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EIP-4844 -- Polynomial Commitments

Table of contents

Introduction

This document specifies basic polynomial operations and KZG polynomial commitment operations as they are needed for the EIP-4844 specification. The implementations are not optimized for performance, but readability. All practical implementations should optimize the polynomial operations.

Custom types

Name SSZ equivalent Description
G1Point Bytes48
G2Point Bytes96
BLSFieldElement uint256 x < BLS_MODULUS
KZGCommitment Bytes48 Same as BLS standard "is valid pubkey" check but also allows 0x00..00 for point-at-infinity
KZGProof Bytes48 Same as for KZGCommitment

Constants

Name Value Notes
BLS_MODULUS 52435875175126190479447740508185965837690552500527637822603658699938581184513 Scalar field modulus of BLS12-381
ROOTS_OF_UNITY Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB] Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field

Preset

Trusted setup

The trusted setup is part of the preset: during testing a minimal insecure variant may be used, but reusing the mainnet settings in public networks is a critical security requirement.

Name Value
KZG_SETUP_G1 Vector[G1Point, FIELD_ELEMENTS_PER_BLOB], contents TBD
KZG_SETUP_G2 Vector[G2Point, FIELD_ELEMENTS_PER_BLOB], contents TBD
KZG_SETUP_LAGRANGE Vector[KZGCommitment, FIELD_ELEMENTS_PER_BLOB], contents TBD

Helper functions

Bit-reversal permutation

All polynomials (which are always given in Lagrange form) should be interpreted as being in bit-reversal permutation. In practice, clients can implement this by storing the lists KZG_SETUP_LAGRANGE and ROOTS_OF_UNITY in bit-reversal permutation, so these functions only have to be called once at startup.

is_power_of_two

def is_power_of_two(value: int) -> bool:
    """
    Check if ``value`` is a power of two integer.
    """
    return (value > 0) and (value & (value - 1) == 0)

reverse_bits

def reverse_bits(n: int, order: int) -> int:
    """
    Reverse the bit order of an integer n
    """
    assert is_power_of_two(order)
    # Convert n to binary with the same number of bits as "order" - 1, then reverse its bit order
    return int(('{:0' + str(order.bit_length() - 1) + 'b}').format(n)[::-1], 2)

bit_reversal_permutation

def bit_reversal_permutation(sequence: Sequence[T]) -> Sequence[T]:
    """
    Return a copy with bit-reversed permutation. The permutation is an involution (inverts itself).

    The input and output are a sequence of generic type ``T`` objects.
    """
    return [sequence[reverse_bits(i, len(sequence))] for i in range(len(sequence))]

BLS12-381 helpers

bytes_to_bls_field

def bytes_to_bls_field(b: Bytes32) -> BLSFieldElement:
    """
    Convert bytes to a BLS field scalar. The output is not uniform over the BLS field.
    """
    return int.from_bytes(b, "little") % BLS_MODULUS

bls_modular_inverse

def bls_modular_inverse(x: BLSFieldElement) -> BLSFieldElement:
    """
    Compute the modular inverse of x
    i.e. return y such that x * y % BLS_MODULUS == 1 and return 0 for x == 0
    """
    return pow(x, -1, BLS_MODULUS) if x != 0 else 0

div

def div(x: BLSFieldElement, y: BLSFieldElement) -> BLSFieldElement:
    """Divide two field elements: `x` by `y`"""
    return (int(x) * int(bls_modular_inverse(y))) % BLS_MODULUS

g1_lincomb

def g1_lincomb(points: Sequence[KZGCommitment], scalars: Sequence[BLSFieldElement]) -> KZGCommitment:
    """
    BLS multiscalar multiplication. This function can be optimized using Pippenger's algorithm and variants.
    """
    assert len(points) == len(scalars)
    result = bls.Z1
    for x, a in zip(points, scalars):
        result = bls.add(result, bls.multiply(bls.bytes48_to_G1(x), a))
    return KZGCommitment(bls.G1_to_bytes48(result))

vector_lincomb

def vector_lincomb(vectors: Sequence[Sequence[BLSFieldElement]],
                   scalars: Sequence[BLSFieldElement]) -> Sequence[BLSFieldElement]:
    """
    Given a list of ``vectors``, interpret it as a 2D matrix and compute the linear combination
    of each column with `scalars`: return the resulting vector.
    """
    result = [0] * len(vectors[0])
    for v, s in zip(vectors, scalars):
        for i, x in enumerate(v):
            result[i] = (result[i] + int(s) * int(x)) % BLS_MODULUS
    return [BLSFieldElement(x) for x in result]

KZG

KZG core functions. These are also defined in EIP-4844 execution specs.

blob_to_kzg_commitment

def blob_to_kzg_commitment(blob: Blob) -> KZGCommitment:
    return g1_lincomb(bit_reversal_permutation(KZG_SETUP_LAGRANGE), blob)

verify_kzg_proof

def verify_kzg_proof(polynomial_kzg: KZGCommitment,
                     z: BLSFieldElement,
                     y: BLSFieldElement,
                     kzg_proof: KZGProof) -> bool:
    """
    Verify KZG proof that ``p(z) == y`` where ``p(z)`` is the polynomial represented by ``polynomial_kzg``.
    """
    # Verify: P - y = Q * (X - z)
    X_minus_z = bls.add(bls.bytes96_to_G2(KZG_SETUP_G2[1]), bls.multiply(bls.G2, BLS_MODULUS - z))
    P_minus_y = bls.add(bls.bytes48_to_G1(polynomial_kzg), bls.multiply(bls.G1, BLS_MODULUS - y))
    return bls.pairing_check([
        [P_minus_y, bls.neg(bls.G2)],
        [bls.bytes48_to_G1(kzg_proof), X_minus_z]
    ])

compute_kzg_proof

def compute_kzg_proof(polynomial: Sequence[BLSFieldElement], z: BLSFieldElement) -> KZGProof:
    """
    Compute KZG proof at point `z` with `polynomial` being in evaluation form
    """

    # To avoid SSZ overflow/underflow, convert element into int
    polynomial = [int(i) for i in polynomial]
    z = int(z)

    # Shift our polynomial first (in evaluation form we can't handle the division remainder)
    y = evaluate_polynomial_in_evaluation_form(polynomial, z)
    polynomial_shifted = [(p - int(y)) % BLS_MODULUS for p in polynomial]

    # Make sure we won't divide by zero during division
    assert z not in ROOTS_OF_UNITY
    denominator_poly = [(x - z) % BLS_MODULUS for x in bit_reversal_permutation(ROOTS_OF_UNITY)]

    # Calculate quotient polynomial by doing point-by-point division
    quotient_polynomial = [div(a, b) for a, b in zip(polynomial_shifted, denominator_poly)]
    return KZGProof(g1_lincomb(bit_reversal_permutation(KZG_SETUP_LAGRANGE), quotient_polynomial))

Polynomials

evaluate_polynomial_in_evaluation_form

def evaluate_polynomial_in_evaluation_form(polynomial: Sequence[BLSFieldElement],
                                           z: BLSFieldElement) -> BLSFieldElement:
    """
    Evaluate a polynomial (in evaluation form) at an arbitrary point `z`
    Uses the barycentric formula:
       f(z) = (1 - z**WIDTH) / WIDTH  *  sum_(i=0)^WIDTH  (f(DOMAIN[i]) * DOMAIN[i]) / (z - DOMAIN[i])
    """
    width = len(polynomial)
    assert width == FIELD_ELEMENTS_PER_BLOB
    inverse_width = bls_modular_inverse(width)

    # Make sure we won't divide by zero during division
    assert z not in ROOTS_OF_UNITY

    roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY)

    result = 0
    for i in range(width):
        result += div(int(polynomial[i]) * int(roots_of_unity_brp[i]), (z - roots_of_unity_brp[i]))
    result = result * (pow(z, width, BLS_MODULUS) - 1) * inverse_width % BLS_MODULUS
    return result