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EIP-4844 -- Polynomial Commitments

Introduction
Custom types
Constants
Preset
Helper functions

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.

Functions flagged as "Public method" MUST be provided by the underlying KZG library as public functions. All other functions are private functions used internally by the KZG library.

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`
`Polynomial`	`Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB]`	a polynomial in evaluation form
`Blob`	`ByteVector[BYTES_PER_FIELD_ELEMENT * FIELD_ELEMENTS_PER_BLOB]`	a basic blob data

Constants

Name	Value	Notes
`BLS_MODULUS`	`52435875175126190479447740508185965837690552500527637822603658699938581184513`	Scalar field modulus of BLS12-381
`BYTES_PER_FIELD_ELEMENT`	`uint64(32)`	Bytes used to encode a BLS scalar field element

Preset

Blob

Name	Value
`FIELD_ELEMENTS_PER_BLOB`	`uint64(4096)`
`FIAT_SHAMIR_PROTOCOL_DOMAIN`	`b'FSBLOBVERIFY_V1_'`

Crypto

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

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 32-byte value to a BLS field scalar. The output is not uniform over the BLS field.
    """
    return int.from_bytes(b, ENDIANNESS) % BLS_MODULUS

`blob_to_polynomial`

def blob_to_polynomial(blob: Blob) -> Polynomial:
    """
    Convert a blob to list of BLS field scalars.
    """
    polynomial = Polynomial()
    for i in range(FIELD_ELEMENTS_PER_BLOB):
        value = int.from_bytes(blob[i * BYTES_PER_FIELD_ELEMENT: (i + 1) * BYTES_PER_FIELD_ELEMENT], ENDIANNESS)
        assert value < BLS_MODULUS
        polynomial[i] = value
    return polynomial

`compute_challenges`

def compute_challenges(polynomials: Sequence[Polynomial],
                       commitments: Sequence[KZGCommitment]) -> BLSFieldElement:
    """
    Return the Fiat-Shamir challenges required by the rest of the protocol.
    The Fiat-Shamir logic works as per the following pseudocode:

       hashed_data = hash(DOMAIN_SEPARATOR, polynomials, commitments)
       r = hash(hashed_data, 0)
       r_powers = [r, r**2, r**3, ...]
       eval_challenge = hash(hashed_data, 1)

    Then return `r_powers` and `eval_challenge` after converting them to BLS field elements.
    The resulting field elements are not uniform over the BLS field.
    """
    # Append the number of polynomials and the degree of each polynomial as a domain separator
    num_polynomials = int.to_bytes(len(polynomials), 8, ENDIANNESS)
    degree_poly = int.to_bytes(FIELD_ELEMENTS_PER_BLOB, 8, ENDIANNESS)
    data = FIAT_SHAMIR_PROTOCOL_DOMAIN + degree_poly + num_polynomials

    # Append each polynomial which is composed by field elements
    for poly in polynomials:
        for field_element in poly:
            data += int.to_bytes(field_element, BYTES_PER_FIELD_ELEMENT, ENDIANNESS)

    # Append serialized G1 points
    for commitment in commitments:
        data += commitment

    # Transcript has been prepared: time to create the challenges
    hashed_data = hash(data)
    r = hash(hashed_data + b'\x00')
    r_powers = compute_powers(bytes_to_bls_field(r), len(commitments))
    eval_challenge = hash(hashed_data + b'\x01')

    return r_powers, bytes_to_bls_field(eval_challenge)

`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))

`poly_lincomb`

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

`compute_powers`

def compute_powers(x: BLSFieldElement, n: uint64) -> Sequence[BLSFieldElement]:
    """
    Return ``x`` to power of [0, n-1].
    """
    current_power = 1
    powers = []
    for _ in range(n):
        powers.append(BLSFieldElement(current_power))
        current_power = current_power * int(x) % BLS_MODULUS
    return powers

Polynomials

`evaluate_polynomial_in_evaluation_form`

def evaluate_polynomial_in_evaluation_form(polynomial: Polynomial,
                                           z: BLSFieldElement) -> BLSFieldElement:
    """
    Evaluate a polynomial (in evaluation form) at an arbitrary point ``z``.
    Uses the barycentric formula:
       f(z) = (z**WIDTH - 1) / 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]), (int(z) - int(roots_of_unity_brp[i])))
    result = result * (pow(z, width, BLS_MODULUS) - 1) * inverse_width % BLS_MODULUS
    return 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:
    """
    Public method.
    """
    return g1_lincomb(bit_reversal_permutation(KZG_SETUP_LAGRANGE), blob_to_polynomial(blob))

`verify_kzg_proof`

def verify_kzg_proof(polynomial_kzg: KZGCommitment,
                     z: Bytes32,
                     y: Bytes32,
                     kzg_proof: KZGProof) -> bool:
    """
    Verify KZG proof that ``p(z) == y`` where ``p(z)`` is the polynomial represented by ``polynomial_kzg``.
    Receives inputs as bytes.
    Public method.
    """
    return verify_kzg_proof_impl(polynomial_kzg, bytes_to_bls_field(z), bytes_to_bls_field(y), kzg_proof)

`verify_kzg_proof_impl`

def verify_kzg_proof_impl(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: Polynomial, z: BLSFieldElement) -> KZGProof:
    """
    Compute KZG proof at point `z` with `polynomial` being in evaluation form
    Do this by computing the quotient polynomial in evaluation form: q(x) = (p(x) - p(z)) / (x - z)
    """

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

    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 = [(int(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))

`compute_aggregated_poly_and_commitment`

def compute_aggregated_poly_and_commitment(
        blobs: Sequence[Blob],
        kzg_commitments: Sequence[KZGCommitment]) -> Tuple[Polynomial, KZGCommitment, BLSFieldElement]:
    """
    Return (1) the aggregated polynomial, (2) the aggregated KZG commitment,
    and (3) the polynomial evaluation random challenge.
    This function should also work with blobs == [] and kzg_commitments == []
    """
    assert len(blobs) == len(kzg_commitments)

    # Convert blobs to polynomials
    polynomials = [blob_to_polynomial(blob) for blob in blobs]

    # Generate random linear combination and evaluation challenges
    r_powers, evaluation_challenge = compute_challenges(polynomials, kzg_commitments)

    # Create aggregated polynomial in evaluation form
    aggregated_poly = Polynomial(poly_lincomb(polynomials, r_powers))

    # Compute commitment to aggregated polynomial
    aggregated_poly_commitment = KZGCommitment(g1_lincomb(kzg_commitments, r_powers))

    return aggregated_poly, aggregated_poly_commitment, evaluation_challenge

`compute_aggregate_kzg_proof`

def compute_aggregate_kzg_proof(blobs: Sequence[Blob]) -> KZGProof:
    """
    Given a list of blobs, return the aggregated KZG proof that is used to verify them against their commitments.
    Public method.
    """
    commitments = [blob_to_kzg_commitment(blob) for blob in blobs]
    aggregated_poly, aggregated_poly_commitment, evaluation_challenge = compute_aggregated_poly_and_commitment(
        blobs,
        commitments
    )
    return compute_kzg_proof(aggregated_poly, evaluation_challenge)

`verify_aggregate_kzg_proof`

def verify_aggregate_kzg_proof(blobs: Sequence[Blob],
                               expected_kzg_commitments: Sequence[KZGCommitment],
                               kzg_aggregated_proof: KZGProof) -> bool:
    """
    Given a list of blobs and an aggregated KZG proof, verify that they correspond to the provided commitments.
    
    Public method.
    """
    aggregated_poly, aggregated_poly_commitment, evaluation_challenge = compute_aggregated_poly_and_commitment(
        blobs,
        expected_kzg_commitments,
    )

    # Evaluate aggregated polynomial at `evaluation_challenge` (evaluation function checks for div-by-zero)
    y = evaluate_polynomial_in_evaluation_form(aggregated_poly, evaluation_challenge)

    # Verify aggregated proof
    return verify_kzg_proof_impl(aggregated_poly_commitment, evaluation_challenge, y, kzg_aggregated_proof)

15 KiB Raw Blame History

EIP-4844 -- Polynomial Commitments

Table of contents

Introduction

Custom types

Constants

Preset

Blob

Crypto

Trusted setup

Helper functions

Bit-reversal permutation

is_power_of_two

reverse_bits

bit_reversal_permutation

BLS12-381 helpers

bytes_to_bls_field

blob_to_polynomial

compute_challenges

bls_modular_inverse

div

g1_lincomb

poly_lincomb

compute_powers