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# Deneb -- Polynomial Commitments
## Table of contents
<!-- TOC -->
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- [Introduction ](#introduction )
- [Custom types ](#custom-types )
- [Constants ](#constants )
- [Preset ](#preset )
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- [Cells ](#cells )
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- [Crypto ](#crypto )
- [Helper functions ](#helper-functions )
- [Linear combinations ](#linear-combinations )
- [`g2_lincomb` ](#g2_lincomb )
- [FFTs ](#ffts )
- [`_fft_field` ](#_fft_field )
- [`fft_field` ](#fft_field )
- [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 )
- [`shift_polynomialcoeff` ](#shift_polynomialcoeff )
- [`interpolate_polynomialcoeff` ](#interpolate_polynomialcoeff )
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- [`vanishing_polynomialcoeff` ](#vanishing_polynomialcoeff )
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- [`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 )
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- [Cell cosets ](#cell-cosets )
- [`coset_for_cell` ](#coset_for_cell )
- [Cells ](#cells-1 )
- [Cell computation ](#cell-computation )
- [`compute_cells_and_proofs` ](#compute_cells_and_proofs )
- [`compute_cells` ](#compute_cells )
- [Cell verification ](#cell-verification )
- [`verify_cell_proof` ](#verify_cell_proof )
- [`verify_cell_proof_batch` ](#verify_cell_proof_batch )
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- [Reconstruction ](#reconstruction )
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- [`recover_polynomial` ](#recover_polynomial )
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## Introduction
This document extends [polynomial-commitments.md ](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` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB]` | A polynomial in coefficient form |
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| `Cell` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]` | The unit of blob data that can come with their own KZG proofs |
| `CellID` | `uint64` | Cell identifier |
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## Constants
| Name | Value | Notes |
| - | - | - |
## Preset
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### 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).
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| Name | Value | Description |
| - | - | - |
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| `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 |
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### Crypto
| Name | Value | Description |
| - | - | - |
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| `ROOT_OF_UNITY_EXTENDED` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(FIELD_ELEMENTS_PER_BLOB * 2), BLS_MODULUS)` | Root of unity of order `FIELD_ELEMENTS_PER_BLOB * 2` over the BLS12-381 field |
| `ROOTS_OF_UNITY_EXTENDED` | `([BLSFieldElement(pow(ROOT_OF_UNITY_EXTENDED, i, BLS_MODULUS)) for i in range(FIELD_ELEMENTS_PER_BLOB * 2)])` | Roots of unity of order `FIELD_ELEMENTS_PER_BLOB * 2` over the BLS12-381 field |
| `ROOT_OF_UNITY_REDUCED` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(CELLS_PER_BLOB), BLS_MODULUS)` | Root of unity of order `CELLS_PER_BLOB` over the BLS12-381 field |
| `ROOTS_OF_UNITY_REDUCED` | `([BLSFieldElement(pow(ROOT_OF_UNITY_REDUCED, i, BLS_MODULUS)) for i in range(CELLS_PER_BLOB)])` | Roots of unity of order `CELLS_PER_BLOB` over the BLS12-381 field |
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## Helper functions
### Linear combinations
#### `g2_lincomb`
```python
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`
```python
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def _fft_field(vals: Sequence[BLSFieldElement],
roots_of_unity: Sequence[BLSFieldElement]) -> Sequence[BLSFieldElement]:
if len(vals) == 1:
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return vals
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L = _fft_field(vals[::2], roots_of_unity[::2])
R = _fft_field(vals[1::2], roots_of_unity[::2])
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o = [BLSFieldElement(0) for _ in vals]
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for i, (x, y) in enumerate(zip(L, R)):
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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)
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return o
```
#### `fft_field`
```python
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def fft_field(vals: Sequence[BLSFieldElement],
roots_of_unity: Sequence[BLSFieldElement],
inv: bool=False) -> Sequence[BLSFieldElement]:
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if inv:
# Inverse FFT
invlen = pow(len(vals), BLS_MODULUS - 2, BLS_MODULUS)
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return [BLSFieldElement((int(x) * invlen) % BLS_MODULUS)
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for x in _fft_field(vals, list(roots_of_unity[0:1]) + list(roots_of_unity[:0:-1]))]
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else:
# Regular FFT
return _fft_field(vals, roots_of_unity)
```
### 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.
"""
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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)
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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)
return [(a[i] + (b[i] if i < len ( b ) else 0 ) ) % BLS_MODULUS for i in range ( len ( 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``
"""
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 = [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`
```python
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def shift_polynomialcoeff(polynomial_coeff: PolynomialCoeff, factor: BLSFieldElement) -> PolynomialCoeff:
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"""
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 = []
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for p in polynomial_coeff:
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o.append(int(p) * factor_power % BLS_MODULUS)
factor_power = factor_power * inv_factor % BLS_MODULUS
return 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, [(- int(weight_adjustment) * int(xs[j])) % BLS_MODULUS, weight_adjustment]
)
r = add_polynomialcoeff(r, summand)
return r
```
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#### `vanishing_polynomialcoeff`
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```python
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def vanishing_polynomialcoeff(xs: Sequence[BLSFieldElement]) -> PolynomialCoeff:
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"""
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Compute the vanishing polynomial on ``xs`` (in coefficient form)
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"""
p = [1]
for x in xs:
p = multiply_polynomialcoeff(p, [-int(x), 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]:
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y = (int(y) * int(z) + int(coef)) % BLS_MODULUS
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return BLSFieldElement(y % BLS_MODULUS)
```
### KZG multiproofs
Extended KZG functions for multiproofs
#### `compute_kzg_proof_multi_impl`
```python
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def compute_kzg_proof_multi_impl(
polynomial_coeff: PolynomialCoeff,
zs: Sequence[BLSFieldElement]) -> Tuple[KZGProof, Sequence[BLSFieldElement]]:
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"""
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)
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denominator_poly = vanishing_polynomialcoeff(zs)
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# Compute the quotient polynomial directly in evaluation form
quotient_polynomial = divide_polynomialcoeff(polynomial_shifted, denominator_poly)
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return KZGProof(g1_lincomb(KZG_SETUP_G1_MONOMIAL[:len(quotient_polynomial)], quotient_polynomial)), ys
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```
#### `verify_kzg_proof_multi_impl`
```python
def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
zs: BLSFieldElement,
ys: BLSFieldElement,
proof: KZGProof) -> bool:
"""
Helper function that verifies a KZG multiproof
"""
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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))
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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))),
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bls.neg(bls.bytes96_to_G2(KZG_SETUP_G2_MONOMIAL[0])),
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],
]))
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```
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### Cell cosets
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#### `coset_for_cell`
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```python
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def coset_for_cell(cell_id: int) -> Cell:
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"""
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Get the coset for a given ``cell_id``
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"""
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assert cell_id < CELLS_PER_BLOB
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roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY_EXTENDED)
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return Cell(roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * cell_id:FIELD_ELEMENTS_PER_CELL * (cell_id + 1)])
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```
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## Cells
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### Cell computation
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#### `compute_cells_and_proofs`
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```python
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def compute_cells_and_proofs(blob: Blob) -> Tuple[
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Vector[Cell, CELLS_PER_BLOB],
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Vector[KZGProof, CELLS_PER_BLOB]]:
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"""
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Compute all the cell proofs for one blob. This is an inefficient O(n^2) algorithm,
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for performant implementation the FK20 algorithm that runs in O(n log n) should be
used instead.
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Public method.
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"""
polynomial = blob_to_polynomial(blob)
polynomial_coeff = polynomial_eval_to_coeff(polynomial)
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cells = []
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proofs = []
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for i in range(CELLS_PER_BLOB):
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coset = coset_for_cell(i)
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proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset)
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cells.append(ys)
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proofs.append(proof)
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return cells, proofs
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```
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#### `compute_cells`
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```python
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def compute_cells(blob: Blob) -> Vector[Cell, CELLS_PER_BLOB]:
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"""
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Compute the cell data for a blob (without computing the proofs).
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Public method.
"""
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polynomial = blob_to_polynomial(blob)
polynomial_coeff = polynomial_eval_to_coeff(polynomial)
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extended_data = fft_field(polynomial_coeff + [0] * FIELD_ELEMENTS_PER_BLOB, ROOTS_OF_UNITY_EXTENDED)
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extended_data_rbo = bit_reversal_permutation(extended_data)
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return [extended_data_rbo[i * FIELD_ELEMENTS_PER_CELL:(i + 1) * FIELD_ELEMENTS_PER_CELL]
for i in range(CELLS_PER_BLOB)]
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```
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### Cell verification
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#### `verify_cell_proof`
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```python
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def verify_cell_proof(commitment: KZGCommitment,
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cell_id: int,
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cell: Cell,
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proof: KZGProof) -> bool:
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"""
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Check a cell proof
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Public method.
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"""
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coset = coset_for_cell(cell_id)
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return verify_kzg_proof_multi_impl(commitment, coset, cell, proof)
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```
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#### `verify_cell_proof_batch`
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```python
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def verify_cell_proof_batch(row_commitments: Sequence[KZGCommitment],
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row_ids: Sequence[int],
column_ids: Sequence[int],
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cells: Sequence[Cell],
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proofs: Sequence[KZGProof]) -> bool:
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"""
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Check multiple cell proofs. This function implements the naive algorithm of checking every cell
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individually; an efficient algorithm can be found here:
https://ethresear.ch/t/a-universal-verification-equation-for-data-availability-sampling/13240
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Public method.
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"""
# Get commitments via row IDs
commitments = [row_commitments[row_id] for row_id in row_ids]
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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)
)
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```
## Reconstruction
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### `recover_polynomial`
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```python
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def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Polynomial:
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"""
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Recovers a polynomial from 2 * FIELD_ELEMENTS_PER_CELL evaluations, half of which can be missing.
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This algorithm uses FFTs to recover cells faster than using Lagrange implementation. However,
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a faster version thanks to Qi Zhou can be found here:
https://github.com/ethereum/research/blob/51b530a53bd4147d123ab3e390a9d08605c2cdb8/polynomial_reconstruction/polynomial_reconstruction_danksharding.py
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Public method.
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"""
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assert len(cell_ids) == len(cells)
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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]
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short_zero_poly = vanishing_polynomialcoeff([
ROOTS_OF_UNITY_REDUCED[reverse_bits(cell_id, CELLS_PER_BLOB)]
for cell_id in missing_cell_ids
])
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full_zero_poly = []
for i in short_zero_poly:
full_zero_poly.append(i)
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full_zero_poly.extend([0] * (FIELD_ELEMENTS_PER_CELL - 1))
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full_zero_poly = full_zero_poly + [0] * (2 * FIELD_ELEMENTS_PER_BLOB - len(full_zero_poly))
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zero_poly_eval = fft_field(full_zero_poly, ROOTS_OF_UNITY_EXTENDED)
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zero_poly_eval_brp = bit_reversal_permutation(zero_poly_eval)
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for cell_id in missing_cell_ids:
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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
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for cell_id in cell_ids:
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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])
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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
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extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
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extended_evaluation_times_zero = [BLSFieldElement(a * b % BLS_MODULUS)
for a, b in zip(zero_poly_eval, extended_evaluation)]
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extended_evaluations_fft = fft_field(extended_evaluation_times_zero, ROOTS_OF_UNITY_EXTENDED, inv=True)
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shift_factor = BLSFieldElement(PRIMITIVE_ROOT_OF_UNITY)
shift_inv = div(BLSFieldElement(1), shift_factor)
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shifted_extended_evaluation = shift_polynomialcoeff(extended_evaluations_fft, shift_factor)
shifted_zero_poly = shift_polynomialcoeff(full_zero_poly, shift_factor)
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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)
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eval_shifted_reconstructed_poly = [
div(a, b)
for a, b in zip(eval_shifted_extended_evaluation, eval_shifted_zero_poly)
]
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shifted_reconstructed_poly = fft_field(eval_shifted_reconstructed_poly, ROOTS_OF_UNITY_EXTENDED, inv=True)
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reconstructed_poly = shift_polynomialcoeff(shifted_reconstructed_poly, shift_inv)
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reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly, ROOTS_OF_UNITY_EXTENDED))
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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
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return reconstructed_data
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```