# EIP-7594 -- Polynomial Commitments Sampling ## Table of contents - [Introduction](#introduction) - [Public Methods](#public-methods) - [Custom types](#custom-types) - [Constants](#constants) - [Preset](#preset) - [Cells](#cells) - [Helper functions](#helper-functions) - [BLS12-381 helpers](#bls12-381-helpers) - [`cell_to_coset_evals`](#cell_to_coset_evals) - [`coset_evals_to_cell`](#coset_evals_to_cell) - [Linear combinations](#linear-combinations) - [`g2_lincomb`](#g2_lincomb) - [FFTs](#ffts) - [`_fft_field`](#_fft_field) - [`fft_field`](#fft_field) - [`coset_fft_field`](#coset_fft_field) - [`compute_verify_cell_kzg_proof_batch_challenge`](#compute_verify_cell_kzg_proof_batch_challenge) - [Polynomials in coefficient form](#polynomials-in-coefficient-form) - [`polynomial_eval_to_coeff`](#polynomial_eval_to_coeff) - [`add_polynomialcoeff`](#add_polynomialcoeff) - [`neg_polynomialcoeff`](#neg_polynomialcoeff) - [`multiply_polynomialcoeff`](#multiply_polynomialcoeff) - [`divide_polynomialcoeff`](#divide_polynomialcoeff) - [`interpolate_polynomialcoeff`](#interpolate_polynomialcoeff) - [`vanishing_polynomialcoeff`](#vanishing_polynomialcoeff) - [`evaluate_polynomialcoeff`](#evaluate_polynomialcoeff) - [KZG multiproofs](#kzg-multiproofs) - [`compute_kzg_proof_multi_impl`](#compute_kzg_proof_multi_impl) - [`verify_kzg_proof_multi_impl`](#verify_kzg_proof_multi_impl) - [`verify_cell_kzg_proof_batch_impl`](#verify_cell_kzg_proof_batch_impl) - [Cell cosets](#cell-cosets) - [`coset_shift_for_cell`](#coset_shift_for_cell) - [`coset_for_cell`](#coset_for_cell) - [Cells](#cells-1) - [Cell computation](#cell-computation) - [`compute_cells_and_kzg_proofs`](#compute_cells_and_kzg_proofs) - [Cell verification](#cell-verification) - [`verify_cell_kzg_proof`](#verify_cell_kzg_proof) - [`verify_cell_kzg_proof_batch`](#verify_cell_kzg_proof_batch) - [Reconstruction](#reconstruction) - [`construct_vanishing_polynomial`](#construct_vanishing_polynomial) - [`recover_data`](#recover_data) - [`recover_cells_and_kzg_proofs`](#recover_cells_and_kzg_proofs) ## Introduction This document extends [polynomial-commitments.md](../../deneb/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. ## Public Methods 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. The following is a list of the public methods: - [`compute_cells_and_kzg_proofs`](#compute_cells_and_kzg_proofs) - [`verify_cell_kzg_proof`](#verify_cell_kzg_proof) - [`verify_cell_kzg_proof_batch`](#verify_cell_kzg_proof_batch) - [`recover_cells_and_kzg_proofs`](#recover_cells_and_kzg_proofs) ## Custom types | Name | SSZ equivalent | Description | | - | - | - | | `PolynomialCoeff` | `List[BLSFieldElement, FIELD_ELEMENTS_PER_EXT_BLOB]` | A polynomial in coefficient form | | `Coset` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]` | The evaluation domain of a cell | | `CosetEvals` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]` | The internal representation of a cell (the evaluations over its Coset) | | `Cell` | `ByteVector[BYTES_PER_FIELD_ELEMENT * FIELD_ELEMENTS_PER_CELL]` | The unit of blob data that can come with its own KZG proof | | `CellIndex` | `uint64` | Validation: `x < CELLS_PER_EXT_BLOB` | ## 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_EXT_BLOB` | `2 * FIELD_ELEMENTS_PER_BLOB` | Number of field elements in a Reed-Solomon extended blob | | `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_EXT_BLOB` | `FIELD_ELEMENTS_PER_EXT_BLOB // FIELD_ELEMENTS_PER_CELL` | The number of cells in an extended blob | | `RANDOM_CHALLENGE_KZG_CELL_BATCH_DOMAIN` | `b'RCKZGCBATCH__V1_'` | ## Helper functions ### BLS12-381 helpers #### `cell_to_coset_evals` ```python def cell_to_coset_evals(cell: Cell) -> CosetEvals: """ Convert an untrusted ``Cell`` into a trusted ``CosetEvals``. """ evals = [] for i in range(FIELD_ELEMENTS_PER_CELL): start = i * BYTES_PER_FIELD_ELEMENT end = (i + 1) * BYTES_PER_FIELD_ELEMENT value = bytes_to_bls_field(cell[start:end]) evals.append(value) return CosetEvals(evals) ``` #### `coset_evals_to_cell` ```python def coset_evals_to_cell(coset_evals: CosetEvals) -> Cell: """ Convert a trusted ``CosetEval`` into an untrusted ``Cell``. """ cell = [] for i in range(FIELD_ELEMENTS_PER_CELL): cell += bls_field_to_bytes(coset_evals[i]) return Cell(cell) ``` ### Linear combinations #### `g2_lincomb` ```python def g2_lincomb(points: Sequence[G2Point], scalars: Sequence[BLSFieldElement]) -> Bytes96: """ BLS multiscalar multiplication in G2. This can be naively implemented using double-and-add. """ assert len(points) == len(scalars) if len(points) == 0: return bls.G2_to_bytes96(bls.Z2()) points_g2 = [] for point in points: points_g2.append(bls.bytes96_to_G2(point)) result = bls.multi_exp(points_g2, scalars) return Bytes96(bls.G2_to_bytes96(result)) ``` ### FFTs #### `_fft_field` ```python 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` ```python 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) ``` #### `coset_fft_field` ```python def coset_fft_field(vals: Sequence[BLSFieldElement], roots_of_unity: Sequence[BLSFieldElement], inv: bool=False) -> Sequence[BLSFieldElement]: """ Computes an FFT/IFFT over a coset of the roots of unity. This is useful for when one wants to divide by a polynomial which vanishes on one or more elements in the domain. """ vals = vals.copy() def shift_vals(vals: Sequence[BLSFieldElement], factor: BLSFieldElement) -> Sequence[BLSFieldElement]: """ Multiply each entry in `vals` by succeeding powers of `factor` i.e., [vals[0] * factor^0, vals[1] * factor^1, ..., vals[n] * factor^n] """ shift = 1 for i in range(len(vals)): vals[i] = BLSFieldElement((int(vals[i]) * shift) % BLS_MODULUS) shift = (shift * int(factor)) % BLS_MODULUS return vals # This is the coset generator; it is used to compute a FFT/IFFT over a coset of # the roots of unity. shift_factor = BLSFieldElement(PRIMITIVE_ROOT_OF_UNITY) if inv: vals = fft_field(vals, roots_of_unity, inv) shift_inv = bls_modular_inverse(shift_factor) return shift_vals(vals, shift_inv) else: vals = shift_vals(vals, shift_factor) return fft_field(vals, roots_of_unity, inv) ``` #### `compute_verify_cell_kzg_proof_batch_challenge` ```python def compute_verify_cell_kzg_proof_batch_challenge(row_commitments: Sequence[KZGCommitment], row_indices: Sequence[RowIndex], column_indices: Sequence[ColumnIndex], cosets_evals: Sequence[CosetEvals], proofs: Sequence[KZGProof]) -> BLSFieldElement: """ Compute a random challenge r used in the universal verification equation. This is used in verify_cell_kzg_proof_batch_impl. To compute the challenge, `RANDOM_CHALLENGE_KZG_CELL_BATCH_DOMAIN` is used as a hash prefix. """ # input the domain separator hashinput = RANDOM_CHALLENGE_KZG_CELL_BATCH_DOMAIN # input the degree bound of the polynomial hashinput += int.to_bytes(FIELD_ELEMENTS_PER_BLOB, 8, KZG_ENDIANNESS) # input the field elements per cell hashinput += int.to_bytes(FIELD_ELEMENTS_PER_CELL, 8, KZG_ENDIANNESS) # input the number of commitments num_commitments = len(row_commitments) hashinput += int.to_bytes(num_commitments, 8, KZG_ENDIANNESS) # input the number of cells num_cells = len(row_indices) hashinput += int.to_bytes(num_cells, 8, KZG_ENDIANNESS) # input all commitments for commitment in row_commitments: hashinput += commitment # input each cell with its indices and proof for k in range(num_cells): hashinput += int.to_bytes(row_indices[k], 8, KZG_ENDIANNESS) hashinput += int.to_bytes(column_indices[k], 8, KZG_ENDIANNESS) for eval in cosets_evals[k]: hashinput += bls_field_to_bytes(eval) hashinput += proofs[k] return hash_to_bls_field(hashinput) ``` ### Polynomials in coefficient form #### `polynomial_eval_to_coeff` ```python 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` ```python 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) length_a = len(a) length_b = len(b) return [(a[i] + (b[i] if i < length_b else 0)) % BLS_MODULUS for i in range(length_a)] ``` #### `neg_polynomialcoeff` ```python def neg_polynomialcoeff(a: PolynomialCoeff) -> PolynomialCoeff: """ Negative of coefficient form polynomial ``a`` """ return [(BLS_MODULUS - x) % BLS_MODULUS for x in a] ``` #### `multiply_polynomialcoeff` ```python def multiply_polynomialcoeff(a: PolynomialCoeff, b: PolynomialCoeff) -> PolynomialCoeff: """ Multiplies the coefficient form polynomials ``a`` and ``b`` """ assert len(a) + len(b) <= FIELD_ELEMENTS_PER_EXT_BLOB 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` ```python def divide_polynomialcoeff(a: PolynomialCoeff, b: PolynomialCoeff) -> PolynomialCoeff: """ Long polynomial division for two coefficient form polynomials ``a`` and ``b`` """ a = a.copy() # Make a copy since `a` is passed by reference o: List[BLSFieldElement] = [] 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] + BLS_MODULUS) * int(quot)) % BLS_MODULUS apos -= 1 diff -= 1 return [x % BLS_MODULUS for x in o] ``` #### `interpolate_polynomialcoeff` ```python 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, [((BLS_MODULUS - int(weight_adjustment)) * int(xs[j])) % BLS_MODULUS, weight_adjustment] ) r = add_polynomialcoeff(r, summand) return r ``` #### `vanishing_polynomialcoeff` ```python 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) + BLS_MODULUS, 1]) return p ``` #### `evaluate_polynomialcoeff` ```python 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` ```python def compute_kzg_proof_multi_impl( polynomial_coeff: PolynomialCoeff, zs: Coset) -> Tuple[KZGProof, CosetEvals]: """ Compute a KZG multi-evaluation proof for a set of `k` points. This is done by committing to the following quotient polynomial: Q(X) = f(X) - I(X) / Z(X) Where: - I(X) is the degree `k-1` polynomial that agrees with f(x) at all `k` points - Z(X) is the degree `k` polynomial that evaluates to zero on all `k` points We further note that since the degree of I(X) is less than the degree of Z(X), the computation can be simplified in monomial form to Q(X) = f(X) / Z(X) """ # For all points, compute the evaluation of those points ys = [evaluate_polynomialcoeff(polynomial_coeff, z) for z in zs] # Compute Z(X) denominator_poly = vanishing_polynomialcoeff(zs) # Compute the quotient polynomial directly in monomial form quotient_polynomial = divide_polynomialcoeff(polynomial_coeff, denominator_poly) return KZGProof(g1_lincomb(KZG_SETUP_G1_MONOMIAL[:len(quotient_polynomial)], quotient_polynomial)), ys ``` #### `verify_kzg_proof_multi_impl` ```python def verify_kzg_proof_multi_impl(commitment: KZGCommitment, zs: Coset, ys: CosetEvals, proof: KZGProof) -> bool: """ Verify a KZG multi-evaluation proof for a set of `k` points. This is done by checking if the following equation holds: Q(x) Z(x) = f(X) - I(X) Where: f(X) is the polynomial that we want to verify opens at `k` points to `k` values Q(X) is the quotient polynomial computed by the prover I(X) is the degree k-1 polynomial that evaluates to `ys` at all `zs`` points Z(X) is the polynomial that evaluates to zero on all `k` points The verifier receives the commitments to Q(X) and f(X), so they check the equation holds by using the following pairing equation: e([Q(X)]_1, [Z(X)]_2) == e([f(X)]_1 - [I(X)]_1, [1]_2) """ assert len(zs) == len(ys) # Compute [Z(X)]_2 zero_poly = g2_lincomb(KZG_SETUP_G2_MONOMIAL[:len(zs) + 1], vanishing_polynomialcoeff(zs)) # Compute [I(X)]_1 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])), ], ])) ``` #### `verify_cell_kzg_proof_batch_impl` ```python def verify_cell_kzg_proof_batch_impl(row_commitments: Sequence[KZGCommitment], row_indices: Sequence[RowIndex], column_indices: Sequence[ColumnIndex], cosets_evals: Sequence[CosetEvals], proofs: Sequence[KZGProof]) -> bool: """ Verify a set of cells, given their corresponding proofs and their coordinates (row_index, column_index) in the blob matrix. The i-th cell is in row row_indices[i] and in column column_indices[i]. The list of all commitments is provided in row_commitments_bytes. This function is the internal implementation of verify_cell_kzg_proof_batch. """ # The verification equation that we will check is pairing (LL, LR) = pairing (RL, [1]), where # LL = sum_k r^k proofs[k], # LR = [s^n] # RL = RLC - RLI + RLP, where # RLC = sum_i weights[i] commitments[i] # RLI = [sum_k r^k interpolation_poly_k(s)] # RLP = sum_k (r^k * h_k^n) proofs[k] # # Here, the variables have the following meaning: # - k < len(row_indices) is an index iterating over all cells in the input # - r is a random coefficient, derived from hashing all data provided by the prover # - s is the secret embedded in the KZG setup # - n = FIELD_ELEMENTS_PER_CELL is the size of the evaluation domain # - i ranges over all rows that are touched # - weights[i] is a weight computed for row i. It depends on r and on which cells are in row i # - interpolation_poly_k is the interpolation polynomial for the kth cell # - h_k is the coset shift specifying the evaluation domain of the kth cell # Preparation num_cells = len(row_indices) n = FIELD_ELEMENTS_PER_CELL num_rows = len(row_commitments) # Step 1: Compute a challenge r and its powers r^0, ..., r^{num_cells-1} r = compute_verify_cell_kzg_proof_batch_challenge( row_commitments, row_indices, column_indices, cosets_evals, proofs ) r_powers = compute_powers(r, num_cells) # Step 2: Compute LL = sum_k r^k proofs[k] ll = bls.bytes48_to_G1(g1_lincomb(proofs, r_powers)) # Step 3: Compute LR = [s^n] lr = bls.bytes96_to_G2(KZG_SETUP_G2_MONOMIAL[n]) # Step 4: Compute RL = RLC - RLI + RLP # Step 4.1: Compute RLC = sum_i weights[i] commitments[i] # Step 4.1a: Compute weights[i]: the sum of all r^k for which cell k is in row i. # Note: we do that by iterating over all k and updating the correct weights[i] accordingly weights = [0] * num_rows for k in range(num_cells): i = row_indices[k] weights[i] = (weights[i] + int(r_powers[k])) % BLS_MODULUS # Step 4.1b: Linearly combine the weights with the commitments to get RLC rlc = bls.bytes48_to_G1(g1_lincomb(row_commitments, weights)) # Step 4.2: Compute RLI = [sum_k r^k interpolation_poly_k(s)] # Note: an efficient implementation would use the IDFT based method explained in the blog post sum_interp_polys_coeff = [0] for k in range(num_cells): interp_poly_coeff = interpolate_polynomialcoeff(coset_for_cell(column_indices[k]), cosets_evals[k]) interp_poly_scaled_coeff = multiply_polynomialcoeff([r_powers[k]], interp_poly_coeff) sum_interp_polys_coeff = add_polynomialcoeff(sum_interp_polys_coeff, interp_poly_scaled_coeff) rli = bls.bytes48_to_G1(g1_lincomb(KZG_SETUP_G1_MONOMIAL[:n], sum_interp_polys_coeff)) # Step 4.3: Compute RLP = sum_k (r^k * h_k^n) proofs[k] weighted_r_powers = [] for k in range(num_cells): h_k = int(coset_shift_for_cell(column_indices[k])) h_k_pow = pow(h_k, n, BLS_MODULUS) wrp = (int(r_powers[k]) * h_k_pow) % BLS_MODULUS weighted_r_powers.append(wrp) rlp = bls.bytes48_to_G1(g1_lincomb(proofs, weighted_r_powers)) # Step 4.4: Compute RL = RLC - RLI + RLP rl = bls.add(rlc, bls.neg(rli)) rl = bls.add(rl, rlp) # Step 5: Check pairing (LL, LR) = pairing (RL, [1]) return (bls.pairing_check([ [ll, lr], [rl, bls.neg(bls.bytes96_to_G2(KZG_SETUP_G2_MONOMIAL[0]))], ])) ``` ### Cell cosets #### `coset_shift_for_cell` ```python def coset_shift_for_cell(cell_index: CellIndex) -> BLSFieldElement: """ Get the shift that determines the coset for a given ``cell_index``. Precisely, consider the group of roots of unity of order FIELD_ELEMENTS_PER_CELL * CELLS_PER_EXT_BLOB. Let G = {1, g, g^2, ...} denote its subgroup of order FIELD_ELEMENTS_PER_CELL. Then, the coset is defined as h * G = {h, hg, hg^2, ...} for an element h. This function returns h. """ assert cell_index < CELLS_PER_EXT_BLOB roots_of_unity_brp = bit_reversal_permutation( compute_roots_of_unity(FIELD_ELEMENTS_PER_EXT_BLOB) ) return roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * cell_index] ``` #### `coset_for_cell` ```python def coset_for_cell(cell_index: CellIndex) -> Coset: """ Get the coset for a given ``cell_index``. Precisely, consider the group of roots of unity of order FIELD_ELEMENTS_PER_CELL * CELLS_PER_EXT_BLOB. Let G = {1, g, g^2, ...} denote its subgroup of order FIELD_ELEMENTS_PER_CELL. Then, the coset is defined as h * G = {h, hg, hg^2, ...}. This function, returns the coset. """ assert cell_index < CELLS_PER_EXT_BLOB roots_of_unity_brp = bit_reversal_permutation( compute_roots_of_unity(FIELD_ELEMENTS_PER_EXT_BLOB) ) return Coset(roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * cell_index:FIELD_ELEMENTS_PER_CELL * (cell_index + 1)]) ``` ## Cells ### Cell computation #### `compute_cells_and_kzg_proofs` ```python def compute_cells_and_kzg_proofs(blob: Blob) -> Tuple[ Vector[Cell, CELLS_PER_EXT_BLOB], Vector[KZGProof, CELLS_PER_EXT_BLOB]]: """ Compute all the cell proofs for an extended 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. """ assert len(blob) == BYTES_PER_BLOB polynomial = blob_to_polynomial(blob) polynomial_coeff = polynomial_eval_to_coeff(polynomial) cells = [] proofs = [] for i in range(CELLS_PER_EXT_BLOB): coset = coset_for_cell(CellIndex(i)) proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset) cells.append(coset_evals_to_cell(ys)) proofs.append(proof) return cells, proofs ``` ### Cell verification #### `verify_cell_kzg_proof` ```python def verify_cell_kzg_proof(commitment_bytes: Bytes48, cell_index: CellIndex, cell: Cell, proof_bytes: Bytes48) -> bool: """ Check a cell proof Public method. """ assert len(commitment_bytes) == BYTES_PER_COMMITMENT assert cell_index < CELLS_PER_EXT_BLOB assert len(cell) == BYTES_PER_CELL assert len(proof_bytes) == BYTES_PER_PROOF coset = coset_for_cell(cell_index) return verify_kzg_proof_multi_impl( bytes_to_kzg_commitment(commitment_bytes), coset, cell_to_coset_evals(cell), bytes_to_kzg_proof(proof_bytes)) ``` #### `verify_cell_kzg_proof_batch` ```python def verify_cell_kzg_proof_batch(row_commitments_bytes: Sequence[Bytes48], row_indices: Sequence[RowIndex], column_indices: Sequence[ColumnIndex], cells: Sequence[Cell], proofs_bytes: Sequence[Bytes48]) -> bool: """ Verify a set of cells, given their corresponding proofs and their coordinates (row_index, column_index) in the blob matrix. The i-th cell is in row = row_indices[i] and in column = column_indices[i]. The list of all commitments is provided in row_commitments_bytes. This function implements the universal verification equation that has been introduced here: https://ethresear.ch/t/a-universal-verification-equation-for-data-availability-sampling/13240 Public method. """ assert len(cells) == len(proofs_bytes) == len(row_indices) == len(column_indices) for commitment_bytes in row_commitments_bytes: assert len(commitment_bytes) == BYTES_PER_COMMITMENT for row_index in row_indices: assert row_index < len(row_commitments_bytes) for column_index in column_indices: assert column_index < CELLS_PER_EXT_BLOB for cell in cells: assert len(cell) == BYTES_PER_CELL for proof_bytes in proofs_bytes: assert len(proof_bytes) == BYTES_PER_PROOF # Get objects from bytes row_commitments = [bytes_to_kzg_commitment(commitment_bytes) for commitment_bytes in row_commitments_bytes] cosets_evals = [cell_to_coset_evals(cell) for cell in cells] proofs = [bytes_to_kzg_proof(proof_bytes) for proof_bytes in proofs_bytes] # Do the actual verification return verify_cell_kzg_proof_batch_impl(row_commitments, row_indices, column_indices, cosets_evals, proofs) ``` ## Reconstruction ### `construct_vanishing_polynomial` ```python def construct_vanishing_polynomial(missing_cell_indices: Sequence[CellIndex]) -> Sequence[BLSFieldElement]: """ Given the cells indices that are missing from the data, compute the polynomial that vanishes at every point that corresponds to a missing field element. This method assumes that all of the cells cannot be missing. In this case the vanishing polynomial could be computed as Z(x) = x^n - 1, where `n` is FIELD_ELEMENTS_PER_EXT_BLOB. We never encounter this case however because this method is used solely for recovery and recovery only works if at least half of the cells are available. """ # Get the small domain roots_of_unity_reduced = compute_roots_of_unity(CELLS_PER_EXT_BLOB) # Compute polynomial that vanishes at all the missing cells (over the small domain) short_zero_poly = vanishing_polynomialcoeff([ roots_of_unity_reduced[reverse_bits(missing_cell_index, CELLS_PER_EXT_BLOB)] for missing_cell_index in missing_cell_indices ]) # Extend vanishing polynomial to full domain using the closed form of the vanishing polynomial over a coset zero_poly_coeff = [BLSFieldElement(0)] * FIELD_ELEMENTS_PER_EXT_BLOB for i, coeff in enumerate(short_zero_poly): zero_poly_coeff[i * FIELD_ELEMENTS_PER_CELL] = coeff return zero_poly_coeff ``` ### `recover_data` ```python def recover_data(cell_indices: Sequence[CellIndex], cells: Sequence[Cell], ) -> Sequence[BLSFieldElement]: """ Recover the missing evaluations for the extended blob, given at least half of the evaluations. """ # Get the extended domain. This will be referred to as the FFT domain. roots_of_unity_extended = compute_roots_of_unity(FIELD_ELEMENTS_PER_EXT_BLOB) # Flatten the cells into evaluations. # If a cell is missing, then its evaluation is zero. extended_evaluation_rbo = [0] * FIELD_ELEMENTS_PER_EXT_BLOB for cell_index, cell in zip(cell_indices, cells): start = cell_index * FIELD_ELEMENTS_PER_CELL end = (cell_index + 1) * FIELD_ELEMENTS_PER_CELL extended_evaluation_rbo[start:end] = cell extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo) # Compute Z(x) in monomial form # Z(x) is the polynomial which vanishes on all of the evaluations which are missing missing_cell_indices = [CellIndex(cell_index) for cell_index in range(CELLS_PER_EXT_BLOB) if cell_index not in cell_indices] zero_poly_coeff = construct_vanishing_polynomial(missing_cell_indices) # Convert Z(x) to evaluation form over the FFT domain zero_poly_eval = fft_field(zero_poly_coeff, roots_of_unity_extended) # Compute (E*Z)(x) = E(x) * Z(x) in evaluation form over the FFT domain extended_evaluation_times_zero = [BLSFieldElement(int(a) * int(b) % BLS_MODULUS) for a, b in zip(zero_poly_eval, extended_evaluation)] # Convert (E*Z)(x) to monomial form extended_evaluation_times_zero_coeffs = fft_field(extended_evaluation_times_zero, roots_of_unity_extended, inv=True) # Convert (E*Z)(x) to evaluation form over a coset of the FFT domain extended_evaluations_over_coset = coset_fft_field(extended_evaluation_times_zero_coeffs, roots_of_unity_extended) # Convert Z(x) to evaluation form over a coset of the FFT domain zero_poly_over_coset = coset_fft_field(zero_poly_coeff, roots_of_unity_extended) # Compute Q_3(x) = (E*Z)(x) / Z(x) in evaluation form over a coset of the FFT domain reconstructed_poly_over_coset = [ div(a, b) for a, b in zip(extended_evaluations_over_coset, zero_poly_over_coset) ] # Convert Q_3(x) to monomial form reconstructed_poly_coeff = coset_fft_field(reconstructed_poly_over_coset, roots_of_unity_extended, inv=True) # Convert Q_3(x) to evaluation form over the FFT domain and bit reverse the result reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly_coeff, roots_of_unity_extended)) return reconstructed_data ``` ### `recover_cells_and_kzg_proofs` ```python def recover_cells_and_kzg_proofs(cell_indices: Sequence[CellIndex], cells: Sequence[Cell]) -> Tuple[ Vector[Cell, CELLS_PER_EXT_BLOB], Vector[KZGProof, CELLS_PER_EXT_BLOB]]: """ Given at least 50% of cells/proofs for a blob, recover all the cells/proofs. This algorithm uses FFTs to recover cells faster than using Lagrange implementation, as can be seen here: https://ethresear.ch/t/reed-solomon-erasure-code-recovery-in-n-log-2-n-time-with-ffts/3039 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_indices) == len(cells) # Check we have enough cells to be able to perform the reconstruction assert CELLS_PER_EXT_BLOB / 2 <= len(cell_indices) <= CELLS_PER_EXT_BLOB # Check for duplicates assert len(cell_indices) == len(set(cell_indices)) # Check that the cell indices are within bounds for cell_index in cell_indices: assert cell_index < CELLS_PER_EXT_BLOB # Check that each cell is the correct length for cell in cells: assert len(cell) == BYTES_PER_CELL # Convert cells to coset evals cosets_evals = [cell_to_coset_evals(cell) for cell in cells] reconstructed_data = recover_data(cell_indices, cosets_evals) for cell_index, coset_evals in zip(cell_indices, cosets_evals): start = cell_index * FIELD_ELEMENTS_PER_CELL end = (cell_index + 1) * FIELD_ELEMENTS_PER_CELL assert reconstructed_data[start:end] == coset_evals recovered_cells = [ coset_evals_to_cell(reconstructed_data[i * FIELD_ELEMENTS_PER_CELL:(i + 1) * FIELD_ELEMENTS_PER_CELL]) for i in range(CELLS_PER_EXT_BLOB)] polynomial_eval = reconstructed_data[:FIELD_ELEMENTS_PER_BLOB] polynomial_coeff = polynomial_eval_to_coeff(polynomial_eval) recovered_proofs = [None] * CELLS_PER_EXT_BLOB for i in range(CELLS_PER_EXT_BLOB): coset = coset_for_cell(CellIndex(i)) proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset) assert coset_evals_to_cell(ys) == recovered_cells[i] recovered_proofs[i] = proof return recovered_cells, recovered_proofs ```