630337
+//! * Scalar field: r =
+//!   28948022309329048855892746252171976963363056481941647379679742748393362948097
+//! * Curve equation: y^2 = x^3 + 5
+//! * Valuation(q - 1, 2) = 32
+//! * Valuation(r - 1, 2) = 32
+
+#[cfg(feature = "r1cs")]
+pub mod constraints;
+#[cfg(feature = "curve")]
+mod curves;
+#[cfg(any(feature = "scalar_field", feature = "base_field"))]
+mod fields;
+
+#[cfg(feature = "curve")]
+pub use curves::*;
+#[cfg(any(feature = "scalar_field", feature = "base_field"))]
+pub use fields::*;

vesta/Cargo.toml

@@ -0,0 +1,33 @@
+[package]
+name = "ark-vesta"
+version = "0.1.0"
+authors = [ "Ying Tong Lai", "Daira Hopwood", "O(1) Labs", "arkworks contributors" ]
+description = "The Vesta prime-order elliptic curve"
+homepage = "https://arkworks.rs"
+repository = "https://github.com/arkworks-rs/curves"
+documentation = "https://docs.rs/ark-vesta/"
+keywords = ["cryptography", "finite fields", "elliptic curves" ]
+categories = ["cryptography"]
+include = ["Cargo.toml", "src"]
+license = "MIT/Apache-2.0"
+edition = "2018"
+
+[dependencies]
+ark-ff = { git = "https://github.com/arkworks-rs/algebra", default-features = false }
+ark-ec = { git = "https://github.com/arkworks-rs/algebra", default-features = false }
+ark-r1cs-std = {  git = "https://github.com/arkworks-rs/r1cs-std", default-features = false, optional = true }
+ark-std = { git = "https://github.com/arkworks-rs/utils", default-features = false }
+ark-pallas = { path = "../pallas", default-features = false, features = [ "scalar_field", "base_field" ] }
+
+[dev-dependencies]
+ark-relations = { git = "https://github.com/arkworks-rs/snark", default-features = false }
+ark-serialize = { git = "https://github.com/arkworks-rs/algebra", default-features = false }
+ark-curve-tests = { path = "../curve-tests", default-features = false }
+ark-curve-constraint-tests = { path = "../curve-constraint-tests", default-features = false }
+rand = { version = "0.7", default-features = false }
+rand_xorshift = "0.2"
+
+[features]
+default = []
+std = [ "ark-std/std", "ark-ff/std", "ark-ec/std" ]
+r1cs = [ "ark-r1cs-std" ]

vesta/src/constraints/curves.rs

@@ -0,0 +1,12 @@
+use crate::*;
+use ark_r1cs_std::groups::curves::short_weierstrass::ProjectiveVar;
+
+use crate::constraints::FBaseVar;
+
+/// A group element in the Vesta prime-order group.
+pub type GVar = ProjectiveVar<VestaParameters, FBaseVar>;
+
+#[test]
+fn test() {
+    ark_curve_constraint_tests::curves::sw_test::<VestaParameters, GVar>().unwrap();
+}

vesta/src/constraints/fields.rs

@@ -0,0 +1,10 @@
+use crate::fq::Fq;
+use ark_r1cs_std::fields::fp::FpVar;
+
+/// A variable that is the R1CS equivalent of `crate::Fq`.
+pub type FBaseVar = FpVar<Fq>;
+
+#[test]
+fn test() {
+    ark_curve_constraint_tests::fields::field_test::<_, _, FBaseVar>().unwrap();
+}

vesta/src/constraints/mod.rs

@@ -0,0 +1,107 @@
+//! This module implements the R1CS equivalent of `ark_vesta`.
+//!
+//! 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 `FBaseVar`:
+//!
+//! ```
+//! # fn main() -> Result<(), ark_relations::r1cs::SynthesisError> {
+//! use ark_std::UniformRand;
+//! use ark_relations::r1cs::*;
+//! use ark_r1cs_std::prelude::*;
+//! use ark_vesta::{*, 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 = FBaseVar::new_witness(ark_relations::ns!(cs, "generate_a"), || Ok(a_native))?;
+//! let b = FBaseVar::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 = FBaseVar::new_constant(ark_relations::ns!(cs, "a_as_constant"), a_native)?;
+//! let b_const = FBaseVar::new_constant(ark_relations::ns!(cs, "b_as_constant"), b_native)?;
+//!
+//! let one = FBaseVar::one();
+//! let zero = FBaseVar::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 `GVar`:
+//!
+//! ```
+//! # fn main() -> Result<(), ark_relations::r1cs::SynthesisError> {
+//! # use ark_std::UniformRand;
+//! # use ark_relations::r1cs::*;
+//! # use ark_r1cs_std::prelude::*;
+//! # use ark_vesta::{*, constraints::*};
+//!
+//! # let cs = ConstraintSystem::<Fq>::new_ref();
+//! # let mut rng = ark_std::test_rng();
+//!
+//! // Generate some random `Projective` elements.
+//! let a_native = Projective::rand(&mut rng);
+//! let b_native = Projective::rand(&mut rng);
+//!
+//! // Allocate `a_native` and `b_native` as witness variables in `cs`.
+//! let a = GVar::new_witness(ark_relations::ns!(cs, "a"), || Ok(a_native))?;
+//! let b = GVar::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 = GVar::new_constant(ark_relations::ns!(cs, "a_as_constant"), a_native)?;
+//! let b_const = GVar::new_constant(ark_relations::ns!(cs, "b_as_constant"), b_native)?;
+//!
+//! // This returns the identity.
+//! let zero = GVar::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::*;

vesta/src/curves/mod.rs

@@ -0,0 +1,51 @@
+use crate::{fq::Fq, fr::Fr};
+use ark_ec::{
+    models::{ModelParameters, SWModelParameters},
+    short_weierstrass_jacobian::{GroupAffine, GroupProjective},
+};
+use ark_ff::{field_new, Zero};
+
+#[cfg(test)]
+mod tests;
+
+#[derive(Copy, Clone, Default, PartialEq, Eq)]
+pub struct VestaParameters;
+
+impl ModelParameters for VestaParameters {
+    type BaseField = Fq;
+    type ScalarField = Fr;
+}
+
+pub type Affine = GroupAffine<VestaParameters>;
+pub type Projective = GroupProjective<VestaParameters>;
+
+impl SWModelParameters for VestaParameters {
+    /// COEFF_A = 0
+    const COEFF_A: Fq = field_new!(Fq, "0");
+
+    /// COEFF_B = 5
+    const COEFF_B: Fq = field_new!(Fq, "5");
+
+    /// COFACTOR = 1
+    const COFACTOR: &'static [u64] = &[0x1];
+
+    /// COFACTOR_INV = 1
+    const COFACTOR_INV: Fr = field_new!(Fr, "1");
+
+    /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
+    const AFFINE_GENERATOR_COEFFS: (Self::BaseField, Self::BaseField) =
+        (G_GENERATOR_X, G_GENERATOR_Y);
+
+    #[inline(always)]
+    fn mul_by_a(_: &Self::BaseField) -> Self::BaseField {
+        Self::BaseField::zero()
+    }
+}
+
+/// G_GENERATOR_X = -1
+/// Encoded in Montgomery form, so the value here is -R mod p.
+pub const G_GENERATOR_X: Fq = field_new!(Fq, "-1");
+
+/// G_GENERATOR_Y = 2
+/// Encoded in Montgomery form, so the value here is 2R mod p.
+pub const G_GENERATOR_Y: Fq = field_new!(Fq, "2");

vesta/src/curves/tests.rs

@@ -0,0 +1,39 @@
+#![allow(unused_imports)]
+use ark_ff::{
+    fields::{Field, FpParameters, PrimeField, SquareRootField},
+    One, Zero,
+};
+use ark_serialize::CanonicalSerialize;
+use ark_std::test_rng;
+
+use ark_ec::{models::SWModelParameters, AffineCurve, PairingEngine, ProjectiveCurve};
+use core::ops::{AddAssign, MulAssign};
+use rand::Rng;
+
+use crate::{Affine, Projective, VestaParameters};
+
+use ark_curve_tests::{
+    curves::{curve_tests, sw_tests},
+    groups::group_test,
+};
+
+#[test]
+fn test_projective_curve() {
+    curve_tests::<Projective>();
+    sw_tests::<VestaParameters>();
+}
+
+#[test]
+fn test_projective_group() {
+    let mut rng = test_rng();
+    let a: Projective = rng.gen();
+    let b: Projective = rng.gen();
+    group_test(a, b);
+}
+
+#[test]
+fn test_generator() {
+    let generator = Affine::prime_subgroup_generator();
+    assert!(generator.is_on_curve());
+    assert!(generator.is_in_correct_subgroup_assuming_on_curve());
+}

vesta/src/fields/fq.rs

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

vesta/src/fields/fr.rs

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

vesta/src/fields/mod.rs

@@ -0,0 +1,8 @@
+pub mod fq;
+pub use self::fq::*;
+
+pub mod fr;
+pub use self::fr::*;
+
+#[cfg(test)]
+mod tests;

vesta/src/fields/tests.rs

@@ -0,0 +1,26 @@
+use ark_std::test_rng;
+use rand::Rng;
+
+use crate::*;
+
+use ark_curve_tests::fields::*;
+
+#[test]
+fn test_fr() {
+    let mut rng = test_rng();
+    let a: Fr = rng.gen();
+    let b: Fr = rng.gen();
+    field_test(a, b);
+    sqrt_field_test(a);
+    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);
+    sqrt_field_test(a);
+    primefield_test::<Fq>();
+}

vesta/src/lib.rs

@@ -0,0 +1,33 @@
+#![cfg_attr(not(feature = "std"), no_std)]
+#![deny(
+    warnings,
+    unused,
+    future_incompatible,
+    nonstandard_style,
+    rust_2018_idioms
+)]
+#![forbid(unsafe_code)]
+
+//! This library implements the prime-order curve Vesta, generated by
+//! [Daira Hopwood](https://github.com/zcash/pasta). The main feature of this
+//! curve is that it forms a cycle with Pallas, i.e. its scalar field and base
+//! field respectively are the base field and scalar field of Pallas.
+//!
+//!
+//! Curve information:
+//! Vesta:
+//! * Base field: q =
+//!   28948022309329048855892746252171976963363056481941647379679742748393362948097
+//! * Scalar field: r =
+//!   28948022309329048855892746252171976963363056481941560715954676764349967630337
+//! * Curve equation: y^2 = x^3 + 5
+//! * Valuation(q - 1, 2) = 32
+//! * Valuation(r - 1, 2) = 32
+
+#[cfg(feature = "r1cs")]
+pub mod constraints;
+mod curves;
+mod fields;
+
+pub use curves::*;
+pub use fields::*;