BN254 Elliptic Curve Pairing Library
- This implementation has not been reviewed or audited. Use at your own risk.
- This implementation targets Noir version 0.9.0 or higher
- This implementation currently requires a specific fork https://github.com/onurinanc/noir-bigint` of noir-bigint library https://github.com/shuklaayush/noir-bigint, workspace dependency will be added in the future for a better use
- Use following commands to use two libraries in your local
git clone https://github.com/onurinanc/noir-bn254
- To add the noir-bigint dependency, use a forked version of https://github.com/shuklaayush/noir-bigint,
git clone https://github.com/onurinanc/noir-bigint
- Test the files using the following commands inside the noir-bn254.
nargo test
BN254 is a pairing-friendly elliptic curve construction. This includes two elliptic curve constructions which are G1: E(Fp) and G2: E(Fp^2).
G1 curve is y^2 = x^3 + 3
Field modulus is 21888242871839275222246405745257275088696311157297823662689037894645226208583
The order of the elliptic curves is 21888242871839275222246405745257275088548364400416034343698204186575808495617
The generator of the G1 is the (1, 2)
The method and parameters for constructing the G1 curve is in the src/bn254.nr
// Construct G1 curve for BN254
fn bn254() -> BN254 {
BN254 {
curve: Curve::new(
Fp::zero(),
Fp::from_u56(3),
Point::from_affine(
Fp::from_u56(1),
Fp::from_u56(2),
),
),
}
}
G2 is constructed using the Fp^2 extension field. The method and the parameters for constructing the G2 curve is in the src/bn254.nr
// Construct G2 curve for BN254
fn bn254_g2() -> BN254G2 {
// Construct a, b, and gen for G2 Curve
// G2_A, G2_B_ G2_GENERATOR_X, and G2_GENERATOR_Y
let G2_A = Fp2::zero();
let G2_B: Fp2 = Fp2::new(
Fp::from_bytes(
[
0xe5, 0x38, 0xa1, 0x24, 0xdc, 0xe6, 0x67, 0x32,
0xa3, 0xef, 0xdb, 0x59, 0xe5, 0xc5, 0xb4, 0xb5,
0xc3, 0x6a, 0xe0, 0x1b, 0x99, 0x18, 0xbe, 0x81,
0xae, 0xaa, 0xb8, 0xce, 0x40, 0x9d, 0x14, 0x2b
]
),
Fp::from_bytes(
[
0xd2, 0x15, 0xc3, 0x85, 0x06, 0xbd, 0xa2, 0xe4,
0x52, 0x18, 0x2d, 0xe5, 0x84, 0xa0, 0x4f, 0xa7,
0xf4, 0xfd, 0xd8, 0xee, 0xad, 0xaf, 0x2c, 0xcd,
0xd4, 0xfe, 0xf0, 0x3a, 0xb0, 0x13, 0x97, 0x00
]
)
);
let G2_GENERATOR_X: Fp2 = Fp2::new(
Fp::from_bytes(
[
0xed, 0xf6, 0x92, 0xd9, 0x5c, 0xbd, 0xde, 0x46,
0xdd, 0xda, 0x5e, 0xf7, 0xd4, 0x22, 0x43, 0x67,
0x79, 0x44, 0x5c, 0x5e, 0x66, 0x00, 0x6a, 0x42,
0x76, 0x1e, 0x1f, 0x12, 0xef, 0xde, 0x00, 0x18
]
),
Fp::from_bytes(
[
0xc2, 0x12, 0xf3, 0xae, 0xb7, 0x85, 0xe4, 0x97,
0x12, 0xe7, 0xa9, 0x35, 0x33, 0x49, 0xaa, 0xf1,
0x25, 0x5d, 0xfb, 0x31, 0xb7, 0xbf, 0x60, 0x72,
0x3a, 0x48, 0x0d, 0x92, 0x93, 0x93, 0x8e, 0x19
]
)
);
let G2_GENERATOR_Y: Fp2 = Fp2::new(
Fp::from_bytes(
[
0xaa, 0x7d, 0xfa, 0x66, 0x01, 0xcc, 0xe6, 0x4c,
0x7b, 0xd3, 0x43, 0x0c, 0x69, 0xe7, 0xd1, 0xe3,
0x8f, 0x40, 0xcb, 0x8d, 0x80, 0x71, 0xab, 0x4a,
0xeb, 0x6d, 0x8c, 0xdb, 0xa5, 0x5e, 0xc8, 0x12
]
),
Fp::from_bytes(
[
0x5b, 0x97, 0x22, 0xd1, 0xdc, 0xda, 0xac, 0x55,
0xf3, 0x8e, 0xb3, 0x70, 0x33, 0x31, 0x4b, 0xbc,
0x95, 0x33, 0x0c, 0x69, 0xad, 0x99, 0x9e, 0xec,
0x75, 0xf0, 0x5f, 0x58, 0xd0, 0x89, 0x06, 0x09
]
)
);
BN254G2 {
curve: G2Curve::new(
Fp2::zero(),
G2_B,
G2Point::from_affine(
G2_GENERATOR_X,
G2_GENERATOR_Y,
)
)
}
}
You can create instances of the elliptic curves using
let g2: BN254G2 = bn254_g2();
let g1: BN254 = bn254();
You can test additions for G1 and G2 seperately using the following command in Noir
nargo test test_bn254_add1
nargo test test_bn254g2_add1
- fn pair(Q: G2Point, P: Point) -> Fp12
It's the pairing function. Q is an element of G2, and P is an element of G1.
-
The
paircalculates the pairing in Fp12 -
Pointis a point on the G1 curve -
G2Pointis a point on the G2 curve
You can create Point and G2Point as the following function signatures
Point::from_affine(x: Fp, y: Fp)G2Point::from_affine(x: Fp2, y:Fp2)
You can create Fp, Fp2, Fp6, Fp12 elements using the following examples
let fp_1 = Fp::from_bytes([
0x61, 0x22, 0xfe, 0xd9,
0x3d, 0xff, 0xf1, 0xcd,
0x57, 0x5b, 0x9c, 0x0b,
0xb4, 0x63, 0x9e, 0x31,
0x75, 0x64, 0x08, 0x8d,
0x7c, 0xdb, 0x4f, 0x55,
0x29, 0x94, 0x48, 0xe0,
0xbe, 0x99, 0xb7, 0x2a,
]);
let fp_2 = Fp::from_u56(8);
let a: Fp2 = Fp2::new(Fp::from_u56(1), Fp::from_u56(2));
let b: Fp2 = Fp2::new(Fp::from_u56(3), Fp::from_u56(5));
let c: Fp2 = Fp2::new(Fp::from_u56(8), Fp::from_u56(11));
let a1: Fp6 = Fp6::new(a, b, c);
let a2: Fp6 = Fp6::new(a, a, b);
let a3: Fp6 = Fp6::new(b, c, c);
let a4: Fp6 = Fp6::new(c, b, a);
let x1: Fp12 = Fp12::new(a1, a4);
let x2: Fp12 = Fp12::new(a3, a2);
let x3: Fp12 = Fp12::new(a4, a2);
Compilation time for one pairing function takes ~0.5 h for 16 GB RAM, Intel Core i5-12450H
Contributions are welcome! Please adhere to the following guidelines:
- Open a pull request with a clear description of your changes.
- Changes should aim to improve code efficiency or readability.
- Add appropriate tests, ensuring all pass before submission.
This is an experimental software and is provided on an "as is" and "as available" basis. We do not give any warranties and will not be liable for any losses incurred through any use of this code base.
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