BLS12_381 Elliptic Curve Pairing and Signature Verification 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-bls12_381
- To add the noir-bigint dependency, use a forked version of https://github.com/shuklaayush/noir-bigint, specifically built for bls12_381 prime
git clone https://github.com/onurinanc/noir-bigint-bls12_381
- Test the files using the following commands inside the noir-bls12_381.
nargo test
BLS12_381 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 + 4
Field modulus is 4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787
The order of the elliptic curves is 52435875175126190479447740508185965837690552500527637822603658699938581184513
The generator of the G1 is the (3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507, 1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569)
The method and parameters for constructing the G1 curve is in the src/bls12_381.nr
// Construct G1 curve for BLS12_381
fn bls12_381() -> BLS12_381 {
BLS12_381 {
curve: Curve::new(
Fp::zero(),
Fp::from_u56(4),
Point::from_affine(
Fp::from_bytes(
[
0xbb, 0xc6, 0x22, 0xdb, 0x0a, 0xf0, 0x3a, 0xfb,
0xef, 0x1a, 0x7a, 0xf9, 0x3f, 0xe8, 0x55, 0x6c,
0x58, 0xac, 0x1b, 0x17, 0x3f, 0x3a, 0x4e, 0xa1,
0x05, 0xb9, 0x74, 0x97, 0x4f, 0x8c, 0x68, 0xc3,
0x0f, 0xac, 0xa9, 0x4f, 0x8c, 0x63, 0x95, 0x26,
0x94, 0xd7, 0x97, 0x31, 0xa7, 0xd3, 0xf1, 0x17
]
),
Fp::from_bytes(
[
0xe1, 0xe7, 0xc5, 0x46, 0x29, 0x23, 0xaa, 0x0c,
0xe4, 0x8a, 0x88, 0xa2, 0x44, 0xc7, 0x3c, 0xd0,
0xed, 0xb3, 0x04, 0x2c, 0xcb, 0x18, 0xdb, 0x00,
0xf6, 0x0a, 0xd0, 0xd5, 0x95, 0xe0, 0xf5, 0xfc,
0xe4, 0x8a, 0x1d, 0x74, 0xed, 0x30, 0x9e, 0xa0,
0xf1, 0xa0, 0xaa, 0xe3, 0x81, 0xf4, 0xb3, 0x08
]
),
),
),
}
}
G2 is constructed using the Fp^2 extension field. The method and the parameters for constructing the G2 curve is in the src/bls12_381.nr
// Construct G2 curve for BLS12_381
fn bls12_381_g2() -> BLS12_381G2 {
// 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_u56(4),
Fp::from_u56(4)
);
// Part X of the generator
// 352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160
// 3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758
let G2_GENERATOR_X: Fp2 = Fp2::new(
Fp::from_bytes(
[
0xb8, 0xbd, 0x21, 0xc1, 0xc8, 0x56, 0x80, 0xd4,
0xef, 0xbb, 0x05, 0xa8, 0x26, 0x03, 0xac, 0x0b,
0x77, 0xd1, 0xe3, 0x7a, 0x64, 0x0b, 0x51, 0xb4,
0x02, 0x3b, 0x40, 0xfa, 0xd4, 0x7a, 0xe4, 0xc6,
0x51, 0x10, 0xc5, 0x2d, 0x27, 0x05, 0x08, 0x26,
0x91, 0x0a, 0x8f, 0xf0, 0xb2, 0xa2, 0x4a, 0x02
]
),
Fp::from_bytes(
[
0x7e, 0x2b, 0x04, 0x5d, 0x05, 0x7d, 0xac, 0xe5,
0x57, 0x5d, 0x94, 0x13, 0x12, 0xf1, 0x4c, 0x33,
0x49, 0x50, 0x7f, 0xdc, 0xbb, 0x61, 0xda, 0xb5,
0x1a, 0xb6, 0x20, 0x99, 0xd0, 0xd0, 0x6b, 0x59,
0x65, 0x4f, 0x27, 0x88, 0xa0, 0xd3, 0xac, 0x7d,
0x60, 0x9f, 0x71, 0x52, 0x60, 0x2b, 0xe0, 0x13
]
)
);
// Part Y of the generator
// 1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905
// 927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582
let G2_GENERATOR_Y: Fp2 = Fp2::new(
Fp::from_bytes(
[
0x01, 0x28, 0xb8, 0x08, 0x86, 0x54, 0x93, 0xe1,
0x89, 0xa2, 0xac, 0x3b, 0xcc, 0xc9, 0x3a, 0x92,
0x2c, 0xd1, 0x60, 0x51, 0x69, 0x9a, 0x42, 0x6d,
0xa7, 0xd3, 0xbd, 0x8c, 0xaa, 0x9b, 0xfd, 0xad,
0x1a, 0x35, 0x2e, 0xda, 0xc6, 0xcd, 0xc9, 0x8c,
0x11, 0x6e, 0x7d, 0x72, 0x27, 0xd5, 0xe5, 0x0c,
]
),
Fp::from_bytes(
[
0xbe, 0x79, 0x5f, 0xf0, 0x5f, 0x07, 0xa9, 0xaa,
0xa1, 0x1d, 0xec, 0x5c, 0x27, 0x0d, 0x37, 0x3f,
0xab, 0x99, 0x2e, 0x57, 0xab, 0x92, 0x74, 0x26,
0xaf, 0x63, 0xa7, 0x85, 0x7e, 0x28, 0x3e, 0xcb,
0x99, 0x8b, 0xc2, 0x2b, 0xb0, 0xd2, 0xac, 0x32,
0xcc, 0x34, 0xa7, 0x2e, 0xa0, 0xc4, 0x06, 0x06
]
)
);
BLS12_381G2 {
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: BLS12_381G2 = bls12_381_g2();
let g1: BLS12_381 = bls12_381();
You can test additions for G1 and G2 seperately using the following command in Noir
nargo test test_bls12_381_add1
nargo test test_bls12_381_g2_add1
- You can verify bls signatures in a circuit by using the following function as it is specified
signature::verify_bls_signature(signature: G2Point, public_key: Point, message_hash: G2Point)
- 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([
0x29, 0x0b, 0x0e, 0x34, 0x32, 0x50, 0x16, 0x12,
0x27, 0x7a, 0xca, 0x7b, 0x36, 0x15, 0xe1, 0xa5,
0x2d, 0xed, 0x21, 0x22, 0x0c, 0x2a, 0xc7, 0xf3,
0x33, 0x22, 0xea, 0xe2, 0x8d, 0x43, 0x48, 0x7e,
0x34, 0x2d, 0xd5, 0xe7, 0x90, 0xb2, 0x60, 0x53,
0x51, 0x06, 0x6b, 0xd8, 0xc1, 0x2e, 0x26, 0x09
]);
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);
The project is compiled by the machine has the following properties:
16 GB RAM, Intel Core i5-12450H
Compilation time for the pairing function (optimal ate), which uses BLS12_381 Pairing Friendly Curve implemented in Noir, is 1.2 h.
The compilation time for the pairing function is less than [circom-pairing][https://github.com/yi-sun/circom-pairing#benchmarks],which is computed with a machine with better properties stated as 1.9 h
The compilation time for BLS12_381 Signature Verification is approximately 2.4 h, it is also less than [circom-pairing][https://github.com/yi-sun/circom-pairing#benchmarks], which is computed with a machine with better properties states that it is 3.2 h
For testing purposes:
- Compilation time for the
final_exponentiationtakes~8 minutes - Compilation time for the miller_loop takes
~64 minutes: so you can reduce to loop to test. So, the following are the smaller loops for testingmiller_loop:for i in 0..2->~0.5 minutesfor i in 0..8->~4 minutesfor i in 0..10->~6 minutesfor i in 0.12->~8 minutes
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.