consensus-specs/specs/_features/eip7594/polynomial-commitments-sampling.md

EIP-7594 -- Polynomial Commitments

Table of contents

• Introduction
• Custom types
• Constants
• Preset
  • Cells
• Helper functions
  • Linear combinations
    • g2_lincomb
  • FFTs
    • _fft_field
    • fft_field
  • Polynomials in coefficient form
    • polynomial_eval_to_coeff
    • add_polynomialcoeff
    • neg_polynomialcoeff
    • multiply_polynomialcoeff
    • divide_polynomialcoeff
    • shift_polynomialcoeff
    • interpolate_polynomialcoeff
    • vanishing_polynomialcoeff
    • evaluate_polynomialcoeff
  • KZG multiproofs
    • compute_kzg_proof_multi_impl
    • verify_kzg_proof_multi_impl
  • Cell cosets
    • coset_for_cell
• Cells
  • Cell computation
    • compute_cells_and_proofs
    • compute_cells
  • Cell verification
    • verify_cell_proof
    • verify_cell_proof_batch
• Reconstruction
  • recover_polynomial

Introduction

This document extends polynomial-commitments.md with the functions required for data availability sampling (DAS). It is not part of the core Deneb spec but an extension that can be optionally implemented to allow nodes to reduce their load using DAS.

For any KZG library extended to support DAS, 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.

Public functions MUST accept raw bytes as input and perform the required cryptographic normalization before invoking any internal functions.

Custom types

Name	SSZ equivalent	Description
PolynomialCoeff	List[BLSFieldElement, 2 * FIELD_ELEMENTS_PER_BLOB]	A polynomial in coefficient form
Cell	Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]	The unit of blob data that can come with their own KZG proofs
CellID	uint64	Cell identifier

Constants

Name	Value	Notes

Preset

Cells

Cells are the smallest unit of blob data that can come with their own KZG proofs. Samples can be constructed from one or several cells (e.g. an individual cell or line).

Name	Value	Description
FIELD_ELEMENTS_PER_CELL	uint64(64)	Number of field elements in a cell
BYTES_PER_CELL	FIELD_ELEMENTS_PER_CELL * BYTES_PER_FIELD_ELEMENT	The number of bytes in a cell
CELLS_PER_BLOB	((2 * FIELD_ELEMENTS_PER_BLOB) // FIELD_ELEMENTS_PER_CELL)	The number of cells in a blob
RANDOM_CHALLENGE_KZG_CELL_BATCH_DOMAIN	b'RCKZGCBATCH__V1_'

Helper functions

Linear combinations

g2_lincomb

def g2_lincomb(points: Sequence[KZGCommitment], scalars: Sequence[BLSFieldElement]) -> Bytes96:
    """
    BLS multiscalar multiplication in G2. This function can be optimized using Pippenger's algorithm and variants.
    """
    assert len(points) == len(scalars)
    result = bls.Z2()
    for x, a in zip(points, scalars):
        result = bls.add(result, bls.multiply(bls.bytes96_to_G2(x), a))
    return Bytes96(bls.G2_to_bytes96(result))

FFTs

_fft_field

def _fft_field(vals: Sequence[BLSFieldElement],
               roots_of_unity: Sequence[BLSFieldElement]) -> Sequence[BLSFieldElement]:
    if len(vals) == 1:
        return vals
    L = _fft_field(vals[::2], roots_of_unity[::2])
    R = _fft_field(vals[1::2], roots_of_unity[::2])
    o = [BLSFieldElement(0) for _ in vals]
    for i, (x, y) in enumerate(zip(L, R)):
        y_times_root = (int(y) * int(roots_of_unity[i])) % BLS_MODULUS
        o[i] = BLSFieldElement((int(x) + y_times_root) % BLS_MODULUS)
        o[i + len(L)] = BLSFieldElement((int(x) - y_times_root + BLS_MODULUS) % BLS_MODULUS)
    return o

fft_field

def fft_field(vals: Sequence[BLSFieldElement],
              roots_of_unity: Sequence[BLSFieldElement],
              inv: bool=False) -> Sequence[BLSFieldElement]:
    if inv:
        # Inverse FFT
        invlen = pow(len(vals), BLS_MODULUS - 2, BLS_MODULUS)
        return [BLSFieldElement((int(x) * invlen) % BLS_MODULUS)
                for x in _fft_field(vals, list(roots_of_unity[0:1]) + list(roots_of_unity[:0:-1]))]
    else:
        # Regular FFT
        return _fft_field(vals, roots_of_unity)

Polynomials in coefficient form

polynomial_eval_to_coeff

def polynomial_eval_to_coeff(polynomial: Polynomial) -> PolynomialCoeff:
    """
    Interpolates a polynomial (given in evaluation form) to a polynomial in coefficient form.
    """
    roots_of_unity = compute_roots_of_unity(FIELD_ELEMENTS_PER_BLOB)
    polynomial_coeff = fft_field(bit_reversal_permutation(list(polynomial)), roots_of_unity, inv=True)

    return polynomial_coeff

add_polynomialcoeff

def add_polynomialcoeff(a: PolynomialCoeff, b: PolynomialCoeff) -> PolynomialCoeff:
    """
    Sum the coefficient form polynomials ``a`` and ``b``.
    """
    a, b = (a, b) if len(a) >= len(b) else (b, a)
    return [(a[i] + (b[i] if i < len(b) else 0)) % BLS_MODULUS for i in range(len(a))]

neg_polynomialcoeff

def neg_polynomialcoeff(a: PolynomialCoeff) -> PolynomialCoeff:
    """
    Negative of coefficient form polynomial ``a``
    """
    return [(BLS_MODULUS - x) % BLS_MODULUS for x in a]

multiply_polynomialcoeff

def multiply_polynomialcoeff(a: PolynomialCoeff, b: PolynomialCoeff) -> PolynomialCoeff:
    """
    Multiplies the coefficient form polynomials ``a`` and ``b``
    """
    r = [0]
    for power, coef in enumerate(a):
        summand = [0] * power + [int(coef) * int(x) % BLS_MODULUS for x in b]
        r = add_polynomialcoeff(r, summand)
    return r

divide_polynomialcoeff

def divide_polynomialcoeff(a: PolynomialCoeff, b: PolynomialCoeff) -> PolynomialCoeff:
    """
    Long polynomial division for two coefficient form polynomials ``a`` and ``b``
    """
    a = [x for x in a]
    o = []
    apos = len(a) - 1
    bpos = len(b) - 1
    diff = apos - bpos
    while diff >= 0:
        quot = div(a[apos], b[bpos])
        o.insert(0, quot)
        for i in range(bpos, -1, -1):
            a[diff + i] = (int(a[diff + i]) - int(b[i]) * int(quot)) % BLS_MODULUS
        apos -= 1
        diff -= 1
    return [x % BLS_MODULUS for x in o]

shift_polynomialcoeff

def shift_polynomialcoeff(polynomial_coeff: PolynomialCoeff, factor: BLSFieldElement) -> PolynomialCoeff:
    """
    Shift the evaluation of a polynomial in coefficient form by factor.
    This results in a new polynomial g(x) = f(factor * x)
    """
    factor_power = 1
    inv_factor = pow(int(factor), BLS_MODULUS - 2, BLS_MODULUS)
    o = []
    for p in polynomial_coeff:
        o.append(int(p) * factor_power % BLS_MODULUS)
        factor_power = factor_power * inv_factor % BLS_MODULUS
    return o

interpolate_polynomialcoeff

def interpolate_polynomialcoeff(xs: Sequence[BLSFieldElement], ys: Sequence[BLSFieldElement]) -> PolynomialCoeff:
    """
    Lagrange interpolation: Finds the lowest degree polynomial that takes the value ``ys[i]`` at ``x[i]``
    for all i.
    Outputs a coefficient form polynomial. Leading coefficients may be zero.
    """
    assert len(xs) == len(ys)
    r = [0]

    for i in range(len(xs)):
        summand = [ys[i]]
        for j in range(len(ys)):
            if j != i:
                weight_adjustment = bls_modular_inverse(int(xs[i]) - int(xs[j]))
                summand = multiply_polynomialcoeff(
                    summand, [(- int(weight_adjustment) * int(xs[j])) % BLS_MODULUS, weight_adjustment]
                )
        r = add_polynomialcoeff(r, summand)
    
    return r

vanishing_polynomialcoeff

def vanishing_polynomialcoeff(xs: Sequence[BLSFieldElement]) -> PolynomialCoeff:
    """
    Compute the vanishing polynomial on ``xs`` (in coefficient form)
    """
    p = [1]
    for x in xs:
        p = multiply_polynomialcoeff(p, [-int(x), 1])
    return p

evaluate_polynomialcoeff

def evaluate_polynomialcoeff(polynomial_coeff: PolynomialCoeff, z: BLSFieldElement) -> BLSFieldElement:
    """
    Evaluate a coefficient form polynomial at ``z`` using Horner's schema
    """
    y = 0
    for coef in polynomial_coeff[::-1]:
        y = (int(y) * int(z) + int(coef)) % BLS_MODULUS
    return BLSFieldElement(y % BLS_MODULUS)

KZG multiproofs

Extended KZG functions for multiproofs

compute_kzg_proof_multi_impl

def compute_kzg_proof_multi_impl(
        polynomial_coeff: PolynomialCoeff,
        zs: Sequence[BLSFieldElement]) -> Tuple[KZGProof, Sequence[BLSFieldElement]]:
    """
    Helper function that computes multi-evaluation KZG proofs.
    """

    # For all x_i, compute p(x_i) - p(z)
    ys = [evaluate_polynomialcoeff(polynomial_coeff, z) for z in zs]
    interpolation_polynomial = interpolate_polynomialcoeff(zs, ys)
    polynomial_shifted = add_polynomialcoeff(polynomial_coeff, neg_polynomialcoeff(interpolation_polynomial))

    # For all x_i, compute (x_i - z)
    denominator_poly = vanishing_polynomialcoeff(zs)

    # Compute the quotient polynomial directly in evaluation form
    quotient_polynomial = divide_polynomialcoeff(polynomial_shifted, denominator_poly)

    return KZGProof(g1_lincomb(KZG_SETUP_G1_MONOMIAL[:len(quotient_polynomial)], quotient_polynomial)), ys

verify_kzg_proof_multi_impl

def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
                                zs: Sequence[BLSFieldElement],
                                ys: Sequence[BLSFieldElement],
                                proof: KZGProof) -> bool:
    """
    Helper function that verifies a KZG multiproof
    """

    assert len(zs) == len(ys)

    zero_poly = g2_lincomb(KZG_SETUP_G2_MONOMIAL[:len(zs) + 1], vanishing_polynomialcoeff(zs))
    interpolated_poly = g1_lincomb(KZG_SETUP_G1_MONOMIAL[:len(zs)], interpolate_polynomialcoeff(zs, ys))

    return (bls.pairing_check([
        [bls.bytes48_to_G1(proof), bls.bytes96_to_G2(zero_poly)],
        [
            bls.add(bls.bytes48_to_G1(commitment), bls.neg(bls.bytes48_to_G1(interpolated_poly))),
            bls.neg(bls.bytes96_to_G2(KZG_SETUP_G2_MONOMIAL[0])),
        ],
    ]))

Cell cosets

coset_for_cell

def coset_for_cell(cell_id: int) -> Cell:
    """
    Get the coset for a given ``cell_id``
    """
    assert cell_id < CELLS_PER_BLOB
    roots_of_unity_brp = bit_reversal_permutation(
        compute_roots_of_unity(2 * FIELD_ELEMENTS_PER_BLOB)
    )
    return Cell(roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * cell_id:FIELD_ELEMENTS_PER_CELL * (cell_id + 1)])

Cells

Cell computation

compute_cells_and_proofs

def compute_cells_and_proofs(blob: Blob) -> Tuple[
        Vector[Cell, CELLS_PER_BLOB],
        Vector[KZGProof, CELLS_PER_BLOB]]:
    """
    Compute all the cell proofs for one blob. This is an inefficient O(n^2) algorithm,
    for performant implementation the FK20 algorithm that runs in O(n log n) should be
    used instead.

    Public method.
    """
    polynomial = blob_to_polynomial(blob)
    polynomial_coeff = polynomial_eval_to_coeff(polynomial)

    cells = []
    proofs = []

    for i in range(CELLS_PER_BLOB):
        coset = coset_for_cell(i)
        proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset)
        cells.append(ys)
        proofs.append(proof)

    return cells, proofs

compute_cells

def compute_cells(blob: Blob) -> Vector[Cell, CELLS_PER_BLOB]:
    """
    Compute the cell data for a blob (without computing the proofs).

    Public method.
    """
    polynomial = blob_to_polynomial(blob)
    polynomial_coeff = polynomial_eval_to_coeff(polynomial)

    extended_data = fft_field(polynomial_coeff + [0] * FIELD_ELEMENTS_PER_BLOB, 
                              compute_roots_of_unity(2 * FIELD_ELEMENTS_PER_BLOB))
    extended_data_rbo = bit_reversal_permutation(extended_data)
    return [extended_data_rbo[i * FIELD_ELEMENTS_PER_CELL:(i + 1) * FIELD_ELEMENTS_PER_CELL]
            for i in range(CELLS_PER_BLOB)]

Cell verification

verify_cell_proof

def verify_cell_proof(commitment: KZGCommitment,
                      cell_id: int,
                      cell: Cell,
                      proof: KZGProof) -> bool:
    """
    Check a cell proof

    Public method.
    """
    coset = coset_for_cell(cell_id)

    return verify_kzg_proof_multi_impl(commitment, coset, cell, proof)

verify_cell_proof_batch

def verify_cell_proof_batch(row_commitments: Sequence[KZGCommitment],
                            row_ids: Sequence[int],
                            column_ids: Sequence[int],
                            cells: Sequence[Cell],
                            proofs: Sequence[KZGProof]) -> bool:
    """
    Check multiple cell proofs. This function implements the naive algorithm of checking every cell
    individually; an efficient algorithm can be found here:
    https://ethresear.ch/t/a-universal-verification-equation-for-data-availability-sampling/13240

    This implementation does not require randomness, but for the algorithm that
    requires it, `RANDOM_CHALLENGE_KZG_CELL_BATCH_DOMAIN` should be used to compute
    the challenge value.

    Public method.
    """

    # Get commitments via row IDs
    commitments = [row_commitments[row_id] for row_id in row_ids]
    
    return all(
        verify_kzg_proof_multi_impl(commitment, coset_for_cell(column_id), cell, proof)
        for commitment, column_id, cell, proof in zip(commitments, column_ids, cells, proofs)
    )

Reconstruction

recover_polynomial

def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Polynomial:
    """
    Recovers a polynomial from 2 * FIELD_ELEMENTS_PER_CELL evaluations, half of which can be missing.

    This algorithm uses FFTs to recover cells faster than using Lagrange implementation. However,
    a faster version thanks to Qi Zhou can be found here:
    https://github.com/ethereum/research/blob/51b530a53bd4147d123ab3e390a9d08605c2cdb8/polynomial_reconstruction/polynomial_reconstruction_danksharding.py

    Public method.
    """
    assert len(cell_ids) == len(cells)
    assert len(cells) >= CELLS_PER_BLOB // 2
    missing_cell_ids = [cell_id for cell_id in range(CELLS_PER_BLOB) if cell_id not in cell_ids]
    roots_of_unity_reduced = compute_roots_of_unity(CELLS_PER_BLOB)
    short_zero_poly = vanishing_polynomialcoeff([
        roots_of_unity_reduced[reverse_bits(cell_id, CELLS_PER_BLOB)]
        for cell_id in missing_cell_ids
    ])

    full_zero_poly = []
    for i in short_zero_poly:
        full_zero_poly.append(i)
        full_zero_poly.extend([0] * (FIELD_ELEMENTS_PER_CELL - 1))
    full_zero_poly = full_zero_poly + [0] * (2 * FIELD_ELEMENTS_PER_BLOB - len(full_zero_poly))

    zero_poly_eval = fft_field(full_zero_poly,
                               compute_roots_of_unity(2 * FIELD_ELEMENTS_PER_BLOB))
    zero_poly_eval_brp = bit_reversal_permutation(zero_poly_eval)
    for cell_id in missing_cell_ids:
        start = cell_id * FIELD_ELEMENTS_PER_CELL
        end = (cell_id + 1) * FIELD_ELEMENTS_PER_CELL
        assert zero_poly_eval_brp[start:end] == [0] * FIELD_ELEMENTS_PER_CELL
    for cell_id in cell_ids:
        start = cell_id * FIELD_ELEMENTS_PER_CELL
        end = (cell_id + 1) * FIELD_ELEMENTS_PER_CELL
        assert all(a != 0 for a in zero_poly_eval_brp[start:end])

    extended_evaluation_rbo = [0] * (FIELD_ELEMENTS_PER_BLOB * 2)
    for cell_id, cell in zip(cell_ids, cells):
        start = cell_id * FIELD_ELEMENTS_PER_CELL
        end = (cell_id + 1) * FIELD_ELEMENTS_PER_CELL
        extended_evaluation_rbo[start:end] = cell
    extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)

    extended_evaluation_times_zero = [BLSFieldElement(int(a) * int(b) % BLS_MODULUS)
                                      for a, b in zip(zero_poly_eval, extended_evaluation)]

    roots_of_unity_extended = compute_roots_of_unity(2 * FIELD_ELEMENTS_PER_BLOB)

    extended_evaluations_fft = fft_field(extended_evaluation_times_zero, roots_of_unity_extended, inv=True)

    shift_factor = BLSFieldElement(PRIMITIVE_ROOT_OF_UNITY)
    shift_inv = div(BLSFieldElement(1), shift_factor)

    shifted_extended_evaluation = shift_polynomialcoeff(extended_evaluations_fft, shift_factor)
    shifted_zero_poly = shift_polynomialcoeff(full_zero_poly, shift_factor)

    eval_shifted_extended_evaluation = fft_field(shifted_extended_evaluation, roots_of_unity_extended)
    eval_shifted_zero_poly = fft_field(shifted_zero_poly, roots_of_unity_extended)
    
    eval_shifted_reconstructed_poly = [
        div(a, b)
        for a, b in zip(eval_shifted_extended_evaluation, eval_shifted_zero_poly)
    ]

    shifted_reconstructed_poly = fft_field(eval_shifted_reconstructed_poly, roots_of_unity_extended, inv=True)

    reconstructed_poly = shift_polynomialcoeff(shifted_reconstructed_poly, shift_inv)

    reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly, roots_of_unity_extended))

    for cell_id, cell in zip(cell_ids, cells):
        start = cell_id * FIELD_ELEMENTS_PER_CELL
        end = (cell_id + 1) * FIELD_ELEMENTS_PER_CELL
        assert reconstructed_data[start:end] == cell

    return reconstructed_data