BN254 G2 Arithmetic Syscalls

Summary

Extend the existing sol_alt_bn128_group_op syscall to support native G2 curve point arithmetic (addition, subtraction, and scalar multiplication) on the BN254 curve.

Motivation

Solana today provides syscalls for Alt‑BN128 G1 group operations (ALT_BN128_ADD, ALT_BN128_SUB, ALT_BN128_MUL) and for pairing checks, as well as compression/decompression for G2 points. However, there is no direct support for arithmetic on G2 points themselves, as they were not included in the Ethereum Precompile on which the syscalls were based.

Adding native G2 syscalls will enable:

• Batch Groth16 verification: Multiple proofs could be aggregated into single verification calls
• KZG polynomial commitments: Enable efficient multi-point opening proofs and batch verification
• Other more advanced ZK systems for batch or direct verification

New Terminology

• G2 point: A point on the twist subgroup of BN254, represented as two Fq₂ elements (each element is a pair of 32‑byte field coefficients).

Detailed Design

New Constants

// Field and point sizes
pub const ALT_BN128_FIELD_SIZE:       u64 = 32;
    // bytes per Fq element
pub const ALT_BN128_G2_POINT_SIZE:    u64 = ALT_BN128_FIELD_SIZE * 4;
    // x=(x0,x1), y=(y0,y1) each 32‑byte

// G2 addition/subtraction
pub const ALT_BN128_G2_ADDITION_INPUT_LEN:   u64 = ALT_BN128_G2_POINT_SIZE * 2;
    // 256
pub const ALT_BN128_G2_ADDITION_OUTPUT_LEN:  u64 = ALT_BN128_G2_POINT_SIZE;
    // 128

// G2 scalar multiplication
pub const ALT_BN128_G2_MULTIPLICATION_INPUT_LEN:  u64 = ALT_BN128_G2_POINT_SIZE
    + ALT_BN128_FIELD_SIZE; // 160
pub const ALT_BN128_G2_MULTIPLICATION_OUTPUT_LEN: u64 = ALT_BN128_G2_POINT_SIZE;
    // 128

### New Opcodes
pub const ALT_BN128_G2_ADD: u64 = 4;
pub const ALT_BN128_G2_SUB: u64 = 5;
pub const ALT_BN128_G2_MUL: u64 = 6;

Endianness

Additional G2 operation support for little endian formats should also be included use the little-endian encoding conventions as specified in SIMD-0284, consistent with existing BN128 syscalls.

Validation and Subgroup Checks

The validation requirements for the new G2 operations are defined as follows to balance safety and compute cost, consistent with the approach for BLS12-381:

1. G2 Addition (ALT_BN128_G2_ADD) and Subtraction (ALT_BN128_G2_SUB):

   • Input points MUST be validated to ensure they are on the curve (satisfy the G2 curve equation).
   • The subgroup check is skipped. The addition formulas are valid for any point on the curve (including those not in the prime-order subgroup). Skipping this costly check allows for cheaper accumulation of points.

2. G2 Scalar Multiplication (ALT_BN128_G2_MUL):

   • Input points MUST undergo full validation, including the field check, curve equation check, and the subgroup check.
   • Enforcing the subgroup check is required to safely support faster endomorphism-based scalar multiplication algorithms.

Note on G1: For the existing G1 operations, the BN254 (Alt-BN128) curve has a cofactor of 1. This means that every point satisfying the curve equation is automatically in the prime-order subgroup. Therefore, no separate subgroup check is needed for G1 as the on-curve check is sufficient.

Alternatives Considered

• Separate G2‑only syscall entrypoint rather than extending sol_alt_bn128_group_op.
• Drawback: Increases API surface, doubles maintenance.
• Library‑level intrinsics (BPF C/Rust) for G2 math as fallback.
• Drawback: Slower, higher compute cost than native syscalls.

Impact

• Native G2 ops reduce compute units by approximately 10×–20× compared to pure‑BPF implementations.
• Enables newer and more advanced ZK verification methods as well as batch proof verification

Security Considerations

None