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G2 Operations

G2 is the extension field elliptic curve group used for BLS public keys. Points are 96 bytes compressed or 192 bytes uncompressed.

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G2 Curve Equation

y² = x³ + 4(1 + i) over Fp2
Extension Field: Fp2 = Fp[i] / (i² + 1) Group Order: r (same as G1) Cofactor: h2 = 0x5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5

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Point Formats

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Uncompressed (192 bytes, padded to 256 for precompiles)

| x.c0 | x.c1 | y.c0 | y.c1 |
|  48  |  48  |  48  |  48  |
| (64) | (64) | (64) | (64) |
Each coordinate is Fp2 element: a + bi where a, b ∈ Fp

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Compressed (96 bytes)

| x.c1 (with flags) | x.c0 |
|     48 bytes      | 48 bytes |

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Operations

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Point Addition

import { bls12_381 } from '@tevm/voltaire/crypto';

const p1 = new Uint8Array(256);
const p2 = new Uint8Array(256);

const input = new Uint8Array(512);
input.set(p1, 0);
input.set(p2, 256);

const output = new Uint8Array(256);
await bls12_381.g2Add(input, output);
Gas Cost: 800 (EIP-2537) Time: ~25 μs

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Scalar Multiplication

const point = new Uint8Array(256);
const scalar = Bytes32();

const input = new Uint8Array(288);
input.set(point, 0);
input.set(scalar, 256);

const output = new Uint8Array(256);
await bls12_381.g2Mul(input, output);
Gas Cost: 45,000 (EIP-2537) Time: ~200 μs

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Multi-Scalar Multiplication

const n = 50;
const input = new Uint8Array(288 * n);

for (let i = 0; i < n; i++) {
  const offset = 288 * i;
  input.set(publicKeys[i], offset);
  input.set(scalars[i], offset + 256);
}

const output = new Uint8Array(256);
await bls12_381.g2Msm(input, output);
Use Case: Aggregate validator public keys

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Subgroup Membership

Critical: G2 has large cofactor - must verify subgroup membership! Attack: Invalid curve attack if subgroup not checked BLST automatically validates:
  • Point on curve
  • In prime-order subgroup
  • Coordinates in field
// Will throw if point not in subgroup
await bls12_381.g2Add(input, output);

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Public Key Aggregation

async function aggregatePublicKeys(
  publicKeys: Uint8Array[]
): Promise<Uint8Array> {
  let aggregated = publicKeys[0];

  for (let i = 1; i < publicKeys.length; i++) {
    const input = new Uint8Array(512);
    input.set(aggregated, 0);
    input.set(publicKeys[i], 256);

    const output = new Uint8Array(256);
    await bls12_381.g2Add(input, output);
    aggregated = output;
  }

  return aggregated;
}
Ethereum: Aggregate 512 sync committee public keys

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Performance

Native (BLST):
  • Addition: ~25 μs
  • Scalar mul: ~200 μs
  • MSM (50): ~6 ms
  • MSM (512): ~50 ms
Optimization: MSM much faster than individual multiplications for validator key aggregation