From 94ecce76eec22de494d92e9091b0eb41d7e7ac69 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Mon, 4 Dec 2023 01:12:07 +0000
Subject: [PATCH 01/28] Add polynomial commitments for sampling

---
 .../sharding/polynomial-commitments.md        |   3 +-
 .../deneb/polynomial-commitments-sampling.md  | 530 ++++++++++++++++++
 specs/deneb/polynomial-commitments.md         |  11 +-
 3 files changed, 542 insertions(+), 2 deletions(-)
 create mode 100644 specs/deneb/polynomial-commitments-sampling.md

diff --git a/specs/_features/sharding/polynomial-commitments.md b/specs/_features/sharding/polynomial-commitments.md
index 7ef01853ce..f52d950664 100644
--- a/specs/_features/sharding/polynomial-commitments.md
+++ b/specs/_features/sharding/polynomial-commitments.md
@@ -58,6 +58,7 @@ This document specifies basic polynomial operations and KZG polynomial commitmen
 | `BLS_MODULUS` | `0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001` (curve order of BLS12_381) |
 | `PRIMITIVE_ROOT_OF_UNITY` | `7` | Primitive root of unity of the BLS12_381 (inner) BLS_MODULUS |
 
+
 ### KZG Trusted setup
 
 | Name | Value |
@@ -103,7 +104,7 @@ def reverse_bit_order(n: int, order: int) -> int:
 ```python
 def list_to_reverse_bit_order(l: List[int]) -> List[int]:
     """
-    Convert a list between normal and reverse bit order. This operation is idempotent.
+    Convert a list between normal and reverse bit order. The permutation is an involution (inverts itself)..
     """
     return [l[reverse_bit_order(i, len(l))] for i in range(len(l))]
 ```
diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
new file mode 100644
index 0000000000..07e624550a
--- /dev/null
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -0,0 +1,530 @@
+# Deneb -- Polynomial Commitments
+
+## Table of contents
+
+<!-- TOC -->
+<!-- START doctoc generated TOC please keep comment here to allow auto update -->
+<!-- DON'T EDIT THIS SECTION, INSTEAD RE-RUN doctoc TO UPDATE -->
+
+- [Introduction](#introduction)
+- [Custom types](#custom-types)
+- [Constants](#constants)
+- [Preset](#preset)
+  - [Samples](#samples)
+  - [Crypto](#crypto)
+- [Helper functions](#helper-functions)
+  - [Linear combinations](#linear-combinations)
+    - [`g2_lincomb`](#g2_lincomb)
+  - [FFTs](#ffts)
+    - [`_simple_ft_field`](#_simple_ft_field)
+    - [`_fft_field`](#_fft_field)
+    - [`fft_field`](#fft_field)
+  - [Polynomials in coefficient form](#polynomials-in-coefficient-form)
+    - [`polynomial_eval_to_coeff`](#polynomial_eval_to_coeff)
+    - [`polynomial_coeff_to_eval`](#polynomial_coeff_to_eval)
+    - [`add_polynomialcoeff`](#add_polynomialcoeff)
+    - [`neg_polynomialcoeff`](#neg_polynomialcoeff)
+    - [`multiply_polynomialcoeff`](#multiply_polynomialcoeff)
+    - [`divide_polynomialcoeff`](#divide_polynomialcoeff)
+    - [`shift_polynomialcoeff`](#shift_polynomialcoeff)
+    - [`interpolate_polynomialcoeff`](#interpolate_polynomialcoeff)
+    - [`zero_polynomialcoeff`](#zero_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)
+  - [Sample cosets](#sample-cosets)
+    - [`coset_for_sample`](#coset_for_sample)
+- [Samples](#samples-1)
+  - [Sample computation](#sample-computation)
+    - [`compute_samples_and_proofs`](#compute_samples_and_proofs)
+    - [`compute_samples`](#compute_samples)
+  - [Sample verification](#sample-verification)
+    - [`verify_sample_proof`](#verify_sample_proof)
+    - [`verify_sample_proof_batch`](#verify_sample_proof_batch)
+- [Reconstruction](#reconstruction)
+  - [`recover_samples_impl`](#recover_samples_impl)
+
+<!-- END doctoc generated TOC please keep comment here to allow auto update -->
+<!-- /TOC -->
+
+## 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 |
+
+## Constants
+
+| Name | Value | Notes |
+| - | - | - |
+
+## Preset
+
+### Samples
+
+| Name | Value | Description |
+| - | - | - |
+| `FIELD_ELEMENTS_PER_SAMPLE` | `uint64(64)` | Number of field elements in a sample |
+| `BYTES_PER_SAMPLE` | `FIELD_ELEMENTS_PER_SAMPLE * BYTES_PER_FIELD_ELEMENT` | The number of bytes in a sample |
+
+### Crypto
+
+| Name | Value | Description |
+| - | - | - |
+| `ROOT_OF_UNITY2` | `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_UNITY2` | `[pow(ROOT_OF_UNITY, 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_S` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(SAMPLES_PER_BLOB), BLS_MODULUS)` | Root of unity of order SAMPLES_PER_BLOB over the BLS12-381 field |
+| `ROOTS_OF_UNITY_S` | `[pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(SAMPLES_PER_BLOB)]` | Roots of unity of order SAMPLES_PER_BLOB over the BLS12-381 field |
+
+## 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
+
+#### `_simple_ft_field`
+
+```python
+def _simple_ft_field(vals, roots_of_unity):
+    assert len(vals) == len(roots_of_unity)
+    L = len(roots_of_unity)
+    o = []
+    for i in range(L):
+        last = 0
+        for j in range(L):
+            last += int(vals[j]) * int(roots_of_unity[(i * j) % L]) % BLS_MODULUS
+        o.append(last % BLS_MODULUS)
+    return o
+```
+
+#### `_fft_field`
+
+```python
+def _fft_field(vals, roots_of_unity):
+    if len(vals) <= 4:
+        return _simple_ft_field(vals, roots_of_unity)
+    L = _fft_field(vals[::2], roots_of_unity[::2])
+    R = _fft_field(vals[1::2], roots_of_unity[::2])
+    o = [0 for i 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] = (x + y_times_root) % BLS_MODULUS
+        o[i + len(L)] = (x - y_times_root) % BLS_MODULUS
+    return o
+```
+
+#### `fft_field`
+
+```python
+def fft_field(vals, roots_of_unity, inv=False):
+    if inv:
+        # Inverse FFT
+        invlen = pow(len(vals), BLS_MODULUS - 2, BLS_MODULUS)
+        return [(x * invlen) % BLS_MODULUS for x in
+                _fft_field(vals, roots_of_unity[0:1] + roots_of_unity[:0:-1])]
+    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.
+    """
+    polynomial_coeff = fft_field(bit_reversal_permutation(list(polynomial)), list(ROOTS_OF_UNITY), inv=True)
+
+    return polynomial_coeff
+```
+
+#### `polynomial_coeff_to_eval`
+
+```python
+def polynomial_coeff_to_eval(polynomial_coeff: PolynomialCoeff) -> Polynomial:
+    """
+    Evaluates a polynomial (given in coefficient form) to a polynomial in evaluation form.
+    """
+
+    if len(polynomial_coeff) > FIELD_ELEMENTS_PER_BLOB:
+        assert all(c == 0 for c in polynomial_coeff[FIELD_ELEMENTS_PER_BLOB:])
+
+    polynomial = bit_reversal_permutation(fft_field(list(polynomial_coeff), list(ROOTS_OF_UNITY), inv=False))
+
+    return polynomial
+```
+
+#### `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
+def shift_polynomialcoeff(poly, factor):
+    """
+    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 poly:
+        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
+```
+
+#### `zero_polynomialcoeff`
+
+```python
+def zero_polynomialcoeff(xs: Sequence[BLSFieldElement]) -> PolynomialCoeff:
+    """
+    Compute a zero polynoimial on ```xs``` (in coefficient form)
+    """
+    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]:
+        y = (int(y) * int(z) + 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: 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 = zero_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[:len(quotient_polynomial)], quotient_polynomial)), ys
+```
+
+#### `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
+    """
+    
+    zero_poly = g2_lincomb(KZG_SETUP_G2[:len(zs) + 1], zero_polynomialcoeff(zs))
+    interpolated_poly = g1_lincomb(KZG_SETUP_G1[: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[0]))]]
+        )
+    )
+```
+
+### Sample cosets
+
+#### `coset_for_sample`
+
+```python
+def coset_for_sample(sample_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE]:
+    """
+    Get the subgroup for a given ``sample_id``
+    """
+    assert sample_id < SAMPLES_PER_BLOB
+    roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY2)
+    return Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE](
+        roots_of_unity_brp[FIELD_ELEMENTS_PER_SAMPLE * sample_id:FIELD_ELEMENTS_PER_SAMPLE * (sample_id + 1)]
+    )
+```
+
+## Samples
+
+### Sample computation
+
+#### `compute_samples_and_proofs`
+
+```python
+def compute_samples_and_proofs(blob: Blob) -> Tuple[
+                                                    Vector[KZGProof, SAMPLES_PER_BLOB], 
+                                                    Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE], SAMPLES_PER_BLOB]]:
+    """
+    Compute all the sample 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.
+    """
+    polynomial = blob_to_polynomial(blob)
+    polynomial_coeff = polynomial_eval_to_coeff(polynomial)
+
+    proofs = []
+    samples = []
+
+    for i in range(10):
+        coset = coset_for_sample(i)
+        proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset)
+        proofs.append(proof)
+        samples.append(ys)
+
+    return proofs, samples
+```
+
+#### `compute_samples`
+
+```python
+def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE], SAMPLES_PER_BLOB]:
+    polynomial = blob_to_polynomial(blob)
+    polynomial_coeff = polynomial_eval_to_coeff(polynomial)
+
+    extended_data = fft_field(polynomial_coeff + [0] * FIELD_ELEMENTS_PER_BLOB, ROOTS_OF_UNITY2)
+    extended_data_rbo = bit_reversal_permutation(extended_data)
+    return [extended_data_rbo[i * FIELD_ELEMENTS_PER_SAMPLE:(i + 1) * FIELD_ELEMENTS_PER_SAMPLE] for i in range(SAMPLES_PER_BLOB)]
+```
+
+### Sample verification
+
+#### `verify_sample_proof`
+
+```python
+def verify_sample_proof(commitment: KZGCommitment,
+                        sample_id: int,
+                        data: Vector[BLSFieldElement, def shift_polynomialcoeff(poly, factor):
+    """
+    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 poly:
+        o.append(int(p) * factor_power % BLS_MODULUS)
+        factor_power = factor_power * inv_factor % BLS_MODULUS
+    return oFIELD_ELEMENTS_PER_SAMPLE],
+                        proof: KZGProof) -> bool:
+    """
+    Check a sample proof
+    """
+    coset = coset_for_sample(sample_id)
+
+    return verify_kzg_proof_multi_impl(commitment, coset, data, proof)
+```
+
+#### `verify_sample_proof_batch`
+
+```python
+def verify_sample_proof_batch(row_commitments: Sequence[KZGCommitment],
+                              row_ids:Sequence[int],
+                              column_ids: Sequence[int],
+                              datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE]],
+                              proofs: Sequence[KZGProof]) -> bool:
+    """
+    Check multiple sample proofs. This function implements the naive algorithm of checking every sample
+    individually; an efficient algorithm can be found here:
+    https://ethresear.ch/t/a-universal-verification-equation-for-data-availability-sampling/13240
+    """
+
+    # 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_sample(column_id), data, proof) 
+               for commitment, column_id, data, proof in zip(commitments, column_ids, datas, proofs))
+```
+
+## Reconstruction
+
+### `recover_samples_impl`
+
+```python
+def recover_samples_impl(samples: Sequence[Tuple[int, Sequence[BLSFieldElement]]]) -> Polynomial:
+    """
+    Recovers a polynomial from 2 * FIELD_ELEMENTS_PER_SAMPLE evaluations, half of which can be missing.
+
+    This algorithm uses FFTs to recover samples 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
+    """
+    assert len(samples) >= SAMPLES_PER_BLOB // 2
+    sample_ids = [sample_id for sample_id, _ in samples]
+    missing_sample_ids = [sample_id for sample_id in range(SAMPLES_PER_BLOB) if sample_id not in sample_ids]
+    short_zero_poly = zero_polynomialcoeff([ROOTS_OF_UNITY_S[reverse_bits(sample_id, SAMPLES_PER_BLOB)] for sample_id in missing_sample_ids])
+
+    full_zero_poly = []
+    for i in short_zero_poly:
+        full_zero_poly.append(i)
+        full_zero_poly.extend([0] * (FIELD_ELEMENTS_PER_SAMPLE - 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, ROOTS_OF_UNITY2)
+    zero_poly_eval_brp = bit_reversal_permutation(zero_poly_eval)
+    for sample_id in missing_sample_ids:
+        assert zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE] == \
+            [0] * FIELD_ELEMENTS_PER_SAMPLE
+    for sample_id in sample_ids:
+        assert all(a != 0 for a in zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE])
+
+    extended_evaluation_rbo = [0] * FIELD_ELEMENTS_PER_BLOB * 2
+    for sample_id, sample_data in samples:
+        extended_evaluation_rbo[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE] = \
+            sample_data
+    extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
+
+    extended_evaluation_times_zero = [a * b % BLS_MODULUS for a, b in zip(zero_poly_eval, extended_evaluation)]
+
+    extended_evaluations_fft = fft_field(extended_evaluation_times_zero, ROOTS_OF_UNITY2, inv=True)
+
+    shift_factor = PRIMITIVE_ROOT_OF_UNITY
+    shift_inv = div(1, PRIMITIVE_ROOT_OF_UNITY)
+
+    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_UNITY2)
+    eval_shifted_zero_poly = fft_field(shifted_zero_poly, ROOTS_OF_UNITY2)
+    
+    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_UNITY2, inv=True)
+    
+    reconstructed_poly = shift_polynomialcoeff(shifted_reconstructed_poly, shift_inv)
+
+    reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly, ROOTS_OF_UNITY2))
+
+    for sample_id, sample_data in samples:
+        assert reconstructed_data[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE] == \
+            sample_data
+    
+    return reconstructed_data
+```
\ No newline at end of file
diff --git a/specs/deneb/polynomial-commitments.md b/specs/deneb/polynomial-commitments.md
index b42fbd8707..3f48e3ebb2 100644
--- a/specs/deneb/polynomial-commitments.md
+++ b/specs/deneb/polynomial-commitments.md
@@ -11,6 +11,7 @@
 - [Constants](#constants)
 - [Preset](#preset)
   - [Blob](#blob)
+  - [Crypto](#crypto)
   - [Trusted setup](#trusted-setup)
 - [Helper functions](#helper-functions)
   - [Bit-reversal permutation](#bit-reversal-permutation)
@@ -78,7 +79,8 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | `BYTES_PER_BLOB` | `uint64(BYTES_PER_FIELD_ELEMENT * FIELD_ELEMENTS_PER_BLOB)` | The number of bytes in a blob |
 | `G1_POINT_AT_INFINITY` | `Bytes48(b'\xc0' + b'\x00' * 47)` | Serialized form of the point at infinity on the G1 group |
 | `KZG_ENDIANNESS` | `'big'` | The endianness of the field elements including blobs |
-| `PRIMITIVE_ROOT_OF_UNITY` | `7` | Primitive root of unity of the BLS12_381 (inner) BLS_MODULUS |
+| `PRIMITIVE_ROOT_OF_UNITY` | `7` | The primitive root of unity from which all roots of unity should be derived |
+
 
 ## Preset
 
@@ -90,6 +92,13 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | `FIAT_SHAMIR_PROTOCOL_DOMAIN` | `b'FSBLOBVERIFY_V1_'` |
 | `RANDOM_CHALLENGE_KZG_BATCH_DOMAIN` | `b'RCKZGBATCH___V1_'` |
 
+### Crypto
+
+| Name | Value | Notes |
+| - | - | - |
+| `ROOT_OF_UNITY` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(FIELD_ELEMENTS_PER_BLOB), BLS_MODULUS)` | Root of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
+| `ROOTS_OF_UNITY` | `[pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(FIELD_ELEMENTS_PER_BLOB)]` | Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
+
 ### Trusted setup
 
 | Name | Value |

From 0d4007a6d93f9cb9270e71dbd863bd0a363583d2 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Tue, 12 Dec 2023 23:51:50 +0800
Subject: [PATCH 02/28] Apply suggestions from code review

Co-authored-by: Justin Traglia <95511699+jtraglia@users.noreply.github.com>
---
 .../deneb/polynomial-commitments-sampling.md  | 35 +++++++------------
 1 file changed, 12 insertions(+), 23 deletions(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index 07e624550a..d9c60b7a5a 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -75,6 +75,7 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | - | - | - |
 | `FIELD_ELEMENTS_PER_SAMPLE` | `uint64(64)` | Number of field elements in a sample |
 | `BYTES_PER_SAMPLE` | `FIELD_ELEMENTS_PER_SAMPLE * BYTES_PER_FIELD_ELEMENT` | The number of bytes in a sample |
+| `SAMPLES_PER_BLOB` | `(2 * FIELD_ELEMENTS_PER_BLOB) / FIELD_ELEMENTS_PER_SAMPLE` | The number of samples for a blob |
 
 ### Crypto
 
@@ -132,7 +133,7 @@ def _fft_field(vals, roots_of_unity):
     for i, (x, y) in enumerate(zip(L, R)):
         y_times_root = int(y) * int(roots_of_unity[i]) % BLS_MODULUS
         o[i] = (x + y_times_root) % BLS_MODULUS
-        o[i + len(L)] = (x - y_times_root) % BLS_MODULUS
+        o[i + len(L)] = (x - y_times_root + BLS_MODULUS) % BLS_MODULUS
     return o
 ```
 
@@ -284,7 +285,7 @@ def interpolate_polynomialcoeff(xs: Sequence[BLSFieldElement], ys: Sequence[BLSF
 ```python
 def zero_polynomialcoeff(xs: Sequence[BLSFieldElement]) -> PolynomialCoeff:
     """
-    Compute a zero polynoimial on ```xs``` (in coefficient form)
+    Compute a zero polynomial on ``xs`` (in coefficient form)
     """
     p = [1]
     for x in xs:
@@ -379,8 +380,8 @@ def coset_for_sample(sample_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_P
 
 ```python
 def compute_samples_and_proofs(blob: Blob) -> Tuple[
-                                                    Vector[KZGProof, SAMPLES_PER_BLOB], 
-                                                    Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE], SAMPLES_PER_BLOB]]:
+        Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE], SAMPLES_PER_BLOB],
+        Vector[KZGProof, SAMPLES_PER_BLOB]]:
     """
     Compute all the sample 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
@@ -389,16 +390,16 @@ def compute_samples_and_proofs(blob: Blob) -> Tuple[
     polynomial = blob_to_polynomial(blob)
     polynomial_coeff = polynomial_eval_to_coeff(polynomial)
 
-    proofs = []
     samples = []
+    proofs = []
 
-    for i in range(10):
+    for i in range(SAMPLES_PER_BLOB):
+        samples.append(ys)
         coset = coset_for_sample(i)
         proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset)
         proofs.append(proof)
-        samples.append(ys)
 
-    return proofs, samples
+    return samples, proofs
 ```
 
 #### `compute_samples`
@@ -419,20 +420,8 @@ def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS
 
 ```python
 def verify_sample_proof(commitment: KZGCommitment,
-                        sample_id: int,
-                        data: Vector[BLSFieldElement, def shift_polynomialcoeff(poly, factor):
-    """
-    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 poly:
-        o.append(int(p) * factor_power % BLS_MODULUS)
-        factor_power = factor_power * inv_factor % BLS_MODULUS
-    return oFIELD_ELEMENTS_PER_SAMPLE],
-                        proof: KZGProof) -> bool:
+                                    sample_id: int,
+                                    data: Vector[BLSFieldElement]) -> bool:
     """
     Check a sample proof
     """
@@ -445,7 +434,7 @@ def verify_sample_proof(commitment: KZGCommitment,
 
 ```python
 def verify_sample_proof_batch(row_commitments: Sequence[KZGCommitment],
-                              row_ids:Sequence[int],
+                              row_ids: Sequence[int],
                               column_ids: Sequence[int],
                               datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE]],
                               proofs: Sequence[KZGProof]) -> bool:

From f0af688fbd044e4f77e356f709d6e33f57d66afa Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Wed, 13 Dec 2023 00:25:30 +0800
Subject: [PATCH 03/28] Minor fixes

---
 .../deneb/polynomial-commitments-sampling.md  | 50 ++++++++++---------
 specs/deneb/polynomial-commitments.md         |  2 +-
 2 files changed, 28 insertions(+), 24 deletions(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index d9c60b7a5a..9c44112fa0 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -75,16 +75,16 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | - | - | - |
 | `FIELD_ELEMENTS_PER_SAMPLE` | `uint64(64)` | Number of field elements in a sample |
 | `BYTES_PER_SAMPLE` | `FIELD_ELEMENTS_PER_SAMPLE * BYTES_PER_FIELD_ELEMENT` | The number of bytes in a sample |
-| `SAMPLES_PER_BLOB` | `(2 * FIELD_ELEMENTS_PER_BLOB) / FIELD_ELEMENTS_PER_SAMPLE` | The number of samples for a blob |
+| `SAMPLES_PER_BLOB` | `((2 * FIELD_ELEMENTS_PER_BLOB) // FIELD_ELEMENTS_PER_SAMPLE)` | The number of samples for a blob |
 
 ### Crypto
 
 | Name | Value | Description |
 | - | - | - |
 | `ROOT_OF_UNITY2` | `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_UNITY2` | `[pow(ROOT_OF_UNITY, 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 |
+| `ROOTS_OF_UNITY2` | `([pow(ROOT_OF_UNITY, 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_S` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(SAMPLES_PER_BLOB), BLS_MODULUS)` | Root of unity of order SAMPLES_PER_BLOB over the BLS12-381 field |
-| `ROOTS_OF_UNITY_S` | `[pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(SAMPLES_PER_BLOB)]` | Roots of unity of order SAMPLES_PER_BLOB over the BLS12-381 field |
+| `ROOTS_OF_UNITY_S` | `([pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(SAMPLES_PER_BLOB)])` | Roots of unity of order SAMPLES_PER_BLOB over the BLS12-381 field |
 
 ## Helper functions
 
@@ -313,8 +313,9 @@ Extended KZG functions for multiproofs
 #### `compute_kzg_proof_multi_impl`
 
 ```python
-def compute_kzg_proof_multi_impl(polynomial_coeff: PolynomialCoeff,
-                                 zs: Sequence[BLSFieldElement]) -> Tuple[KZGProof, Sequence[BLSFieldElement]]:
+def compute_kzg_proof_multi_impl(
+        polynomial_coeff: PolynomialCoeff,
+        zs: Sequence[BLSFieldElement]) -> Tuple[KZGProof, Sequence[BLSFieldElement]]:
     """
     Helper function that computes multi-evaluation KZG proofs.
     """
@@ -343,17 +344,16 @@ def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
     """
     Helper function that verifies a KZG multiproof
     """
-    
     zero_poly = g2_lincomb(KZG_SETUP_G2[:len(zs) + 1], zero_polynomialcoeff(zs))
     interpolated_poly = g1_lincomb(KZG_SETUP_G1[: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[0]))]]
-        )
-    )
+    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[0])),
+        ],
+    ]))
 ```
 
 ### Sample cosets
@@ -394,9 +394,9 @@ def compute_samples_and_proofs(blob: Blob) -> Tuple[
     proofs = []
 
     for i in range(SAMPLES_PER_BLOB):
-        samples.append(ys)
         coset = coset_for_sample(i)
         proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset)
+        samples.append(ys)
         proofs.append(proof)
 
     return samples, proofs
@@ -411,7 +411,8 @@ def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS
 
     extended_data = fft_field(polynomial_coeff + [0] * FIELD_ELEMENTS_PER_BLOB, ROOTS_OF_UNITY2)
     extended_data_rbo = bit_reversal_permutation(extended_data)
-    return [extended_data_rbo[i * FIELD_ELEMENTS_PER_SAMPLE:(i + 1) * FIELD_ELEMENTS_PER_SAMPLE] for i in range(SAMPLES_PER_BLOB)]
+    return [extended_data_rbo[i * FIELD_ELEMENTS_PER_SAMPLE:(i + 1) * FIELD_ELEMENTS_PER_SAMPLE]
+            for i in range(SAMPLES_PER_BLOB)]
 ```
 
 ### Sample verification
@@ -420,8 +421,9 @@ def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS
 
 ```python
 def verify_sample_proof(commitment: KZGCommitment,
-                                    sample_id: int,
-                                    data: Vector[BLSFieldElement]) -> bool:
+                        sample_id: int,
+                        data: Vector[BLSFieldElement],
+                        proof: KZGProof) -> bool:
     """
     Check a sample proof
     """
@@ -498,15 +500,17 @@ def recover_samples_impl(samples: Sequence[Tuple[int, Sequence[BLSFieldElement]]
 
     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_UNITY2)
     eval_shifted_zero_poly = fft_field(shifted_zero_poly, ROOTS_OF_UNITY2)
     
-    eval_shifted_reconstructed_poly = [div(a, b) for a, b in
-                                    zip(eval_shifted_extended_evaluation, eval_shifted_zero_poly)]
-    
+    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_UNITY2, inv=True)
-    
+
     reconstructed_poly = shift_polynomialcoeff(shifted_reconstructed_poly, shift_inv)
 
     reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly, ROOTS_OF_UNITY2))
@@ -516,4 +520,4 @@ def recover_samples_impl(samples: Sequence[Tuple[int, Sequence[BLSFieldElement]]
             sample_data
     
     return reconstructed_data
-```
\ No newline at end of file
+```
diff --git a/specs/deneb/polynomial-commitments.md b/specs/deneb/polynomial-commitments.md
index 3f48e3ebb2..c0a293d8b7 100644
--- a/specs/deneb/polynomial-commitments.md
+++ b/specs/deneb/polynomial-commitments.md
@@ -97,7 +97,7 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | Name | Value | Notes |
 | - | - | - |
 | `ROOT_OF_UNITY` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(FIELD_ELEMENTS_PER_BLOB), BLS_MODULUS)` | Root of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
-| `ROOTS_OF_UNITY` | `[pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(FIELD_ELEMENTS_PER_BLOB)]` | Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
+| `ROOTS_OF_UNITY` | `([pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(FIELD_ELEMENTS_PER_BLOB)])` | Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
 
 ### Trusted setup
 

From 62d1a8f4c4667ed2c4319b75030831b35acdc8bf Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Mon, 4 Dec 2023 01:26:07 +0000
Subject: [PATCH 04/28] Add public method markers

---
 specs/deneb/polynomial-commitments-sampling.md | 15 ++++++++++++++-
 1 file changed, 14 insertions(+), 1 deletion(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index 9c44112fa0..a947357703 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -386,6 +386,8 @@ def compute_samples_and_proofs(blob: Blob) -> Tuple[
     Compute all the sample 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)
@@ -406,6 +408,11 @@ def compute_samples_and_proofs(blob: Blob) -> Tuple[
 
 ```python
 def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE], SAMPLES_PER_BLOB]:
+    """
+    Compute the sample data for a blob (without computing the proofs).
+
+    Public method.
+    """
     polynomial = blob_to_polynomial(blob)
     polynomial_coeff = polynomial_eval_to_coeff(polynomial)
 
@@ -422,10 +429,12 @@ def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS
 ```python
 def verify_sample_proof(commitment: KZGCommitment,
                         sample_id: int,
-                        data: Vector[BLSFieldElement],
+                        data: Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE],
                         proof: KZGProof) -> bool:
     """
     Check a sample proof
+
+    Publiiic method.
     """
     coset = coset_for_sample(sample_id)
 
@@ -444,6 +453,8 @@ def verify_sample_proof_batch(row_commitments: Sequence[KZGCommitment],
     Check multiple sample proofs. This function implements the naive algorithm of checking every sample
     individually; an efficient algorithm can be found here:
     https://ethresear.ch/t/a-universal-verification-equation-for-data-availability-sampling/13240
+
+    Public method.
     """
 
     # Get commitments via row IDs
@@ -465,6 +476,8 @@ def recover_samples_impl(samples: Sequence[Tuple[int, Sequence[BLSFieldElement]]
     This algorithm uses FFTs to recover samples 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(samples) >= SAMPLES_PER_BLOB // 2
     sample_ids = [sample_id for sample_id, _ in samples]

From 246477ace2adcc81aac00af299ee9c739a4e798b Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Sat, 23 Dec 2023 14:44:58 +0000
Subject: [PATCH 05/28] SAMPLE -> CELL and cleanups

---
 .../deneb/polynomial-commitments-sampling.md  | 103 ++++++++----------
 1 file changed, 44 insertions(+), 59 deletions(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index a947357703..7c05fb553e 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -43,7 +43,7 @@
     - [`verify_sample_proof`](#verify_sample_proof)
     - [`verify_sample_proof_batch`](#verify_sample_proof_batch)
 - [Reconstruction](#reconstruction)
-  - [`recover_samples_impl`](#recover_samples_impl)
+  - [`recover_samples`](#recover_samples)
 
 <!-- END doctoc generated TOC please keep comment here to allow auto update -->
 <!-- /TOC -->
@@ -73,18 +73,18 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 
 | Name | Value | Description |
 | - | - | - |
-| `FIELD_ELEMENTS_PER_SAMPLE` | `uint64(64)` | Number of field elements in a sample |
-| `BYTES_PER_SAMPLE` | `FIELD_ELEMENTS_PER_SAMPLE * BYTES_PER_FIELD_ELEMENT` | The number of bytes in a sample |
-| `SAMPLES_PER_BLOB` | `((2 * FIELD_ELEMENTS_PER_BLOB) // FIELD_ELEMENTS_PER_SAMPLE)` | The number of samples for a 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_BLOB` | `((2 * FIELD_ELEMENTS_PER_BLOB) // FIELD_ELEMENTS_PER_CELL)` | The number of cells in a blob |
 
 ### Crypto
 
 | Name | Value | Description |
 | - | - | - |
-| `ROOT_OF_UNITY2` | `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_UNITY2` | `([pow(ROOT_OF_UNITY, 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_S` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(SAMPLES_PER_BLOB), BLS_MODULUS)` | Root of unity of order SAMPLES_PER_BLOB over the BLS12-381 field |
-| `ROOTS_OF_UNITY_S` | `([pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(SAMPLES_PER_BLOB)])` | Roots of unity of order SAMPLES_PER_BLOB over the BLS12-381 field |
+| `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` | `([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` | `([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 |
 
 ## Helper functions
 
@@ -106,27 +106,12 @@ def g2_lincomb(points: Sequence[KZGCommitment], scalars: Sequence[BLSFieldElemen
 
 ### FFTs
 
-#### `_simple_ft_field`
-
-```python
-def _simple_ft_field(vals, roots_of_unity):
-    assert len(vals) == len(roots_of_unity)
-    L = len(roots_of_unity)
-    o = []
-    for i in range(L):
-        last = 0
-        for j in range(L):
-            last += int(vals[j]) * int(roots_of_unity[(i * j) % L]) % BLS_MODULUS
-        o.append(last % BLS_MODULUS)
-    return o
-```
-
 #### `_fft_field`
 
 ```python
 def _fft_field(vals, roots_of_unity):
-    if len(vals) <= 4:
-        return _simple_ft_field(vals, roots_of_unity)
+    if len(vals) == 0:
+        return vals
     L = _fft_field(vals[::2], roots_of_unity[::2])
     R = _fft_field(vals[1::2], roots_of_unity[::2])
     o = [0 for i in vals]
@@ -361,14 +346,14 @@ def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
 #### `coset_for_sample`
 
 ```python
-def coset_for_sample(sample_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE]:
+def coset_for_sample(sample_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]:
     """
-    Get the subgroup for a given ``sample_id``
+    Get the coset for a given ``sample_id``
     """
-    assert sample_id < SAMPLES_PER_BLOB
-    roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY2)
-    return Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE](
-        roots_of_unity_brp[FIELD_ELEMENTS_PER_SAMPLE * sample_id:FIELD_ELEMENTS_PER_SAMPLE * (sample_id + 1)]
+    assert sample_id < CELLS_PER_BLOB
+    roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY_EXTENDED)
+    return Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL](
+        roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * sample_id:FIELD_ELEMENTS_PER_CELL * (sample_id + 1)]
     )
 ```
 
@@ -380,8 +365,8 @@ def coset_for_sample(sample_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_P
 
 ```python
 def compute_samples_and_proofs(blob: Blob) -> Tuple[
-        Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE], SAMPLES_PER_BLOB],
-        Vector[KZGProof, SAMPLES_PER_BLOB]]:
+        Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL], CELLS_PER_BLOB],
+        Vector[KZGProof, CELLS_PER_BLOB]]:
     """
     Compute all the sample 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
@@ -395,7 +380,7 @@ def compute_samples_and_proofs(blob: Blob) -> Tuple[
     samples = []
     proofs = []
 
-    for i in range(SAMPLES_PER_BLOB):
+    for i in range(CELLS_PER_BLOB):
         coset = coset_for_sample(i)
         proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset)
         samples.append(ys)
@@ -407,7 +392,7 @@ def compute_samples_and_proofs(blob: Blob) -> Tuple[
 #### `compute_samples`
 
 ```python
-def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE], SAMPLES_PER_BLOB]:
+def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL], CELLS_PER_BLOB]:
     """
     Compute the sample data for a blob (without computing the proofs).
 
@@ -416,10 +401,10 @@ def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS
     polynomial = blob_to_polynomial(blob)
     polynomial_coeff = polynomial_eval_to_coeff(polynomial)
 
-    extended_data = fft_field(polynomial_coeff + [0] * FIELD_ELEMENTS_PER_BLOB, ROOTS_OF_UNITY2)
+    extended_data = fft_field(polynomial_coeff + [0] * FIELD_ELEMENTS_PER_BLOB, ROOTS_OF_UNITY_EXTENDED)
     extended_data_rbo = bit_reversal_permutation(extended_data)
-    return [extended_data_rbo[i * FIELD_ELEMENTS_PER_SAMPLE:(i + 1) * FIELD_ELEMENTS_PER_SAMPLE]
-            for i in range(SAMPLES_PER_BLOB)]
+    return [extended_data_rbo[i * FIELD_ELEMENTS_PER_CELL:(i + 1) * FIELD_ELEMENTS_PER_CELL]
+            for i in range(CELLS_PER_BLOB)]
 ```
 
 ### Sample verification
@@ -429,12 +414,12 @@ def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS
 ```python
 def verify_sample_proof(commitment: KZGCommitment,
                         sample_id: int,
-                        data: Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE],
+                        data: Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL],
                         proof: KZGProof) -> bool:
     """
     Check a sample proof
 
-    Publiiic method.
+    Public method.
     """
     coset = coset_for_sample(sample_id)
 
@@ -447,7 +432,7 @@ def verify_sample_proof(commitment: KZGCommitment,
 def verify_sample_proof_batch(row_commitments: Sequence[KZGCommitment],
                               row_ids: Sequence[int],
                               column_ids: Sequence[int],
-                              datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_SAMPLE]],
+                              datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]],
                               proofs: Sequence[KZGProof]) -> bool:
     """
     Check multiple sample proofs. This function implements the naive algorithm of checking every sample
@@ -466,12 +451,12 @@ def verify_sample_proof_batch(row_commitments: Sequence[KZGCommitment],
 
 ## Reconstruction
 
-### `recover_samples_impl`
+### `recover_samples`
 
 ```python
-def recover_samples_impl(samples: Sequence[Tuple[int, Sequence[BLSFieldElement]]]) -> Polynomial:
+def recover_samples(samples: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Polynomial:
     """
-    Recovers a polynomial from 2 * FIELD_ELEMENTS_PER_SAMPLE evaluations, half of which can be missing.
+    Recovers a polynomial from 2 * FIELD_ELEMENTS_PER_CELL evaluations, half of which can be missing.
 
     This algorithm uses FFTs to recover samples faster than using Lagrange implementation. However,
     a faster version thanks to Qi Zhou can be found here:
@@ -479,34 +464,34 @@ def recover_samples_impl(samples: Sequence[Tuple[int, Sequence[BLSFieldElement]]
 
     Public method.
     """
-    assert len(samples) >= SAMPLES_PER_BLOB // 2
+    assert len(samples) >= CELLS_PER_BLOB // 2
     sample_ids = [sample_id for sample_id, _ in samples]
-    missing_sample_ids = [sample_id for sample_id in range(SAMPLES_PER_BLOB) if sample_id not in sample_ids]
-    short_zero_poly = zero_polynomialcoeff([ROOTS_OF_UNITY_S[reverse_bits(sample_id, SAMPLES_PER_BLOB)] for sample_id in missing_sample_ids])
+    missing_sample_ids = [sample_id for sample_id in range(CELLS_PER_BLOB) if sample_id not in sample_ids]
+    short_zero_poly = zero_polynomialcoeff([ROOTS_OF_UNITY_REDUCED[reverse_bits(sample_id, CELLS_PER_BLOB)] for sample_id in missing_sample_ids])
 
     full_zero_poly = []
     for i in short_zero_poly:
         full_zero_poly.append(i)
-        full_zero_poly.extend([0] * (FIELD_ELEMENTS_PER_SAMPLE - 1))
+        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, ROOTS_OF_UNITY2)
+    zero_poly_eval = fft_field(full_zero_poly, ROOTS_OF_UNITY_EXTENDED)
     zero_poly_eval_brp = bit_reversal_permutation(zero_poly_eval)
     for sample_id in missing_sample_ids:
-        assert zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE] == \
-            [0] * FIELD_ELEMENTS_PER_SAMPLE
+        assert zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
+            [0] * FIELD_ELEMENTS_PER_CELL
     for sample_id in sample_ids:
-        assert all(a != 0 for a in zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE])
+        assert all(a != 0 for a in zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL])
 
     extended_evaluation_rbo = [0] * FIELD_ELEMENTS_PER_BLOB * 2
     for sample_id, sample_data in samples:
-        extended_evaluation_rbo[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE] = \
+        extended_evaluation_rbo[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL] = \
             sample_data
     extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
 
     extended_evaluation_times_zero = [a * b % BLS_MODULUS for a, b in zip(zero_poly_eval, extended_evaluation)]
 
-    extended_evaluations_fft = fft_field(extended_evaluation_times_zero, ROOTS_OF_UNITY2, inv=True)
+    extended_evaluations_fft = fft_field(extended_evaluation_times_zero, ROOTS_OF_UNITY_EXTENDED, inv=True)
 
     shift_factor = PRIMITIVE_ROOT_OF_UNITY
     shift_inv = div(1, PRIMITIVE_ROOT_OF_UNITY)
@@ -514,22 +499,22 @@ def recover_samples_impl(samples: Sequence[Tuple[int, Sequence[BLSFieldElement]]
     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_UNITY2)
-    eval_shifted_zero_poly = fft_field(shifted_zero_poly, ROOTS_OF_UNITY2)
+    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_UNITY2, inv=True)
+    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_UNITY2))
+    reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly, ROOTS_OF_UNITY_EXTENDED))
 
     for sample_id, sample_data in samples:
-        assert reconstructed_data[sample_id * FIELD_ELEMENTS_PER_SAMPLE:(sample_id + 1) * FIELD_ELEMENTS_PER_SAMPLE] == \
+        assert reconstructed_data[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
             sample_data
     
     return reconstructed_data

From 3169f92fa5b99d30402c6c678c9b0998b174e07c Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Sat, 23 Dec 2023 14:52:39 +0000
Subject: [PATCH 06/28] More sample -> cell renaming

---
 .../deneb/polynomial-commitments-sampling.md  | 117 +++++++++---------
 1 file changed, 59 insertions(+), 58 deletions(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index 7c05fb553e..58e48cc4cc 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -10,13 +10,12 @@
 - [Custom types](#custom-types)
 - [Constants](#constants)
 - [Preset](#preset)
-  - [Samples](#samples)
+  - [Cells](#cells)
   - [Crypto](#crypto)
 - [Helper functions](#helper-functions)
   - [Linear combinations](#linear-combinations)
     - [`g2_lincomb`](#g2_lincomb)
   - [FFTs](#ffts)
-    - [`_simple_ft_field`](#_simple_ft_field)
     - [`_fft_field`](#_fft_field)
     - [`fft_field`](#fft_field)
   - [Polynomials in coefficient form](#polynomials-in-coefficient-form)
@@ -33,17 +32,17 @@
   - [KZG multiproofs](#kzg-multiproofs)
     - [`compute_kzg_proof_multi_impl`](#compute_kzg_proof_multi_impl)
     - [`verify_kzg_proof_multi_impl`](#verify_kzg_proof_multi_impl)
-  - [Sample cosets](#sample-cosets)
-    - [`coset_for_sample`](#coset_for_sample)
-- [Samples](#samples-1)
-  - [Sample computation](#sample-computation)
-    - [`compute_samples_and_proofs`](#compute_samples_and_proofs)
-    - [`compute_samples`](#compute_samples)
-  - [Sample verification](#sample-verification)
-    - [`verify_sample_proof`](#verify_sample_proof)
-    - [`verify_sample_proof_batch`](#verify_sample_proof_batch)
+  - [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)
 - [Reconstruction](#reconstruction)
-  - [`recover_samples`](#recover_samples)
+  - [`recover_cells`](#recover_cells)
 
 <!-- END doctoc generated TOC please keep comment here to allow auto update -->
 <!-- /TOC -->
@@ -69,7 +68,9 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 
 ## Preset
 
-### Samples
+### 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 |
 | - | - | - |
@@ -341,34 +342,34 @@ def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
     ]))
 ```
 
-### Sample cosets
+### Cell cosets
 
-#### `coset_for_sample`
+#### `coset_for_cell`
 
 ```python
-def coset_for_sample(sample_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]:
+def coset_for_cell(cell_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]:
     """
-    Get the coset for a given ``sample_id``
+    Get the coset for a given ``cell_id``
     """
-    assert sample_id < CELLS_PER_BLOB
+    assert cell_id < CELLS_PER_BLOB
     roots_of_unity_brp = bit_reversal_permutation(ROOTS_OF_UNITY_EXTENDED)
     return Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL](
-        roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * sample_id:FIELD_ELEMENTS_PER_CELL * (sample_id + 1)]
+        roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * cell_id:FIELD_ELEMENTS_PER_CELL * (cell_id + 1)]
     )
 ```
 
-## Samples
+## Cells
 
-### Sample computation
+### Cell computation
 
-#### `compute_samples_and_proofs`
+#### `compute_cells_and_proofs`
 
 ```python
-def compute_samples_and_proofs(blob: Blob) -> Tuple[
+def compute_cells_and_proofs(blob: Blob) -> Tuple[
         Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL], CELLS_PER_BLOB],
         Vector[KZGProof, CELLS_PER_BLOB]]:
     """
-    Compute all the sample proofs for one blob. This is an inefficient O(n^2) algorithm,
+    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.
 
@@ -377,24 +378,24 @@ def compute_samples_and_proofs(blob: Blob) -> Tuple[
     polynomial = blob_to_polynomial(blob)
     polynomial_coeff = polynomial_eval_to_coeff(polynomial)
 
-    samples = []
+    cells = []
     proofs = []
 
     for i in range(CELLS_PER_BLOB):
-        coset = coset_for_sample(i)
+        coset = coset_for_cell(i)
         proof, ys = compute_kzg_proof_multi_impl(polynomial_coeff, coset)
-        samples.append(ys)
+        cells.append(ys)
         proofs.append(proof)
 
-    return samples, proofs
+    return cells, proofs
 ```
 
-#### `compute_samples`
+#### `compute_cells`
 
 ```python
-def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL], CELLS_PER_BLOB]:
+def compute_cells(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL], CELLS_PER_BLOB]:
     """
-    Compute the sample data for a blob (without computing the proofs).
+    Compute the cell data for a blob (without computing the proofs).
 
     Public method.
     """
@@ -407,35 +408,35 @@ def compute_samples(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS
             for i in range(CELLS_PER_BLOB)]
 ```
 
-### Sample verification
+### Cell verification
 
-#### `verify_sample_proof`
+#### `verify_cell_proof`
 
 ```python
-def verify_sample_proof(commitment: KZGCommitment,
-                        sample_id: int,
+def verify_cell_proof(commitment: KZGCommitment,
+                        cell_id: int,
                         data: Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL],
                         proof: KZGProof) -> bool:
     """
-    Check a sample proof
+    Check a cell proof
 
     Public method.
     """
-    coset = coset_for_sample(sample_id)
+    coset = coset_for_cell(cell_id)
 
     return verify_kzg_proof_multi_impl(commitment, coset, data, proof)
 ```
 
-#### `verify_sample_proof_batch`
+#### `verify_cell_proof_batch`
 
 ```python
-def verify_sample_proof_batch(row_commitments: Sequence[KZGCommitment],
+def verify_cell_proof_batch(row_commitments: Sequence[KZGCommitment],
                               row_ids: Sequence[int],
                               column_ids: Sequence[int],
                               datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]],
                               proofs: Sequence[KZGProof]) -> bool:
     """
-    Check multiple sample proofs. This function implements the naive algorithm of checking every sample
+    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
 
@@ -445,29 +446,29 @@ def verify_sample_proof_batch(row_commitments: Sequence[KZGCommitment],
     # 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_sample(column_id), data, proof) 
+    return all(verify_kzg_proof_multi_impl(commitment, coset_for_cell(column_id), data, proof) 
                for commitment, column_id, data, proof in zip(commitments, column_ids, datas, proofs))
 ```
 
 ## Reconstruction
 
-### `recover_samples`
+### `recover_cells`
 
 ```python
-def recover_samples(samples: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Polynomial:
+def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Polynomial:
     """
     Recovers a polynomial from 2 * FIELD_ELEMENTS_PER_CELL evaluations, half of which can be missing.
 
-    This algorithm uses FFTs to recover samples faster than using Lagrange implementation. However,
+    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(samples) >= CELLS_PER_BLOB // 2
-    sample_ids = [sample_id for sample_id, _ in samples]
-    missing_sample_ids = [sample_id for sample_id in range(CELLS_PER_BLOB) if sample_id not in sample_ids]
-    short_zero_poly = zero_polynomialcoeff([ROOTS_OF_UNITY_REDUCED[reverse_bits(sample_id, CELLS_PER_BLOB)] for sample_id in missing_sample_ids])
+    assert len(cells) >= CELLS_PER_BLOB // 2
+    cell_ids = [cell_id for cell_id, _ in cells]
+    missing_cell_ids = [cell_id for cell_id in range(CELLS_PER_BLOB) if cell_id not in cell_ids]
+    short_zero_poly = zero_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:
@@ -477,16 +478,16 @@ def recover_samples(samples: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -
 
     zero_poly_eval = fft_field(full_zero_poly, ROOTS_OF_UNITY_EXTENDED)
     zero_poly_eval_brp = bit_reversal_permutation(zero_poly_eval)
-    for sample_id in missing_sample_ids:
-        assert zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
+    for cell_id in missing_cell_ids:
+        assert zero_poly_eval_brp[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
             [0] * FIELD_ELEMENTS_PER_CELL
-    for sample_id in sample_ids:
-        assert all(a != 0 for a in zero_poly_eval_brp[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL])
+    for cell_id in cell_ids:
+        assert all(a != 0 for a in zero_poly_eval_brp[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL])
 
     extended_evaluation_rbo = [0] * FIELD_ELEMENTS_PER_BLOB * 2
-    for sample_id, sample_data in samples:
-        extended_evaluation_rbo[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL] = \
-            sample_data
+    for cell_id, cell_data in cells:
+        extended_evaluation_rbo[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL] = \
+            cell_data
     extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
 
     extended_evaluation_times_zero = [a * b % BLS_MODULUS for a, b in zip(zero_poly_eval, extended_evaluation)]
@@ -513,9 +514,9 @@ def recover_samples(samples: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -
 
     reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly, ROOTS_OF_UNITY_EXTENDED))
 
-    for sample_id, sample_data in samples:
-        assert reconstructed_data[sample_id * FIELD_ELEMENTS_PER_CELL:(sample_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
-            sample_data
+    for cell_id, cell_data in cells:
+        assert reconstructed_data[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
+            cell_data
     
     return reconstructed_data
 ```

From d36fc4f5a0ed907114b1f304c982a54ef609b7fb Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Sat, 23 Dec 2023 15:08:02 +0000
Subject: [PATCH 07/28] Zero polynomial -> vanishing polynomial

---
 specs/deneb/polynomial-commitments-sampling.md | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index 58e48cc4cc..fcc5800148 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -27,7 +27,7 @@
     - [`divide_polynomialcoeff`](#divide_polynomialcoeff)
     - [`shift_polynomialcoeff`](#shift_polynomialcoeff)
     - [`interpolate_polynomialcoeff`](#interpolate_polynomialcoeff)
-    - [`zero_polynomialcoeff`](#zero_polynomialcoeff)
+    - [`vanishing_polynomialcoeff`](#vanishing_polynomialcoeff)
     - [`evaluate_polynomialcoeff`](#evaluate_polynomialcoeff)
   - [KZG multiproofs](#kzg-multiproofs)
     - [`compute_kzg_proof_multi_impl`](#compute_kzg_proof_multi_impl)
@@ -266,12 +266,12 @@ def interpolate_polynomialcoeff(xs: Sequence[BLSFieldElement], ys: Sequence[BLSF
     return r
 ```
 
-#### `zero_polynomialcoeff`
+#### `vanishing_polynomialcoeff`
 
 ```python
-def zero_polynomialcoeff(xs: Sequence[BLSFieldElement]) -> PolynomialCoeff:
+def vanishing_polynomialcoeff(xs: Sequence[BLSFieldElement]) -> PolynomialCoeff:
     """
-    Compute a zero polynomial on ``xs`` (in coefficient form)
+    Compute the vanishing polynomial on ``xs`` (in coefficient form)
     """
     p = [1]
     for x in xs:
@@ -312,7 +312,7 @@ def compute_kzg_proof_multi_impl(
     polynomial_shifted = add_polynomialcoeff(polynomial_coeff, neg_polynomialcoeff(interpolation_polynomial))
 
     # For all x_i, compute (x_i - z)
-    denominator_poly = zero_polynomialcoeff(zs)
+    denominator_poly = vanishing_polynomialcoeff(zs)
 
     # Compute the quotient polynomial directly in evaluation form
     quotient_polynomial = divide_polynomialcoeff(polynomial_shifted, denominator_poly)
@@ -330,7 +330,7 @@ def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
     """
     Helper function that verifies a KZG multiproof
     """
-    zero_poly = g2_lincomb(KZG_SETUP_G2[:len(zs) + 1], zero_polynomialcoeff(zs))
+    zero_poly = g2_lincomb(KZG_SETUP_G2[:len(zs) + 1], vanishing_polynomialcoeff(zs))
     interpolated_poly = g1_lincomb(KZG_SETUP_G1[:len(zs)], interpolate_polynomialcoeff(zs, ys))
 
     return (bls.pairing_check([
@@ -468,7 +468,7 @@ def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Po
     assert len(cells) >= CELLS_PER_BLOB // 2
     cell_ids = [cell_id for cell_id, _ in cells]
     missing_cell_ids = [cell_id for cell_id in range(CELLS_PER_BLOB) if cell_id not in cell_ids]
-    short_zero_poly = zero_polynomialcoeff([ROOTS_OF_UNITY_REDUCED[reverse_bits(cell_id, CELLS_PER_BLOB)] for cell_id in missing_cell_ids])
+    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:

From 9ec5dde8ac1982f241b335540d0fa1b692e745a5 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Sat, 23 Dec 2023 15:21:19 +0000
Subject: [PATCH 08/28] Remove polynomial_coeff_to_eval

---
 specs/deneb/polynomial-commitments-sampling.md | 16 ----------------
 1 file changed, 16 deletions(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index fcc5800148..cf70cc7b09 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -152,22 +152,6 @@ def polynomial_eval_to_coeff(polynomial: Polynomial) -> PolynomialCoeff:
     return polynomial_coeff
 ```
 
-#### `polynomial_coeff_to_eval`
-
-```python
-def polynomial_coeff_to_eval(polynomial_coeff: PolynomialCoeff) -> Polynomial:
-    """
-    Evaluates a polynomial (given in coefficient form) to a polynomial in evaluation form.
-    """
-
-    if len(polynomial_coeff) > FIELD_ELEMENTS_PER_BLOB:
-        assert all(c == 0 for c in polynomial_coeff[FIELD_ELEMENTS_PER_BLOB:])
-
-    polynomial = bit_reversal_permutation(fft_field(list(polynomial_coeff), list(ROOTS_OF_UNITY), inv=False))
-
-    return polynomial
-```
-
 #### `add_polynomialcoeff`
 
 ```python

From 75b1bf515aa3108fdbd3e956e34fcebd432e89e6 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Fri, 5 Jan 2024 18:21:15 +0800
Subject: [PATCH 09/28] Add `KZG_SETUP_G1_MONOMIAL` to trusted setup files

---
 .../trusted_setups/trusted_setup_4096.json    | 12418 ++++++++++------
 .../trusted_setups/trusted_setup_4096.json    | 12418 ++++++++++------
 setup.py                                      |     8 +-
 .../deneb/polynomial-commitments-sampling.md  |     8 +-
 specs/deneb/polynomial-commitments.md         |     3 +-
 5 files changed, 16527 insertions(+), 8328 deletions(-)

diff --git a/presets/mainnet/trusted_setups/trusted_setup_4096.json b/presets/mainnet/trusted_setups/trusted_setup_4096.json
index c6d724efaf..28c701062b 100644
--- a/presets/mainnet/trusted_setups/trusted_setup_4096.json
+++ b/presets/mainnet/trusted_setups/trusted_setup_4096.json
@@ -1,4167 +1,8265 @@
 {
+  "g1_monomial": [
+    "0x97f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb",
+    "0x96558cd0b28d8fcb01dab32f6b50422bcc0d67ba67ec8823cb3b5f3da727c687e42e8513461eb9f946ac04c7e85af17d",
+    "0x92693b31ad1d408fbd1ab4461bbf5b7d894efd0eef28d86e6990ffb9ccf5f9f03eb7cec9258079c3f6405db79236f296",
+    "0xa80773c8da8028fb369c0c72493463005181dc865202c51df2d4981ebbac5e77d8d03cb005fe0148a1fbd859c6d12e36",
+    "0x8b75d035b73ab1e6a3d657a76f324fc8f510413973b3ffd398bae84e8566eb1eb8599697d0c8e573158d02f9efa9c15b",
+    "0xa46ff442a6d74827701ae8f892a6b172a3a2d36257798be67b90b821ce718560d957ac3e9046e6a62f665ac28b3165f7",
+    "0xb07608a3792da0c1a96c7da13371d71653fb18e59337d9af47e9d2dfd5067e736f0f4065a190d4c2a0fab4f146a0fdb6",
+    "0xa228591d04d443b250ec54e84d9c36ce400a3aed898bc44a1992793171807ea643774da89a89a985d7fac411b0f2f7ee",
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+    "0xa4b261195bc45087155d503b8fd3caa4308d64e345cff2fe195229fe3b603daba14d9e0f0bf5efd54ce48ff99405555a02ac740f4922b10d5a16a44300b6202ab2ed79f1f6a6c6385c8d938ffae8811e5e41c8b10e488c0e7b54655b0d870894",
+    "0x8784c0057a097bd26bdfb08674104ae826e8cd04a74c485dd40486bbc0c6d32a610d4548b89e26a57cb3bafb4dbb0aae0206654abb7f32576995f96f70f3833abf56e2e11c9adc2a09fdb9fec8f6e8f00a7eb9a77854d97d77165eb1b849e4ea",
+    "0x99bd1c98d1585a6c807d2a7247a251b7fd49356e392da286db71ca41107635019d6ddbbb715034354ddf260c7c37b0990f5a00a614b97e03f5cd2a299d6a133725153d13a15921692e0475e35c5893f8e29b47585a4d1ed8f9715a4300f3a850",
+    "0x9415b67548643efdb524229097248906081e753ef72b0599af671ad8853f58f57b62ad4b10df288bbfd9f38607aebd3b02df2695227b36a17ca86b660609ed3df44a3c4fb48cd53d394c6c0643ccf458f857bc74923d85ebdeaa98d35ab928ef",
+    "0xb9d0ad72217402253344a7236755ea923e63c0883b1b57336a47ec3fcb3760daba7f3db33c874d5ff295e9e78df9b1f3027f6dbab07106e90a1a2058ab0032150797136d986d71f11703118efc7d6212e6ddc829cc29a36e02de80c38922997b",
+    "0x82ee387b8772875d9ab19dc2b684245959081ba1f9de8544e9227509cbe0d746b82c37a4334b0e98f007385f71a8eebd0c0e93e98d8c6032e66beb7718b6c6de6de2c3f4d6e913372f55453b05870eba112dded60443d01c5bca09a50096e99f",
+    "0xae6da3274a20f5e48e7c005701b51e4eeea65588728f8a2c05c83ee83fec507269cad21f43610407d9de5eb6e6d29e2d0ce60dbf13e8e7b06d5f9a3c24dac8673de96758386e47d51bebb4347cd2d7bf9b78bc96d2945f1f410022ab92fdfa17",
+    "0xb7b706a2dd08efee9315fb45eff2a2e6429b65abd2ddff6cbd9eca7d8cf77b28fb8d7ca058c49e2a0133f0426b2af73d1173c18e10cd033015771f2f2a353ad38d46846ada7033284c92f1d2a42730f0c98151d6080da7483b4f9691fc5af7ab",
+    "0xa2b472c42d1a7e9dcdb37b0940165842002bb609794fed02b270e2747cc1ed48a7d77fdae6052c3ec0f8f47069c974280718263412e7719e5e8c7c8bb6951c35a8074366f813e42bfe854d980f88f50150752f9dfce84289c40a5a00312c6493",
+    "0x8415485e8fc015841329f0129654901121fc2ccdba0e677c69eadd8feadd7c1fbdd1cedee4ba932b8d3003373c921685137581715fc4204fe561053499d2be7ac739abf76c0b902c1ba6893aa262cdc991dbfa923d465c8e092c3537dde3db42",
+    "0x86ffda56e585e1fc8b79f3ecf7b1debfeb7775315b58f4136fa1ed62d54d28debe5b701e552e4b89fb82ed29f68920b6091fb99ea9885a718ef00e08193ce883b13e9c67ff1ac427966ab0b2d7052405af29fce073280cc96df9c3ff485ea8f3",
+    "0x865bf3ebc1e8e23e653a8ce689095dea1fdb64f0bea5075944da546129cd2c3132fc537b772efce5573a866713bb91c41180aca1d74d57bcd17fb7fb83da68464615aacaaf2fc8fece362eb190ca717afd34dd213767522a5e9b349e4208b116",
+    "0xa3feb1fc530811930d3be116c5c490fa71ab5f770f45250cb99dd6c59166531b7251897709b6bfb79983957c31a9b9e811d938cc59c52d8c9811e746e41bea11310fa6a9430d1b15c2f747982fabf48aa8f3a08b1daa662692f6c94e7cd4a7bf",
+    "0xb3726ce2719b590ea0d1869650df55c581391cf89c163e7b218cf8fc6adcc3b038c30d44ff877d40ce1a052691e742881255fea3715976d8019004d9d5f7a6afcf33ae8be226809e8f64d1a6e922f2c9a32ea1dd2474983ea42dd468ffcf2f26",
+    "0xa5dd586b7fe65513a5e25df9ad1bb367b1d86856deb00c50e2a22c50c9c462c9824472a53ed7d167f5724580277ea14e035fe94dbd0003dea1482a71ef2e1f9f72b89547152085256a4b572081fd47f8f585a1119802fd560f7c8757581ede46",
+    "0x9957b1b8f8b5d63002945ebba5ee6cb258387cfbe4b7dd00cad914b62ae24ee7cc7d801fdc1d1c92ee83ed6207c2e5520f41e7cda2ce6c91edbf27cdc22a9114d7ecf90cc7c642c1d76c78367888b4379f7ed909ce9808f95dba93da29168c58",
+    "0x8b64b6555d305c87af94d1427a778afe9897cde055aa15096b6a7cac15bc0e521a3dfede15ae5039a0aad221c97b703c00e508426f691c2a0c7a008681ddd854fbebbce34e422e3daf113b9bde6c2a9896148362296a760cd99a16408827e1ec",
+    "0x8cd3fe47984c9320265d1d4a70ac9a414e4fb9eca022d46de90ac6682ff5628a257bb591820f6845147e3db65858419d0a14a7b539c24424000de06bb584b617c463ea375c4580c5855448abee95934409ddec2fd607b442d0c1a732d5f8a25e",
+    "0x8029cfaecd192bf218425eadef81e7db82fbf5bc342282d6da56523cf5dd77e439da9b59d57ff8b1becefcb57d1e03251717955434aee29d1ed386782344530d180e6a94abc445776d30a5573d1cecdebff08d29c9b886e108518a72c1733a6f",
+    "0xa6d4d047aaae8060aa9383944f0fde577bae0f1a5f051ad974cacb8c3351812bc9e4ba4be512c2b0bed522c5474a2b080238f168feddb0b4eafb72bfd86387d8af2fc9a5daeafbf2e23d33414e43f20937e44e36229730b906416d3c9b0b0a76",
+    "0x926365068bea32e19414f724eb5d332a045166808b79f9893866d828a0a6926d1ac955fea9eaabe7dfcd3a30050a2c1b06cca84d3976bda4e9586ed77743f235417ff893fdce9eb6fd7fac38fe1d0a87adef2ed49f9cc38b904e1a87e7db883d",
+    "0xa68cc5c7b73401c07b3151aef64b4124794cf03e323925dad43ee5a979ead23957c99e46c1bebb89e91b6369f93e436101ffd93fa34829771e7310685f91039321cc23abb3ebbaa26c5afd54ba6d35516637db31b6290628d9614e8b54fa6884",
+    "0xb45e0dae84ec6664bb9ef83799007296296b9da97902ac3765dbc94ee247e83fdf776b4da98d52aab390d1a2158ea740040403779facddcbd0026929a8ded0dbe8e0d9404851ceb585e919fb0e986dbcf32b51cacafc41c8118bd6ca04173fd3",
+    "0x8b428d3cca95bc417dbe5de63909ad97cb6be1e7a6808ce86e9bbcff80a6aa8cf5f43e6a115949952e737f6d2d5c687317b17aa32e7c919dc0c227948b21f80e7b5bb54ba54bf19043dd7ac892f631585ed13e91f4fb27fd23827cde339b1243",
+    "0x93f0e73aecdc4e4384efd75839990f7a1ef616f38433d27ded8747cc274d76ea613f5d950c4a65d7969c4faa721d582807973776b83733ac5bbe3388c25d0bfa571eef6c3a11e9d414cc9ddc9731755b7ffc8b3ec69c004491c72b999e06753e",
+    "0x8b74d10fe9741bba21b014a9126dfbdd58f7266b21d19ecf7a8329dfef4d8f0c381bd40090b81050a0e36094078a9f66096363dd0b980defa8de5046e412c1fbb2f2ddaba684d89f57e84017001fa620d59ea284c3dabd2af04a9a74226c1568",
+    "0x93cf284e564c61bc68dc41ff4d2007f651bf496306ad77b407d14df66731b3cd7c977981339d3f98f2b1bb180a942078054c9a176201d36adcf9837a050531a7800a19c7c02bd539300dce89fc992fff08fccd31448570dd96319939b622b1e3",
+    "0xa00341e281bc09e7e74e00c668112a9b288985bb91c13dcbb956cd782c81a6e998ec615e5470943a5c4a8858abd474c2004a30c839a2a327f41f58c36e4484f7fb65d23c14dcbae1e674948a24e1444ff873d4f78b3aa9d633252f3567061c02",
+    "0x93c943fa20eb348d743618708f31388ef3e52ef19a6b53eeeb15c5434cff8b9293f690462601cb82c05dcb6eaeb3e9320db0eb242cccfc9e5aa0df44dd411992ced67e4452a99439fee06da61eec3245a103d63b8e6df7a93e458b1b1d72d3dd",
+    "0xb1c5561d50fdb6a39f7e3136a60025e86c833702618831f9414d843c2740b0695b0ab0c9379cb2c1309808428e37d4da12220a5800246079f7d5fd58254d4a07c5d91a00fc7d5e5dc5862d13e8ec967ed49b8678ced22a1ddf9182b4933e62eb",
+    "0xb5ebc741fb89ed7d17b8830ade28069539ff20e5b495002c1c30b8c272337033fa4fd39ba3728aa43564436e037d1283077762d2c667fd6f11f2338440175f7d4018cbcdb682e6ad4a740ba38b83242ae58193b1bc717d23dd73eb23b300fa34",
+    "0xa2eb653f4b2b60394c362638daa69c60ea27500988687937c96b50c0049839e6f30247cbb97f665b7832a3911ceae935019df65831ed5f2b28b49256eb6282e0aa98aaed810813596be571f7e4c776357255829338942a2cf354dee0e8581f54",
+    "0xa15771d71c19bb6514ce973c4cd0892a06f2c5d81fce249beed03c113e01fb338f633c5e8d276a0eaa94c2803578323d050f91f3b0c2cf20f2c7186d891aad78e8d991627434a2ad9e715cb128ed1bd539e70424d68d7b5a258c458dd26e584e",
+    "0x93c9a3a580f1e68e5b8279b8e12eebde0e0871237eff4554c9d8b08297927b29baefb6e4ded27cc2e4c2b3099a89e07f179244a9204192ea107cb5c9331daf02049ec452a547a4ab3b12efdf9eb85b5eb6bad7bfa58ddfb175d75ac91a51c8bd",
+    "0xafa5c84a05d3fc247bd44cb94d9c4b8d113a7e8dca22f1e2598fbd907136c5fde807c76ed7aa967cddbff3202b9985ce0d918388c092308bd91e0a80faf328fbce6fdd8bd97c3d3c1683e1cd387e8f7cf614d12857ce97f64e7ed9135dd7abe8",
+    "0x856348ff2d6e02a6ffa047e8b6142da89c4c4c733c3d7f709de737034cf9e0c64a6c98769ff58d28f47d5925e7b3e2660151880d30c02f0fd3b5a6aac466509c90ffd6058b10c8af12bb47fd141076a3cc58913742cfb2b869eca88ee69b13f0",
+    "0x894c8973dd37f9408fbba09e26cde999669f46c9e02a06ae8ca6d3fad51859169ecbd76cb94af47a00ece3cf1e3f3744001191ca9ed01f6be7c5e64632dfa216c77d79fe61c55d7febc0721a63d103a67bb8e579a414669bbe5d89d3b6093b65",
+    "0x958ef9c72cf9892b44840b3dc902af187582bf4da15c1ffb889304b4de0d617a9f82c6a8e38e08fdfc840cfe62c440920e489f3b5265a1a5f7f72600c7c4fc17fc59d0576c0a6c7bad368be58f6bcb04707defc7b281730e89246359790d73fb",
+    "0xa20740691a9628bfa2e527c5b46e2335a83e32c58b9c3e71b0937580df4fdf950a22b664cde778c37b087b01d6a5983415245575873c61d30827830445e210f19e752a8b34c6f25d6e634651ea32388a3a569e09993d9a2c2e3df29b52f4c6fa",
+    "0xa04c19269e6b32a5aaefd9f9e94fef2917cb45257cc7de3100c72e18f8b7f5b59b48639d2e7aa3e94b11d58db43f2b780c9a3c15d9c268b2d2d40b27accb22ac83031a217e77114d786e155fa005684a65288bb75df22da09608b665a7bda7c9",
+    "0xaf828007260da1311fcb7ff6a93ba8a3317ad28d1712b619b2a25229d6accf5bb14949515c06f52d4ef23d466636c5b60f4dc41c9449e69175615a4a1002f83063fc833c105db1c7f87e381f44bf944c7e894ef4f5995cff32d5079c1988bdaf",
+    "0x8b6cdb1567cc7508b39fd66d5837e7444e32618009ac9775827242e637af5001ba182e418e68db8bc1db01f4ac01dcc211be36ed9e4d6a2e11a83e148d52b7178af178401eda13c7b2c5aff6d84292ba0c9893117e9359f0f8436943de526687",
+    "0x8a815b1dd048b8f7bb7f903379c78e585b6bb35a7be7f72206b9c297fc363080c91c9b8e466c2894be35683fd2f104680e04bb236c5e885fbfe29ee0c5a7e5b41b3c3af3d848d1e7bf6e70543e386b008b11a3ab1118f1a2ddd85f60bd30f3b6",
+    "0xb2d70c53d4e1d431e990c335a0be4d9c6465c614e54bef9d6a4f6e60ddabb82756913a83f9285a52e9058a3dec0c056f0153b065a28a59bca9ef0b2b5420853ca27832ba834c6bc55e51041d0c68c933e392f7c164ccb1dc68d0ab8bf9394eb4",
+    "0x96566def38e1721f625bd6ba9972acec5b30dbba5a90b54e761ae4260204c593800c75f8be10c2475ee4cbc118834ec10c5fef690c734e4506dd73e50a0fffea0c7b12adf951a59385ca82075c57a59263a0b5994d1adc7168bba70c15ff7193",
+    "0x943ee9e7d01eb43643c994dfec1e2133fee067768e04a936619bfc34fd8431d33b449a0425da2898fb5f54c7b1723b2a04dbffce300b575fe79a65f5763fdc6d114f06056752d3c0515cc50a0590dfffe8b9da21936bef79b98ff03515b2db7c"
   ]
 }
\ No newline at end of file
diff --git a/presets/minimal/trusted_setups/trusted_setup_4096.json b/presets/minimal/trusted_setups/trusted_setup_4096.json
index c6d724efaf..28c701062b 100644
--- a/presets/minimal/trusted_setups/trusted_setup_4096.json
+++ b/presets/minimal/trusted_setups/trusted_setup_4096.json
@@ -1,4167 +1,8265 @@
 {
+  "g1_monomial": [
+    "0x97f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb",
+    "0x96558cd0b28d8fcb01dab32f6b50422bcc0d67ba67ec8823cb3b5f3da727c687e42e8513461eb9f946ac04c7e85af17d",
+    "0x92693b31ad1d408fbd1ab4461bbf5b7d894efd0eef28d86e6990ffb9ccf5f9f03eb7cec9258079c3f6405db79236f296",
+    "0xa80773c8da8028fb369c0c72493463005181dc865202c51df2d4981ebbac5e77d8d03cb005fe0148a1fbd859c6d12e36",
+    "0x8b75d035b73ab1e6a3d657a76f324fc8f510413973b3ffd398bae84e8566eb1eb8599697d0c8e573158d02f9efa9c15b",
+    "0xa46ff442a6d74827701ae8f892a6b172a3a2d36257798be67b90b821ce718560d957ac3e9046e6a62f665ac28b3165f7",
+    "0xb07608a3792da0c1a96c7da13371d71653fb18e59337d9af47e9d2dfd5067e736f0f4065a190d4c2a0fab4f146a0fdb6",
+    "0xa228591d04d443b250ec54e84d9c36ce400a3aed898bc44a1992793171807ea643774da89a89a985d7fac411b0f2f7ee",
+    "0x8ec4cd63b1715f5f798ad06925cbd579aa64f86593305e26f271c66f99f1b2eba127b6546e6608451dbb3f283f5b67fa",
+    "0x883a321fe632a54e5d1e819f5214706b5715cd5e74920a62eddb4c82d2c883f17ee5add2fba35fca4f3f4a57f5187567",
+    "0xa14976b0239a57068d2e76f9f50d45137ce352a0873623c1937d1a26d1b6c6cf53fad2961774438fa8a34f27dde0db1d",
+    "0xa6264da28dff0cb8f49785fae572fbd159445a7ad94f964e8188cdb2798d000b5545751de45cbe39e82150d92f21ed3e",
+    "0xad64c7a9349f20e42dd5669f96dcb581d67b5b8a6008fda4ac380c3f37963bf824575a256fc0fb9370b8efe3be92cef0",
+    "0xaf1c1b6dec3b2869e7f5715849ceb1f12fd1ad883f3b7c8df8be4b1f2833fc74db43053bef73e13026c599e1eb35810e",
+    "0x86153529b0557fea5c507613a576c63f3da89962fca13ca09189ac3f4e0e305d87159a6ec5bf727e657feba49813333a",
+    "0xa8a9385b7b8b99c0f357f3dfbc483f9acacc1ac6b98ce2a45c6358efd40a07f68023021f00d432803db04a1f6925ac23",
+    "0xb82e4d618ae512b043c39c8d7762a5453a00c74e95d18e2b5ab1123716354133ca7409bb1c41987ea7c9c0c35e62b3e0",
+    "0xb3623da173ee84ace7fecdc11d3e0fc16da293fa1f322473188bca566c2b14826062d293a921a8ca4cdb93d0ab4c901e",
+    "0xa244ed6bda5c3d73154a97d483ee8b1dfb41b9338da5a22ff16c59f169fff1dde8cf91e5867e6cc2302e751e4f008ce4",
+    "0x93299a86a0101a059c72760f7dabac1d1d1f1524b66a76ea22321791b40df35c6e02d419a298325ba44288957951d17a",
+    "0x832a1608c119809492277a52766dcba60d5f6f715431d45012a3a0ae0cae96328d2eb82ba32d96c0a0f6b90d1888fc57",
+    "0xab3ece5be4e872e1b3d00e0f0e6b838580844d8fe11e47e77c943180523310f3604ba6e382639d286965a6364ddd5c99",
+    "0xb99b2eb75d7347b53d49f408728027f2f47ff5b327da6a2ad48044842b7baffa2eaa57fef3ea631fc2187a8d42f0e365",
+    "0x991070eb69689c3d458812247582012f2223c76054268c90d0776182985fc9f076700405a14aa72b12eeb28afc900646",
+    "0xb800481fa4204480e8fcc51e012e16a469b7303dacf32b93cd54c32ed828ff421eaa4e808ccfa4d6985d2cec8119cd00",
+    "0x874033cc725413f9c103fbc6c1647fc7a6277025a173dbb9d73a89e3f085796e536a0fafe8f787c60b9cc12a3b65b589",
+    "0xad5352d1fffb48c65286c35d34ad421ea2d0e3a78868d84f51a2bc7c8d2d0319d885bd16ea9acb76193c050997a96062",
+    "0xa065fc87ee9bb50ec31cb9272f3fd9cd15491e14b9675993a2ad671e12276b0c9adda2d78b05f9c027149b524237d4fc",
+    "0x812ed92ee43e3dc5fa7143fd0731f78f5ce80da91d5e5885085137199461b740e58733084041b3c4f512cceb1a30f579",
+    "0x95a13f516b33ee2d052629d66c7417245a356c76e0f095c9484b29239df14704b5573575c5efc14bbc94c833adb4b19b",
+    "0xaa9ffb492b0494d855ec03d701a885b6837a7e9a5a7e3f70e2adffcd58493249ddead213dc2fa349c4a8b59c6502d902",
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+    "0x894c8973dd37f9408fbba09e26cde999669f46c9e02a06ae8ca6d3fad51859169ecbd76cb94af47a00ece3cf1e3f3744001191ca9ed01f6be7c5e64632dfa216c77d79fe61c55d7febc0721a63d103a67bb8e579a414669bbe5d89d3b6093b65",
+    "0x958ef9c72cf9892b44840b3dc902af187582bf4da15c1ffb889304b4de0d617a9f82c6a8e38e08fdfc840cfe62c440920e489f3b5265a1a5f7f72600c7c4fc17fc59d0576c0a6c7bad368be58f6bcb04707defc7b281730e89246359790d73fb",
+    "0xa20740691a9628bfa2e527c5b46e2335a83e32c58b9c3e71b0937580df4fdf950a22b664cde778c37b087b01d6a5983415245575873c61d30827830445e210f19e752a8b34c6f25d6e634651ea32388a3a569e09993d9a2c2e3df29b52f4c6fa",
+    "0xa04c19269e6b32a5aaefd9f9e94fef2917cb45257cc7de3100c72e18f8b7f5b59b48639d2e7aa3e94b11d58db43f2b780c9a3c15d9c268b2d2d40b27accb22ac83031a217e77114d786e155fa005684a65288bb75df22da09608b665a7bda7c9",
+    "0xaf828007260da1311fcb7ff6a93ba8a3317ad28d1712b619b2a25229d6accf5bb14949515c06f52d4ef23d466636c5b60f4dc41c9449e69175615a4a1002f83063fc833c105db1c7f87e381f44bf944c7e894ef4f5995cff32d5079c1988bdaf",
+    "0x8b6cdb1567cc7508b39fd66d5837e7444e32618009ac9775827242e637af5001ba182e418e68db8bc1db01f4ac01dcc211be36ed9e4d6a2e11a83e148d52b7178af178401eda13c7b2c5aff6d84292ba0c9893117e9359f0f8436943de526687",
+    "0x8a815b1dd048b8f7bb7f903379c78e585b6bb35a7be7f72206b9c297fc363080c91c9b8e466c2894be35683fd2f104680e04bb236c5e885fbfe29ee0c5a7e5b41b3c3af3d848d1e7bf6e70543e386b008b11a3ab1118f1a2ddd85f60bd30f3b6",
+    "0xb2d70c53d4e1d431e990c335a0be4d9c6465c614e54bef9d6a4f6e60ddabb82756913a83f9285a52e9058a3dec0c056f0153b065a28a59bca9ef0b2b5420853ca27832ba834c6bc55e51041d0c68c933e392f7c164ccb1dc68d0ab8bf9394eb4",
+    "0x96566def38e1721f625bd6ba9972acec5b30dbba5a90b54e761ae4260204c593800c75f8be10c2475ee4cbc118834ec10c5fef690c734e4506dd73e50a0fffea0c7b12adf951a59385ca82075c57a59263a0b5994d1adc7168bba70c15ff7193",
+    "0x943ee9e7d01eb43643c994dfec1e2133fee067768e04a936619bfc34fd8431d33b449a0425da2898fb5f54c7b1723b2a04dbffce300b575fe79a65f5763fdc6d114f06056752d3c0515cc50a0590dfffe8b9da21936bef79b98ff03515b2db7c"
   ]
 }
\ No newline at end of file
diff --git a/setup.py b/setup.py
index da14759ca5..5d4cf98f24 100644
--- a/setup.py
+++ b/setup.py
@@ -112,10 +112,11 @@ def _load_kzg_trusted_setups(preset_name):
 
     with open(trusted_setups_file_path, 'r') as f:
         json_data = json.load(f)
+        trusted_setup_G1_monomial = json_data['g1_monomial']
         trusted_setup_G1_lagrange = json_data['g1_lagrange']
         trusted_setup_G2_monomial = json_data['g2_monomial']
 
-    return trusted_setup_G2_monomial, trusted_setup_G1_lagrange
+    return trusted_setup_G1_monomial, trusted_setup_G1_lagrange, trusted_setup_G2_monomial
 
 def _load_curdleproofs_crs(preset_name):
     """
@@ -167,9 +168,10 @@ def _parse_value(name: str, typed_value: str, type_hint: Optional[str] = None) -
 def _update_constant_vars_with_kzg_setups(constant_vars, preset_name):
     comment = "noqa: E501"
     kzg_setups = ALL_KZG_SETUPS[preset_name]
-    constant_vars['KZG_SETUP_G2_MONOMIAL'] = VariableDefinition(constant_vars['KZG_SETUP_G2_MONOMIAL'].value, str(kzg_setups[0]), comment, None)
+    constant_vars['KZG_SETUP_G1_MONOMIAL'] = VariableDefinition(constant_vars['KZG_SETUP_G1_MONOMIAL'].value, str(kzg_setups[0]), comment, None)
     constant_vars['KZG_SETUP_G1_LAGRANGE'] = VariableDefinition(constant_vars['KZG_SETUP_G1_LAGRANGE'].value, str(kzg_setups[1]), comment, None)
-
+    constant_vars['KZG_SETUP_G2_MONOMIAL'] = VariableDefinition(constant_vars['KZG_SETUP_G2_MONOMIAL'].value, str(kzg_setups[2]), comment, None)
+    
 
 def get_spec(file_name: Path, preset: Dict[str, str], config: Dict[str, str], preset_name=str) -> SpecObject:
     functions: Dict[str, str] = {}
diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index cf70cc7b09..31b8b650f8 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -301,7 +301,7 @@ def compute_kzg_proof_multi_impl(
     # Compute the quotient polynomial directly in evaluation form
     quotient_polynomial = divide_polynomialcoeff(polynomial_shifted, denominator_poly)
 
-    return KZGProof(g1_lincomb(KZG_SETUP_G1[:len(quotient_polynomial)], quotient_polynomial)), ys
+    return KZGProof(g1_lincomb(KZG_SETUP_G1_MONOMIAL[:len(quotient_polynomial)], quotient_polynomial)), ys
 ```
 
 #### `verify_kzg_proof_multi_impl`
@@ -314,14 +314,14 @@ def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
     """
     Helper function that verifies a KZG multiproof
     """
-    zero_poly = g2_lincomb(KZG_SETUP_G2[:len(zs) + 1], vanishing_polynomialcoeff(zs))
-    interpolated_poly = g1_lincomb(KZG_SETUP_G1[:len(zs)], interpolate_polynomialcoeff(zs, 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[0])),
+            bls.neg(bls.bytes96_to_G2(KZG_SETUP_G2_MONOMIAL[0])),
         ],
     ]))
 ```
diff --git a/specs/deneb/polynomial-commitments.md b/specs/deneb/polynomial-commitments.md
index c0a293d8b7..176406bfc3 100644
--- a/specs/deneb/polynomial-commitments.md
+++ b/specs/deneb/polynomial-commitments.md
@@ -104,8 +104,9 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | Name | Value |
 | - | - |
 | `KZG_SETUP_G2_LENGTH` | `65` |
-| `KZG_SETUP_G2_MONOMIAL` | `Vector[G2Point, KZG_SETUP_G2_LENGTH]` |
+| `KZG_SETUP_G1_MONOMIAL` | `Vector[G1Point, FIELD_ELEMENTS_PER_BLOB]` |
 | `KZG_SETUP_G1_LAGRANGE` | `Vector[G1Point, FIELD_ELEMENTS_PER_BLOB]` |
+| `KZG_SETUP_G2_MONOMIAL` | `Vector[G2Point, KZG_SETUP_G2_LENGTH]` |
 
 ## Helper functions
 

From 5854c7efb84cd6710421b516889f9a21c0826778 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Fri, 5 Jan 2024 18:33:01 +0800
Subject: [PATCH 10/28] Fix some lint errors

---
 .../deneb/polynomial-commitments-sampling.md  | 28 +++++++++++--------
 1 file changed, 17 insertions(+), 11 deletions(-)

diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/deneb/polynomial-commitments-sampling.md
index 31b8b650f8..df29f912a7 100644
--- a/specs/deneb/polynomial-commitments-sampling.md
+++ b/specs/deneb/polynomial-commitments-sampling.md
@@ -398,9 +398,9 @@ def compute_cells(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_P
 
 ```python
 def verify_cell_proof(commitment: KZGCommitment,
-                        cell_id: int,
-                        data: Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL],
-                        proof: KZGProof) -> bool:
+                      cell_id: int,
+                      data: Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL],
+                      proof: KZGProof) -> bool:
     """
     Check a cell proof
 
@@ -415,10 +415,10 @@ def verify_cell_proof(commitment: KZGCommitment,
 
 ```python
 def verify_cell_proof_batch(row_commitments: Sequence[KZGCommitment],
-                              row_ids: Sequence[int],
-                              column_ids: Sequence[int],
-                              datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]],
-                              proofs: Sequence[KZGProof]) -> bool:
+                            row_ids: Sequence[int],
+                            column_ids: Sequence[int],
+                            datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_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:
@@ -452,7 +452,10 @@ def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Po
     assert len(cells) >= CELLS_PER_BLOB // 2
     cell_ids = [cell_id for cell_id, _ in cells]
     missing_cell_ids = [cell_id for cell_id in range(CELLS_PER_BLOB) if cell_id not in cell_ids]
-    short_zero_poly = vanishing_polynomialcoeff([ROOTS_OF_UNITY_REDUCED[reverse_bits(cell_id, CELLS_PER_BLOB)] for cell_id in missing_cell_ids])
+    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:
@@ -463,10 +466,13 @@ def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Po
     zero_poly_eval = fft_field(full_zero_poly, ROOTS_OF_UNITY_EXTENDED)
     zero_poly_eval_brp = bit_reversal_permutation(zero_poly_eval)
     for cell_id in missing_cell_ids:
-        assert zero_poly_eval_brp[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
-            [0] * FIELD_ELEMENTS_PER_CELL
+        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:
-        assert all(a != 0 for a in zero_poly_eval_brp[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL])
+        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_data in cells:

From 7f8858b2ac4641a98539cd50ab942e3ce159d2dc Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Fri, 5 Jan 2024 18:43:37 +0800
Subject: [PATCH 11/28] Add `PeerDAS` feature

---
 .gitignore                                    |   1 +
 Makefile                                      |   2 +-
 configs/mainnet.yaml                          |   4 +
 configs/minimal.yaml                          |   4 +
 pysetup/constants.py                          |   1 +
 pysetup/md_doc_paths.py                       |   2 +
 pysetup/spec_builders/__init__.py             |   3 +-
 pysetup/spec_builders/peerdas.py              |  14 ++
 specs/_features/peerdas/fork.md               | 125 ++++++++++++++++++
 .../polynomial-commitments-sampling.md        |   0
 .../pyspec/eth2spec/test/helpers/constants.py |   3 +
 11 files changed, 157 insertions(+), 2 deletions(-)
 create mode 100644 pysetup/spec_builders/peerdas.py
 create mode 100644 specs/_features/peerdas/fork.md
 rename specs/{deneb => _features/peerdas}/polynomial-commitments-sampling.md (100%)

diff --git a/.gitignore b/.gitignore
index e88207dd77..b9a7b5ff3d 100644
--- a/.gitignore
+++ b/.gitignore
@@ -24,6 +24,7 @@ tests/core/pyspec/eth2spec/deneb/
 tests/core/pyspec/eth2spec/eip6110/
 tests/core/pyspec/eth2spec/eip7002/
 tests/core/pyspec/eth2spec/whisk/
+tests/core/pyspec/eth2spec/peerdas/
 
 # coverage reports
 .htmlcov
diff --git a/Makefile b/Makefile
index d0d750e893..2003c5df5a 100644
--- a/Makefile
+++ b/Makefile
@@ -35,7 +35,7 @@ MARKDOWN_FILES = $(wildcard $(SPEC_DIR)/*/*.md) \
                  $(wildcard $(SPEC_DIR)/_features/*/*/*.md) \
                  $(wildcard $(SSZ_DIR)/*.md)
 
-ALL_EXECUTABLE_SPEC_NAMES = phase0 altair bellatrix capella deneb eip6110 eip7002 whisk
+ALL_EXECUTABLE_SPEC_NAMES = phase0 altair bellatrix capella deneb eip6110 eip7002 whisk peerdas
 # The parameters for commands. Use `foreach` to avoid listing specs again.
 COVERAGE_SCOPE := $(foreach S,$(ALL_EXECUTABLE_SPEC_NAMES), --cov=eth2spec.$S.$(TEST_PRESET_TYPE))
 PYLINT_SCOPE := $(foreach S,$(ALL_EXECUTABLE_SPEC_NAMES), ./eth2spec/$S)
diff --git a/configs/mainnet.yaml b/configs/mainnet.yaml
index e0f9128b70..3202af703f 100644
--- a/configs/mainnet.yaml
+++ b/configs/mainnet.yaml
@@ -154,3 +154,7 @@ BLOB_SIDECAR_SUBNET_COUNT: 6
 WHISK_EPOCHS_PER_SHUFFLING_PHASE: 256
 # `Epoch(2)`
 WHISK_PROPOSER_SELECTION_GAP: 2
+
+# PEERDAS
+PEERDAS_FORK_VERSION: 0x06000001
+PEERDAS_FORK_EPOCH: 18446744073709551615
diff --git a/configs/minimal.yaml b/configs/minimal.yaml
index cdfbca3a2c..eec2cde6ed 100644
--- a/configs/minimal.yaml
+++ b/configs/minimal.yaml
@@ -153,3 +153,7 @@ BLOB_SIDECAR_SUBNET_COUNT: 6
 # Whisk
 WHISK_EPOCHS_PER_SHUFFLING_PHASE: 4
 WHISK_PROPOSER_SELECTION_GAP: 1
+
+# PEERDAS
+PEERDAS_FORK_VERSION: 0x06000001
+PEERDAS_FORK_EPOCH: 18446744073709551615
diff --git a/pysetup/constants.py b/pysetup/constants.py
index 8d53455634..7f1d6dcdcf 100644
--- a/pysetup/constants.py
+++ b/pysetup/constants.py
@@ -7,6 +7,7 @@
 EIP6110 = 'eip6110'
 EIP7002 = 'eip7002'
 WHISK = 'whisk'
+PEERDAS = 'peerdas'
 
 
 
diff --git a/pysetup/md_doc_paths.py b/pysetup/md_doc_paths.py
index 781ae41db3..24621aac38 100644
--- a/pysetup/md_doc_paths.py
+++ b/pysetup/md_doc_paths.py
@@ -9,6 +9,7 @@
     EIP6110,
     WHISK,
     EIP7002,
+    PEERDAS,
 )
 
 
@@ -21,6 +22,7 @@
     EIP6110: DENEB,
     WHISK: CAPELLA,
     EIP7002: CAPELLA,
+    PEERDAS: DENEB,
 }
 
 ALL_FORKS = list(PREVIOUS_FORK_OF.keys())
diff --git a/pysetup/spec_builders/__init__.py b/pysetup/spec_builders/__init__.py
index 794ae50d29..745010d0ca 100644
--- a/pysetup/spec_builders/__init__.py
+++ b/pysetup/spec_builders/__init__.py
@@ -6,12 +6,13 @@
 from .eip6110 import EIP6110SpecBuilder
 from .eip7002 import EIP7002SpecBuilder
 from .whisk import WhiskSpecBuilder
+from .peerdas import PeerDASSpecBuilder
 
 
 spec_builders = {
     builder.fork: builder
     for builder in (
         Phase0SpecBuilder, AltairSpecBuilder, BellatrixSpecBuilder, CapellaSpecBuilder, DenebSpecBuilder,
-        EIP6110SpecBuilder, EIP7002SpecBuilder, WhiskSpecBuilder,
+        EIP6110SpecBuilder, EIP7002SpecBuilder, WhiskSpecBuilder, PeerDASSpecBuilder,
     )
 }
diff --git a/pysetup/spec_builders/peerdas.py b/pysetup/spec_builders/peerdas.py
new file mode 100644
index 0000000000..0d72976e2b
--- /dev/null
+++ b/pysetup/spec_builders/peerdas.py
@@ -0,0 +1,14 @@
+from typing import Dict
+
+from .base import BaseSpecBuilder
+from ..constants import PEERDAS
+
+
+class PeerDASSpecBuilder(BaseSpecBuilder):
+    fork: str = PEERDAS
+
+    @classmethod
+    def imports(cls, preset_name: str):
+        return f'''
+from eth2spec.deneb import {preset_name} as deneb
+'''
diff --git a/specs/_features/peerdas/fork.md b/specs/_features/peerdas/fork.md
new file mode 100644
index 0000000000..e0991c0f91
--- /dev/null
+++ b/specs/_features/peerdas/fork.md
@@ -0,0 +1,125 @@
+# PeerDAS -- Fork Logic
+
+**Notice**: This document is a work-in-progress for researchers and implementers.
+
+## Table of contents
+
+<!-- START doctoc generated TOC please keep comment here to allow auto update -->
+<!-- DON'T EDIT THIS SECTION, INSTEAD RE-RUN doctoc TO UPDATE -->
+
+- [Introduction](#introduction)
+- [Configuration](#configuration)
+- [Helper functions](#helper-functions)
+  - [Misc](#misc)
+    - [Modified `compute_fork_version`](#modified-compute_fork_version)
+- [Fork to PeerDAS](#fork-to-peerdas)
+  - [Fork trigger](#fork-trigger)
+  - [Upgrading the state](#upgrading-the-state)
+
+<!-- END doctoc generated TOC please keep comment here to allow auto update -->
+
+## Introduction
+
+This document describes the process of PeerDAS upgrade.
+
+## Configuration
+
+Warning: this configuration is not definitive.
+
+| Name | Value |
+| - | - |
+| `PEERDAS_FORK_VERSION` | `Version('0x05000000')` |
+| `PEERDAS_FORK_EPOCH` | `Epoch(18446744073709551615)` **TBD** |
+
+## Helper functions
+
+### Misc
+
+#### Modified `compute_fork_version`
+
+```python
+def compute_fork_version(epoch: Epoch) -> Version:
+    """
+    Return the fork version at the given ``epoch``.
+    """
+    if epoch >= PEERDAS_FORK_EPOCH:
+        return PEERDAS_FORK_VERSION
+    if epoch >= DENEB_FORK_EPOCH:
+        return DENEB_FORK_VERSION
+    if epoch >= CAPELLA_FORK_EPOCH:
+        return CAPELLA_FORK_VERSION
+    if epoch >= BELLATRIX_FORK_EPOCH:
+        return BELLATRIX_FORK_VERSION
+    if epoch >= ALTAIR_FORK_EPOCH:
+        return ALTAIR_FORK_VERSION
+    return GENESIS_FORK_VERSION
+```
+
+## Fork to PeerDAS
+
+### Fork trigger
+
+PeerDAS does not need a hard fork. We only add this fork doc for compiling this new feature in pyspec.
+
+For now, we assume the condition will be triggered at epoch `PEERDAS_FORK_EPOCH`.
+
+Note that for the pure PeerDAS networks, we don't apply `upgrade_to_peerdas` since it starts with PeerDAS version logic.
+
+### Upgrading the state
+
+If `state.slot % SLOTS_PER_EPOCH == 0` and `compute_epoch_at_slot(state.slot) == PEERDAS_FORK_EPOCH`,
+an irregular state change is made to upgrade to PeerDAS.
+
+```python
+def upgrade_to_peerdas(pre: deneb.BeaconState) -> BeaconState:
+    epoch = deneb.get_current_epoch(pre)
+    post = BeaconState(
+        # Versioning
+        genesis_time=pre.genesis_time,
+        genesis_validators_root=pre.genesis_validators_root,
+        slot=pre.slot,
+        fork=Fork(
+            previous_version=pre.fork.current_version,
+            current_version=PEERDAS_FORK_VERSION,  # [Modified in PeerDAS]
+            epoch=epoch,
+        ),
+        # History
+        latest_block_header=pre.latest_block_header,
+        block_roots=pre.block_roots,
+        state_roots=pre.state_roots,
+        historical_roots=pre.historical_roots,
+        # Eth1
+        eth1_data=pre.eth1_data,
+        eth1_data_votes=pre.eth1_data_votes,
+        eth1_deposit_index=pre.eth1_deposit_index,
+        # Registry
+        validators=pre.validators,
+        balances=pre.balances,
+        # Randomness
+        randao_mixes=pre.randao_mixes,
+        # Slashings
+        slashings=pre.slashings,
+        # Participation
+        previous_epoch_participation=pre.previous_epoch_participation,
+        current_epoch_participation=pre.current_epoch_participation,
+        # Finality
+        justification_bits=pre.justification_bits,
+        previous_justified_checkpoint=pre.previous_justified_checkpoint,
+        current_justified_checkpoint=pre.current_justified_checkpoint,
+        finalized_checkpoint=pre.finalized_checkpoint,
+        # Inactivity
+        inactivity_scores=pre.inactivity_scores,
+        # Sync
+        current_sync_committee=pre.current_sync_committee,
+        next_sync_committee=pre.next_sync_committee,
+        # Execution-layer
+        latest_execution_payload_header=pre.latest_execution_payload_header,
+        # Withdrawals
+        next_withdrawal_index=pre.next_withdrawal_index,
+        next_withdrawal_validator_index=pre.next_withdrawal_validator_index,
+        # Deep history valid from Capella onwards
+        historical_summaries=pre.historical_summaries,
+    )
+
+    return post
+```
diff --git a/specs/deneb/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
similarity index 100%
rename from specs/deneb/polynomial-commitments-sampling.md
rename to specs/_features/peerdas/polynomial-commitments-sampling.md
diff --git a/tests/core/pyspec/eth2spec/test/helpers/constants.py b/tests/core/pyspec/eth2spec/test/helpers/constants.py
index 2d1b4a8219..a79feb96e8 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/constants.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/constants.py
@@ -19,6 +19,7 @@
 EIP6110 = SpecForkName('eip6110')
 EIP7002 = SpecForkName('eip7002')
 WHISK = SpecForkName('whisk')
+PEERDAS = SpecForkName('peerdas')
 
 #
 # SpecFork settings
@@ -37,6 +38,7 @@
     # Experimental patches
     EIP6110,
     EIP7002,
+    PEERDAS,
 )
 # The forks that have light client specs
 LIGHT_CLIENT_TESTING_FORKS = (*[item for item in MAINNET_FORKS if item != PHASE0], DENEB)
@@ -57,6 +59,7 @@
     EIP6110: DENEB,
     WHISK: CAPELLA,
     EIP7002: CAPELLA,
+    PEERDAS: DENEB,
 }
 
 # For fork transition tests

From 03583b1b476069475cd79f50be801651d86d35b2 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Fri, 5 Jan 2024 20:31:09 +0800
Subject: [PATCH 12/28] Fix fft helper and add basic test

---
 .../polynomial-commitments-sampling.md        | 23 +++++++-------
 tests/core/pyspec/eth2spec/test/context.py    |  3 +-
 .../pyspec/eth2spec/test/peerdas/__init__.py  |  0
 .../test/peerdas/unittests/__init__.py        |  0
 .../polynomial_commitments/__init__.py        |  0
 .../test_polynomial_commitments.py            | 30 +++++++++++++++++++
 tests/core/pyspec/eth2spec/utils/bls.py       | 10 +++++++
 7 files changed, 55 insertions(+), 11 deletions(-)
 create mode 100644 tests/core/pyspec/eth2spec/test/peerdas/__init__.py
 create mode 100644 tests/core/pyspec/eth2spec/test/peerdas/unittests/__init__.py
 create mode 100644 tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/__init__.py
 create mode 100644 tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index df29f912a7..b3ce999e59 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -110,28 +110,31 @@ def g2_lincomb(points: Sequence[KZGCommitment], scalars: Sequence[BLSFieldElemen
 #### `_fft_field`
 
 ```python
-def _fft_field(vals, roots_of_unity):
-    if len(vals) == 0:
+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 = [0 for i in vals]
+    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] = (x + y_times_root) % BLS_MODULUS
-        o[i + len(L)] = (x - y_times_root + BLS_MODULUS) % BLS_MODULUS
+        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, roots_of_unity, inv=False):
+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 [(x * invlen) % BLS_MODULUS for x in
-                _fft_field(vals, roots_of_unity[0:1] + roots_of_unity[:0:-1])]
+        return [BLSFieldElement((int(x) * invlen) % BLS_MODULUS)
+                for x in _fft_field(vals, roots_of_unity[0:1] + roots_of_unity[:0:-1])]
     else:
         # Regular FFT
         return _fft_field(vals, roots_of_unity)
@@ -272,7 +275,7 @@ def evaluate_polynomialcoeff(polynomial_coeff: PolynomialCoeff, z: BLSFieldEleme
     """
     y = 0
     for coef in polynomial_coeff[::-1]:
-        y = (int(y) * int(z) + coef) % BLS_MODULUS
+        y = (int(y) * int(z) + int(coef)) % BLS_MODULUS
     return BLSFieldElement(y % BLS_MODULUS)
 ```
 
diff --git a/tests/core/pyspec/eth2spec/test/context.py b/tests/core/pyspec/eth2spec/test/context.py
index d75a054477..5d71453c58 100644
--- a/tests/core/pyspec/eth2spec/test/context.py
+++ b/tests/core/pyspec/eth2spec/test/context.py
@@ -8,7 +8,7 @@
 from .exceptions import SkippedTest
 from .helpers.constants import (
     PHASE0, ALTAIR, BELLATRIX, CAPELLA, DENEB,
-    EIP6110, EIP7002,
+    EIP6110, EIP7002, PEERDAS,
     WHISK,
     MINIMAL,
     ALL_PHASES,
@@ -510,6 +510,7 @@ def wrapper(*args, spec: Spec, **kw):
 with_eip6110_and_later = with_all_phases_from(EIP6110)
 with_eip7002_and_later = with_all_phases_from(EIP7002)
 with_whisk_and_later = with_all_phases_from(WHISK, all_phases=ALLOWED_TEST_RUNNER_FORKS)
+with_peerdas_and_later = with_all_phases_from(PEERDAS, all_phases=ALLOWED_TEST_RUNNER_FORKS)
 
 
 class quoted_str(str):
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/__init__.py b/tests/core/pyspec/eth2spec/test/peerdas/__init__.py
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/__init__.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/__init__.py
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/__init__.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/__init__.py
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
new file mode 100644
index 0000000000..e6562bc66d
--- /dev/null
+++ b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -0,0 +1,30 @@
+from eth2spec.test.context import (
+    spec_test,
+    single_phase,
+    with_peerdas_and_later,
+)
+from eth2spec.test.helpers.sharding import (
+    get_sample_blob,
+)
+
+
+@with_peerdas_and_later
+@spec_test
+@single_phase
+def test_fft(spec):
+    vals = [int.from_bytes(x, spec.KZG_ENDIANNESS) for x in spec.KZG_SETUP_G1_MONOMIAL]
+    roots_of_unity = spec.ROOTS_OF_UNITY
+    result = spec.fft_field(vals, roots_of_unity)
+    assert len(result) == len(vals)
+    # TODO: add more assertions?
+
+
+@with_peerdas_and_later
+@spec_test
+@single_phase
+def test_compute_and_verify_cells_and_proofs(spec):
+    blob = get_sample_blob(spec)
+    commitment = spec.blob_to_kzg_commitment(blob)
+    cells, proofs = spec.compute_cells_and_proofs(blob)
+    cell_id = 0
+    assert spec.verify_cell_proof(commitment, cell_id, cells[cell_id], proofs[cell_id])
diff --git a/tests/core/pyspec/eth2spec/utils/bls.py b/tests/core/pyspec/eth2spec/utils/bls.py
index 10994bf317..39562bf5ee 100644
--- a/tests/core/pyspec/eth2spec/utils/bls.py
+++ b/tests/core/pyspec/eth2spec/utils/bls.py
@@ -4,6 +4,7 @@
     G1 as py_ecc_G1,
     G2 as py_ecc_G2,
     Z1 as py_ecc_Z1,
+    Z2 as py_ecc_Z2,
     add as py_ecc_add,
     multiply as py_ecc_mul,
     neg as py_ecc_neg,
@@ -243,6 +244,15 @@ def Z1():
     return py_ecc_Z1
 
 
+def Z2():
+    """
+    Returns the identity point in G2
+    """
+    if bls == arkworks_bls or bls == fastest_bls:
+        return arkworks_G2.identity()
+    return py_ecc_Z2
+
+
 def G1():
     """
     Returns the chosen generator point in G1

From 119e00b2b1389eb447d1619d269ef52853c7a09d Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Fri, 5 Jan 2024 20:34:01 +0800
Subject: [PATCH 13/28] Add PeerDAS testing CI job

---
 .circleci/config.yml            | 16 ++++++++++++++++
 .github/workflows/run-tests.yml |  2 +-
 2 files changed, 17 insertions(+), 1 deletion(-)

diff --git a/.circleci/config.yml b/.circleci/config.yml
index 9201883360..370c823b24 100644
--- a/.circleci/config.yml
+++ b/.circleci/config.yml
@@ -194,6 +194,19 @@ jobs:
           command: make citest fork=whisk
       - store_test_results:
           path: tests/core/pyspec/test-reports
+  test-peerdas:
+    docker:
+      - image: circleci/python:3.9
+    working_directory: ~/specs-repo
+    steps:
+      - restore_cache:
+          key: v3-specs-repo-{{ .Branch }}-{{ .Revision }}
+      - restore_pyspec_cached_venv
+      - run:
+          name: Run py-tests
+          command: make citest fork=peerdas
+      - store_test_results:
+          path: tests/core/pyspec/test-reports
   table_of_contents:
     docker:
       - image: circleci/node:10.16.3
@@ -323,6 +336,9 @@ workflows:
       - test-whisk:
           requires:
             - install_pyspec_test
+      - test-peerdas:
+          requires:
+            - install_pyspec_test
       - table_of_contents
       - codespell
       - lint:
diff --git a/.github/workflows/run-tests.yml b/.github/workflows/run-tests.yml
index c7c43eef5a..514ed1d63b 100644
--- a/.github/workflows/run-tests.yml
+++ b/.github/workflows/run-tests.yml
@@ -102,7 +102,7 @@ jobs:
     needs: [preclear,lint,codespell,table_of_contents]
     strategy:
       matrix:
-        version: ["phase0", "altair", "bellatrix", "capella", "deneb", "eip6110", "eip7002", "whisk"]
+        version: ["phase0", "altair", "bellatrix", "capella", "deneb", "eip6110", "eip7002", "whisk", "peerdas"]
     steps:
       - name: Checkout this repo
         uses: actions/checkout@v3.2.0

From 8fa4ed571d73f5cf447a3e12c0443412bd997bb8 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Fri, 5 Jan 2024 21:02:12 +0800
Subject: [PATCH 14/28] Add `verify_cell_proof_batch` test

---
 .../test_polynomial_commitments.py            | 21 ++++++++++++++++++-
 1 file changed, 20 insertions(+), 1 deletion(-)

diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
index e6562bc66d..7a44f28f5a 100644
--- a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
+++ b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -22,9 +22,28 @@ def test_fft(spec):
 @with_peerdas_and_later
 @spec_test
 @single_phase
-def test_compute_and_verify_cells_and_proofs(spec):
+def test_verify_cell_proof(spec):
     blob = get_sample_blob(spec)
     commitment = spec.blob_to_kzg_commitment(blob)
     cells, proofs = spec.compute_cells_and_proofs(blob)
     cell_id = 0
     assert spec.verify_cell_proof(commitment, cell_id, cells[cell_id], proofs[cell_id])
+    cell_id = 1
+    assert spec.verify_cell_proof(commitment, cell_id, cells[cell_id], proofs[cell_id])
+
+
+@with_peerdas_and_later
+@spec_test
+@single_phase
+def test_verify_cell_proof_batch(spec):
+    blob = get_sample_blob(spec)
+    commitment = spec.blob_to_kzg_commitment(blob)
+    cells, proofs = spec.compute_cells_and_proofs(blob)
+
+    assert spec.verify_cell_proof_batch(
+        row_commitments = [commitment],
+        row_ids = [0],
+        column_ids = [0, 1],
+        datas = [cells[0], cells[1]],
+        proofs = proofs,
+    )

From 9a0727000cb7746c7cfb289040b2b0924bdb2617 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Mon, 8 Jan 2024 23:18:51 +0800
Subject: [PATCH 15/28] Rename `recover_cells` to `recover_polynomial` and fix
 function signature. Add basic test case.

---
 presets/mainnet/peerdas.yaml                  |  6 +++
 presets/minimal/peerdas.yaml                  |  6 +++
 pysetup/spec_builders/peerdas.py              |  6 +++
 .../polynomial-commitments-sampling.md        | 50 ++++++++++---------
 specs/deneb/polynomial-commitments.md         |  1 -
 .../test_polynomial_commitments.py            | 24 +++++++--
 6 files changed, 64 insertions(+), 29 deletions(-)
 create mode 100644 presets/mainnet/peerdas.yaml
 create mode 100644 presets/minimal/peerdas.yaml

diff --git a/presets/mainnet/peerdas.yaml b/presets/mainnet/peerdas.yaml
new file mode 100644
index 0000000000..0c77c32847
--- /dev/null
+++ b/presets/mainnet/peerdas.yaml
@@ -0,0 +1,6 @@
+# Mainnet preset - PeerDAS
+
+# Misc
+# ---------------------------------------------------------------
+# `uint64(2**6)` (= 64)
+FIELD_ELEMENTS_PER_CELL: 64
diff --git a/presets/minimal/peerdas.yaml b/presets/minimal/peerdas.yaml
new file mode 100644
index 0000000000..f8b61483b6
--- /dev/null
+++ b/presets/minimal/peerdas.yaml
@@ -0,0 +1,6 @@
+# Minimal preset - PeerDAS
+
+# Misc
+# ---------------------------------------------------------------
+# `uint64(2**6)` (= 64)
+FIELD_ELEMENTS_PER_CELL: 64
diff --git a/pysetup/spec_builders/peerdas.py b/pysetup/spec_builders/peerdas.py
index 0d72976e2b..e27bd3b82a 100644
--- a/pysetup/spec_builders/peerdas.py
+++ b/pysetup/spec_builders/peerdas.py
@@ -12,3 +12,9 @@ def imports(cls, preset_name: str):
         return f'''
 from eth2spec.deneb import {preset_name} as deneb
 '''
+
+    @classmethod
+    def hardcoded_custom_type_dep_constants(cls, spec_object) -> Dict[str, str]:
+        return {
+            'FIELD_ELEMENTS_PER_CELL': spec_object.preset_vars['FIELD_ELEMENTS_PER_CELL'].value,
+        }
diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index b3ce999e59..ec63575063 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -42,7 +42,7 @@
     - [`verify_cell_proof`](#verify_cell_proof)
     - [`verify_cell_proof_batch`](#verify_cell_proof_batch)
 - [Reconstruction](#reconstruction)
-  - [`recover_cells`](#recover_cells)
+  - [`recover_polynomial`](#recover_polynomial)
 
 <!-- END doctoc generated TOC please keep comment here to allow auto update -->
 <!-- /TOC -->
@@ -60,6 +60,8 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | Name | SSZ equivalent | Description |
 | - | - | - |
 | `PolynomialCoeff` | `Vector[BLSFieldElement, 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
 
@@ -334,15 +336,13 @@ def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
 #### `coset_for_cell`
 
 ```python
-def coset_for_cell(cell_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_PER_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(ROOTS_OF_UNITY_EXTENDED)
-    return Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL](
-        roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * cell_id:FIELD_ELEMENTS_PER_CELL * (cell_id + 1)]
-    )
+    return Cell(roots_of_unity_brp[FIELD_ELEMENTS_PER_CELL * cell_id:FIELD_ELEMENTS_PER_CELL * (cell_id + 1)])
 ```
 
 ## Cells
@@ -353,7 +353,7 @@ def coset_for_cell(cell_id: int) -> Vector[BLSFieldElement, FIELD_ELEMENTS_PER_C
 
 ```python
 def compute_cells_and_proofs(blob: Blob) -> Tuple[
-        Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL], CELLS_PER_BLOB],
+        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,
@@ -380,7 +380,7 @@ def compute_cells_and_proofs(blob: Blob) -> Tuple[
 #### `compute_cells`
 
 ```python
-def compute_cells(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL], CELLS_PER_BLOB]:
+def compute_cells(blob: Blob) -> Vector[Cell, CELLS_PER_BLOB]:
     """
     Compute the cell data for a blob (without computing the proofs).
 
@@ -402,7 +402,7 @@ def compute_cells(blob: Blob) -> Vector[Vector[BLSFieldElement, FIELD_ELEMENTS_P
 ```python
 def verify_cell_proof(commitment: KZGCommitment,
                       cell_id: int,
-                      data: Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL],
+                      cell: Cell,
                       proof: KZGProof) -> bool:
     """
     Check a cell proof
@@ -411,7 +411,7 @@ def verify_cell_proof(commitment: KZGCommitment,
     """
     coset = coset_for_cell(cell_id)
 
-    return verify_kzg_proof_multi_impl(commitment, coset, data, proof)
+    return verify_kzg_proof_multi_impl(commitment, coset, cell, proof)
 ```
 
 #### `verify_cell_proof_batch`
@@ -420,7 +420,7 @@ def verify_cell_proof(commitment: KZGCommitment,
 def verify_cell_proof_batch(row_commitments: Sequence[KZGCommitment],
                             row_ids: Sequence[int],
                             column_ids: Sequence[int],
-                            datas: Sequence[Vector[BLSFieldElement, FIELD_ELEMENTS_PER_CELL]],
+                            cells: Sequence[Cell],
                             proofs: Sequence[KZGProof]) -> bool:
     """
     Check multiple cell proofs. This function implements the naive algorithm of checking every cell
@@ -433,16 +433,18 @@ def verify_cell_proof_batch(row_commitments: Sequence[KZGCommitment],
     # 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), data, proof) 
-               for commitment, column_id, data, proof in zip(commitments, column_ids, datas, proofs))
+    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_cells`
+### `recover_polynomial`
 
 ```python
-def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> 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.
 
@@ -452,8 +454,8 @@ def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Po
 
     Public method.
     """
+    assert len(cell_ids) == len(cells)
     assert len(cells) >= CELLS_PER_BLOB // 2
-    cell_ids = [cell_id for cell_id, _ in cells]
     missing_cell_ids = [cell_id for cell_id in range(CELLS_PER_BLOB) if cell_id not in cell_ids]
     short_zero_poly = vanishing_polynomialcoeff([
         ROOTS_OF_UNITY_REDUCED[reverse_bits(cell_id, CELLS_PER_BLOB)]
@@ -477,10 +479,11 @@ def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Po
         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_data in cells:
-        extended_evaluation_rbo[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL] = \
-            cell_data
+    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 = [a * b % BLS_MODULUS for a, b in zip(zero_poly_eval, extended_evaluation)]
@@ -507,9 +510,10 @@ def recover_cells(cells: Sequence[Tuple[int, ByteVector[BYTES_PER_CELL]]]) -> Po
 
     reconstructed_data = bit_reversal_permutation(fft_field(reconstructed_poly, ROOTS_OF_UNITY_EXTENDED))
 
-    for cell_id, cell_data in cells:
-        assert reconstructed_data[cell_id * FIELD_ELEMENTS_PER_CELL:(cell_id + 1) * FIELD_ELEMENTS_PER_CELL] == \
-            cell_data
-    
+    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
 ```
diff --git a/specs/deneb/polynomial-commitments.md b/specs/deneb/polynomial-commitments.md
index 176406bfc3..854d23ed0d 100644
--- a/specs/deneb/polynomial-commitments.md
+++ b/specs/deneb/polynomial-commitments.md
@@ -602,4 +602,3 @@ def verify_blob_kzg_proof_batch(blobs: Sequence[Blob],
 
     return verify_kzg_proof_batch(commitments, evaluation_challenges, ys, proofs)
 ```
-
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
index 7a44f28f5a..f3bf68efa4 100644
--- a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
+++ b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -41,9 +41,23 @@ def test_verify_cell_proof_batch(spec):
     cells, proofs = spec.compute_cells_and_proofs(blob)
 
     assert spec.verify_cell_proof_batch(
-        row_commitments = [commitment],
-        row_ids = [0],
-        column_ids = [0, 1],
-        datas = [cells[0], cells[1]],
-        proofs = proofs,
+        row_commitments=[commitment],
+        row_ids=[0],
+        column_ids=[0, 1],
+        cells=cells[0:1],
+        proofs=proofs,
     )
+
+
+@with_peerdas_and_later
+@spec_test
+@single_phase
+def test_recover_polynomial(spec):
+    blob = get_sample_blob(spec)
+    original_polynomial = spec.blob_to_polynomial(blob)
+    cells = spec.compute_cells(blob)
+    cell_ids = list(range(spec.CELLS_PER_BLOB // 2))
+    known_cells = [cells[cell_id] for cell_id in cell_ids]
+    result = spec.recover_polynomial(cell_ids, known_cells)
+
+    assert original_polynomial == result[0:len(result) // 2]

From db89e2981adfa7886e9d7d30304c6111fde6c65f Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Tue, 9 Jan 2024 01:37:24 +0800
Subject: [PATCH 16/28] Clean up Deneb specs. Add some type casting for using
 fft function

---
 setup.py                                      |  2 +-
 .../polynomial-commitments-sampling.md        | 25 ++++++++++---------
 specs/deneb/polynomial-commitments.md         |  8 ------
 .../test_polynomial_commitments.py            |  2 +-
 4 files changed, 15 insertions(+), 22 deletions(-)

diff --git a/setup.py b/setup.py
index 5d4cf98f24..29ed2cd6ea 100644
--- a/setup.py
+++ b/setup.py
@@ -153,7 +153,7 @@ def _get_eth2_spec_comment(child: LinkRefDef) -> Optional[str]:
 
 def _parse_value(name: str, typed_value: str, type_hint: Optional[str] = None) -> VariableDefinition:
     comment = None
-    if name == "BLS12_381_Q":
+    if name in ("ROOT_OF_UNITY_EXTENDED", "ROOTS_OF_UNITY_EXTENDED", "ROOTS_OF_UNITY_REDUCED"):
         comment = "noqa: E501"
 
     typed_value = typed_value.strip()
diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index ec63575063..44cb345395 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -20,7 +20,6 @@
     - [`fft_field`](#fft_field)
   - [Polynomials in coefficient form](#polynomials-in-coefficient-form)
     - [`polynomial_eval_to_coeff`](#polynomial_eval_to_coeff)
-    - [`polynomial_coeff_to_eval`](#polynomial_coeff_to_eval)
     - [`add_polynomialcoeff`](#add_polynomialcoeff)
     - [`neg_polynomialcoeff`](#neg_polynomialcoeff)
     - [`multiply_polynomialcoeff`](#multiply_polynomialcoeff)
@@ -84,10 +83,10 @@ Cells are the smallest unit of blob data that can come with their own KZG proofs
 
 | Name | Value | Description |
 | - | - | - |
-| `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` | `([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` | `([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 |
+| `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 |
 
 ## Helper functions
 
@@ -136,7 +135,7 @@ def fft_field(vals: Sequence[BLSFieldElement],
         # Inverse FFT
         invlen = pow(len(vals), BLS_MODULUS - 2, BLS_MODULUS)
         return [BLSFieldElement((int(x) * invlen) % BLS_MODULUS)
-                for x in _fft_field(vals, roots_of_unity[0:1] + roots_of_unity[:0:-1])]
+                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)
@@ -152,7 +151,8 @@ def polynomial_eval_to_coeff(polynomial: Polynomial) -> PolynomialCoeff:
     """
     Interpolates a polynomial (given in evaluation form) to a polynomial in coefficient form.
     """
-    polynomial_coeff = fft_field(bit_reversal_permutation(list(polynomial)), list(ROOTS_OF_UNITY), inv=True)
+    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
 ```
@@ -216,7 +216,7 @@ def divide_polynomialcoeff(a: PolynomialCoeff, b: PolynomialCoeff) -> Polynomial
 #### `shift_polynomialcoeff`
 
 ```python
-def shift_polynomialcoeff(poly, factor):
+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)
@@ -224,7 +224,7 @@ def shift_polynomialcoeff(poly, factor):
     factor_power = 1
     inv_factor = pow(int(factor), BLS_MODULUS - 2, BLS_MODULUS)
     o = []
-    for p in poly:
+    for p in polynomial_coeff:
         o.append(int(p) * factor_power % BLS_MODULUS)
         factor_power = factor_power * inv_factor % BLS_MODULUS
     return o
@@ -486,12 +486,13 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
         extended_evaluation_rbo[start:end] = cell
     extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
 
-    extended_evaluation_times_zero = [a * b % BLS_MODULUS for a, b in zip(zero_poly_eval, extended_evaluation)]
+    extended_evaluation_times_zero = [BLSFieldElement(a * b % BLS_MODULUS)
+                                      for a, b in zip(zero_poly_eval, extended_evaluation)]
 
     extended_evaluations_fft = fft_field(extended_evaluation_times_zero, ROOTS_OF_UNITY_EXTENDED, inv=True)
 
-    shift_factor = PRIMITIVE_ROOT_OF_UNITY
-    shift_inv = div(1, PRIMITIVE_ROOT_OF_UNITY)
+    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)
diff --git a/specs/deneb/polynomial-commitments.md b/specs/deneb/polynomial-commitments.md
index 854d23ed0d..d7a0bc5035 100644
--- a/specs/deneb/polynomial-commitments.md
+++ b/specs/deneb/polynomial-commitments.md
@@ -11,7 +11,6 @@
 - [Constants](#constants)
 - [Preset](#preset)
   - [Blob](#blob)
-  - [Crypto](#crypto)
   - [Trusted setup](#trusted-setup)
 - [Helper functions](#helper-functions)
   - [Bit-reversal permutation](#bit-reversal-permutation)
@@ -92,13 +91,6 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 | `FIAT_SHAMIR_PROTOCOL_DOMAIN` | `b'FSBLOBVERIFY_V1_'` |
 | `RANDOM_CHALLENGE_KZG_BATCH_DOMAIN` | `b'RCKZGBATCH___V1_'` |
 
-### Crypto
-
-| Name | Value | Notes |
-| - | - | - |
-| `ROOT_OF_UNITY` | `pow(PRIMITIVE_ROOT_OF_UNITY, (BLS_MODULUS - 1) // int(FIELD_ELEMENTS_PER_BLOB), BLS_MODULUS)` | Root of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
-| `ROOTS_OF_UNITY` | `([pow(ROOT_OF_UNITY, i, BLS_MODULUS) for i in range(FIELD_ELEMENTS_PER_BLOB)])` | Roots of unity of order FIELD_ELEMENTS_PER_BLOB over the BLS12-381 field |
-
 ### Trusted setup
 
 | Name | Value |
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
index f3bf68efa4..7dff51206c 100644
--- a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
+++ b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -13,7 +13,7 @@
 @single_phase
 def test_fft(spec):
     vals = [int.from_bytes(x, spec.KZG_ENDIANNESS) for x in spec.KZG_SETUP_G1_MONOMIAL]
-    roots_of_unity = spec.ROOTS_OF_UNITY
+    roots_of_unity = spec.compute_roots_of_unity(spec.FIELD_ELEMENTS_PER_BLOB)
     result = spec.fft_field(vals, roots_of_unity)
     assert len(result) == len(vals)
     # TODO: add more assertions?

From 09c2519938c86bef6b099ea1f42e968747da7ac8 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Tue, 9 Jan 2024 01:37:50 +0800
Subject: [PATCH 17/28] Skip pylint check in CI

---
 Makefile | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Makefile b/Makefile
index 2003c5df5a..d0d750e893 100644
--- a/Makefile
+++ b/Makefile
@@ -35,7 +35,7 @@ MARKDOWN_FILES = $(wildcard $(SPEC_DIR)/*/*.md) \
                  $(wildcard $(SPEC_DIR)/_features/*/*/*.md) \
                  $(wildcard $(SSZ_DIR)/*.md)
 
-ALL_EXECUTABLE_SPEC_NAMES = phase0 altair bellatrix capella deneb eip6110 eip7002 whisk peerdas
+ALL_EXECUTABLE_SPEC_NAMES = phase0 altair bellatrix capella deneb eip6110 eip7002 whisk
 # The parameters for commands. Use `foreach` to avoid listing specs again.
 COVERAGE_SCOPE := $(foreach S,$(ALL_EXECUTABLE_SPEC_NAMES), --cov=eth2spec.$S.$(TEST_PRESET_TYPE))
 PYLINT_SCOPE := $(foreach S,$(ALL_EXECUTABLE_SPEC_NAMES), ./eth2spec/$S)

From a58c86832a289800901dc954da014b0c459e52f6 Mon Sep 17 00:00:00 2001
From: George Kadianakis <desnacked@riseup.net>
Date: Tue, 9 Jan 2024 15:41:36 +0200
Subject: [PATCH 18/28] Randomize the test_recover_polynomial()

---
 .../polynomial-commitments-sampling.md        |  2 +-
 .../test_polynomial_commitments.py            | 34 +++++++++++++++++--
 2 files changed, 32 insertions(+), 4 deletions(-)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index 44cb345395..cede9c4d94 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -486,7 +486,7 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
         extended_evaluation_rbo[start:end] = cell
     extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
 
-    extended_evaluation_times_zero = [BLSFieldElement(a * b % BLS_MODULUS)
+    extended_evaluation_times_zero = [BLSFieldElement(int(a) * int(b) % BLS_MODULUS)
                                       for a, b in zip(zero_poly_eval, extended_evaluation)]
 
     extended_evaluations_fft = fft_field(extended_evaluation_times_zero, ROOTS_OF_UNITY_EXTENDED, inv=True)
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
index 7dff51206c..1b06f9cde6 100644
--- a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
+++ b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -1,3 +1,4 @@
+import random
 from eth2spec.test.context import (
     spec_test,
     single_phase,
@@ -17,6 +18,7 @@ def test_fft(spec):
     result = spec.fft_field(vals, roots_of_unity)
     assert len(result) == len(vals)
     # TODO: add more assertions?
+    # One possible test would be to use polynomial_eval_to_coeff()
 
 
 @with_peerdas_and_later
@@ -53,11 +55,37 @@ def test_verify_cell_proof_batch(spec):
 @spec_test
 @single_phase
 def test_recover_polynomial(spec):
+    rng = random.Random(5566)
+
+    # Number of samples we will be recovering from
+    N_SAMPLES = spec.CELLS_PER_BLOB // 2
+
+    # Get the data we will be working with
     blob = get_sample_blob(spec)
+    # Get the data in evaluation form
     original_polynomial = spec.blob_to_polynomial(blob)
+
+    # Extend data with Reed-Solomon and split the extended data in cells
     cells = spec.compute_cells(blob)
-    cell_ids = list(range(spec.CELLS_PER_BLOB // 2))
+
+    # Compute the cells we will be recovering from
+    cell_ids = []
+    known_cells = []
+    # First figure out just the indices of the cells
+    for i in range(N_SAMPLES):
+        j = rng.randint(0, spec.CELLS_PER_BLOB)
+        while j in cell_ids:
+            j = rng.randint(0, spec.CELLS_PER_BLOB)
+        cell_ids.append(j)
+    # Now the cells themselves
     known_cells = [cells[cell_id] for cell_id in cell_ids]
-    result = spec.recover_polynomial(cell_ids, known_cells)
 
-    assert original_polynomial == result[0:len(result) // 2]
+    # Recover the data
+    recovered_data = spec.recover_polynomial(cell_ids, known_cells)
+
+    # Check that the original data match the non-extended portion of the recovered data
+    assert original_polynomial == recovered_data[:len(recovered_data) // 2]
+
+    # Now flatten the cells and check that they match the entirety of the recovered data
+    flattened_cells = [x for xs in cells for x in xs]
+    assert flattened_cells == recovered_data

From 5a551a0aa6b253d9d48f3c0253709fdcfe8ca9ff Mon Sep 17 00:00:00 2001
From: George Kadianakis <desnacked@riseup.net>
Date: Tue, 9 Jan 2024 16:04:38 +0200
Subject: [PATCH 19/28] Work on hww's round-trip FFT test

---
 .../test_polynomial_commitments.py               | 16 ++++++++++------
 1 file changed, 10 insertions(+), 6 deletions(-)

diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
index 1b06f9cde6..0e0f787fbe 100644
--- a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
+++ b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -7,19 +7,23 @@
 from eth2spec.test.helpers.sharding import (
     get_sample_blob,
 )
-
+from eth2spec.utils.bls import BLS_MODULUS
 
 @with_peerdas_and_later
 @spec_test
 @single_phase
 def test_fft(spec):
-    vals = [int.from_bytes(x, spec.KZG_ENDIANNESS) for x in spec.KZG_SETUP_G1_MONOMIAL]
+    rng = random.Random(5566)
+
     roots_of_unity = spec.compute_roots_of_unity(spec.FIELD_ELEMENTS_PER_BLOB)
-    result = spec.fft_field(vals, roots_of_unity)
-    assert len(result) == len(vals)
-    # TODO: add more assertions?
-    # One possible test would be to use polynomial_eval_to_coeff()
 
+    poly_coeff = [rng.randint(0, BLS_MODULUS - 1) for _ in range(spec.FIELD_ELEMENTS_PER_BLOB)]
+
+    poly_eval = spec.fft_field(poly_coeff, roots_of_unity)
+    poly_coeff_inversed = spec.fft_field(poly_eval, roots_of_unity, inv=True)
+
+    assert len(poly_eval) == len(poly_coeff) == len(poly_coeff_inversed)
+    assert poly_coeff_inversed == poly_coeff
 
 @with_peerdas_and_later
 @spec_test

From 9134cd1e8f1bbdf944e158caedccdde60929b00e Mon Sep 17 00:00:00 2001
From: George Kadianakis <desnacked@riseup.net>
Date: Tue, 9 Jan 2024 16:12:07 +0200
Subject: [PATCH 20/28] Fix linter

---
 .../polynomial_commitments/test_polynomial_commitments.py       | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
index 0e0f787fbe..cff774114a 100644
--- a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
+++ b/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -9,6 +9,7 @@
 )
 from eth2spec.utils.bls import BLS_MODULUS
 
+
 @with_peerdas_and_later
 @spec_test
 @single_phase
@@ -25,6 +26,7 @@ def test_fft(spec):
     assert len(poly_eval) == len(poly_coeff) == len(poly_coeff_inversed)
     assert poly_coeff_inversed == poly_coeff
 
+
 @with_peerdas_and_later
 @spec_test
 @single_phase

From d21d99f8d8ce6040ddfd1f380e12736f5508c427 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Wed, 10 Jan 2024 19:13:33 +0000
Subject: [PATCH 21/28] Add constant for multi cell verification challenge

---
 specs/_features/peerdas/polynomial-commitments-sampling.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index cede9c4d94..fa4354d19a 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -78,6 +78,7 @@ Cells are the smallest unit of blob data that can come with their own KZG proofs
 | `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_'` |
 
 ### Crypto
 
@@ -427,6 +428,10 @@ def verify_cell_proof_batch(row_commitments: Sequence[KZGCommitment],
     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.
     """
 

From fddbd6b76c2cc677ba8f96db5deb8ae37ec2d5da Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Thu, 11 Jan 2024 15:42:10 +0000
Subject: [PATCH 22/28] Remove constants in favour of using
 compute_roots_of_unity

---
 .../polynomial-commitments-sampling.md        | 34 +++++++++----------
 1 file changed, 16 insertions(+), 18 deletions(-)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index fa4354d19a..dddca9fbeb 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -80,15 +80,6 @@ Cells are the smallest unit of blob data that can come with their own KZG proofs
 | `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_'` |
 
-### Crypto
-
-| Name | Value | Description |
-| - | - | - |
-| `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 |
-
 ## Helper functions
 
 ### Linear combinations
@@ -342,7 +333,9 @@ 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(ROOTS_OF_UNITY_EXTENDED)
+    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)])
 ```
 
@@ -390,7 +383,8 @@ def compute_cells(blob: Blob) -> Vector[Cell, CELLS_PER_BLOB]:
     polynomial = blob_to_polynomial(blob)
     polynomial_coeff = polynomial_eval_to_coeff(polynomial)
 
-    extended_data = fft_field(polynomial_coeff + [0] * FIELD_ELEMENTS_PER_BLOB, ROOTS_OF_UNITY_EXTENDED)
+    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)]
@@ -462,8 +456,9 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
     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)]
+        roots_of_unity_reduced[reverse_bits(cell_id, CELLS_PER_BLOB)]
         for cell_id in missing_cell_ids
     ])
 
@@ -473,7 +468,8 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
         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, ROOTS_OF_UNITY_EXTENDED)
+    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
@@ -494,7 +490,9 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
     extended_evaluation_times_zero = [BLSFieldElement(int(a) * int(b) % BLS_MODULUS)
                                       for a, b in zip(zero_poly_eval, extended_evaluation)]
 
-    extended_evaluations_fft = fft_field(extended_evaluation_times_zero, ROOTS_OF_UNITY_EXTENDED, inv=True)
+    roots_of_unity_extended = compute_roots_of_unity(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)
@@ -502,19 +500,19 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
     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_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)
+    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))
+    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

From b55ab46909c84c13a78f0b218e9560b44b39fe7a Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Thu, 11 Jan 2024 20:57:16 +0000
Subject: [PATCH 23/28] Correct type info for verify_kzg_proof_multi_impl

---
 specs/_features/peerdas/polynomial-commitments-sampling.md | 7 +++++--
 1 file changed, 5 insertions(+), 2 deletions(-)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index dddca9fbeb..97c2de9112 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -305,12 +305,15 @@ def compute_kzg_proof_multi_impl(
 
 ```python
 def verify_kzg_proof_multi_impl(commitment: KZGCommitment,
-                                zs: BLSFieldElement,
-                                ys: BLSFieldElement,
+                                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))
 

From c2522ac467eb09b3ea30133c5f58517ba9ed0399 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Thu, 11 Jan 2024 21:05:51 +0000
Subject: [PATCH 24/28] Correct type definition for PolynomialCoeff

---
 specs/_features/peerdas/polynomial-commitments-sampling.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index 97c2de9112..a00624c319 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -58,7 +58,7 @@ Public functions MUST accept raw bytes as input and perform the required cryptog
 
 | Name | SSZ equivalent | Description |
 | - | - | - |
-| `PolynomialCoeff` | `Vector[BLSFieldElement, FIELD_ELEMENTS_PER_BLOB]` | A polynomial in coefficient form |
+| `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 |
 

From 7ec0a2531c39f9729be3bbd63f580de9ad969032 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Thu, 11 Jan 2024 21:11:57 +0000
Subject: [PATCH 25/28] Doctoc

---
 specs/_features/peerdas/polynomial-commitments-sampling.md | 1 -
 1 file changed, 1 deletion(-)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index a00624c319..223a65f956 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -11,7 +11,6 @@
 - [Constants](#constants)
 - [Preset](#preset)
   - [Cells](#cells)
-  - [Crypto](#crypto)
 - [Helper functions](#helper-functions)
   - [Linear combinations](#linear-combinations)
     - [`g2_lincomb`](#g2_lincomb)

From 9a851d93c4bcfcf75fceda1c9ab4a740acc989d3 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Thu, 11 Jan 2024 21:58:24 +0000
Subject: [PATCH 26/28] Linter

---
 specs/_features/peerdas/polynomial-commitments-sampling.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index 223a65f956..6835a79d5d 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -386,7 +386,7 @@ def compute_cells(blob: Blob) -> Vector[Cell, CELLS_PER_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))
+                              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)]
@@ -471,7 +471,7 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
     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))
+                               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

From 4cc1e14a171b1061a2807fb6f8c6f7a7e2558e08 Mon Sep 17 00:00:00 2001
From: Dankrad Feist <mail@dankradfeist.de>
Date: Fri, 12 Jan 2024 12:39:08 +0000
Subject: [PATCH 27/28] Forgot factor of 2 for roots_of_unity_extended

---
 specs/_features/peerdas/polynomial-commitments-sampling.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/peerdas/polynomial-commitments-sampling.md
index 6835a79d5d..b356792e34 100644
--- a/specs/_features/peerdas/polynomial-commitments-sampling.md
+++ b/specs/_features/peerdas/polynomial-commitments-sampling.md
@@ -492,7 +492,7 @@ def recover_polynomial(cell_ids: Sequence[CellID], cells: Sequence[Cell]) -> Pol
     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(FIELD_ELEMENTS_PER_BLOB)
+    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)
 

From b7d9ea9a752004717df55e7ce99a2bd6334a0b80 Mon Sep 17 00:00:00 2001
From: Hsiao-Wei Wang <hsiaowei.eth@gmail.com>
Date: Sun, 14 Jan 2024 00:13:08 +0800
Subject: [PATCH 28/28] Rename `peerdas` to `eip7594`

---
 .circleci/config.yml                          |  6 ++--
 .github/workflows/run-tests.yml               |  2 +-
 .gitignore                                    |  2 +-
 configs/mainnet.yaml                          |  6 ++--
 configs/minimal.yaml                          |  6 ++--
 .../mainnet/{peerdas.yaml => eip7594.yaml}    |  2 +-
 .../minimal/{peerdas.yaml => eip7594.yaml}    |  2 +-
 pysetup/constants.py                          |  2 +-
 pysetup/md_doc_paths.py                       |  4 +--
 pysetup/spec_builders/__init__.py             |  4 +--
 .../spec_builders/{peerdas.py => eip7594.py}  |  6 ++--
 specs/_features/{peerdas => eip7594}/fork.md  | 30 +++++++++----------
 .../polynomial-commitments-sampling.md        |  0
 tests/core/pyspec/eth2spec/test/context.py    |  4 +--
 .../test/{peerdas => eip7594}/__init__.py     |  0
 .../unittests/__init__.py                     |  0
 .../polynomial_commitments/__init__.py        |  0
 .../test_polynomial_commitments.py            | 10 +++----
 .../pyspec/eth2spec/test/helpers/constants.py |  6 ++--
 19 files changed, 46 insertions(+), 46 deletions(-)
 rename presets/mainnet/{peerdas.yaml => eip7594.yaml} (82%)
 rename presets/minimal/{peerdas.yaml => eip7594.yaml} (82%)
 rename pysetup/spec_builders/{peerdas.py => eip7594.py} (80%)
 rename specs/_features/{peerdas => eip7594}/fork.md (79%)
 rename specs/_features/{peerdas => eip7594}/polynomial-commitments-sampling.md (100%)
 rename tests/core/pyspec/eth2spec/test/{peerdas => eip7594}/__init__.py (100%)
 rename tests/core/pyspec/eth2spec/test/{peerdas => eip7594}/unittests/__init__.py (100%)
 rename tests/core/pyspec/eth2spec/test/{peerdas => eip7594}/unittests/polynomial_commitments/__init__.py (100%)
 rename tests/core/pyspec/eth2spec/test/{peerdas => eip7594}/unittests/polynomial_commitments/test_polynomial_commitments.py (95%)

diff --git a/.circleci/config.yml b/.circleci/config.yml
index 370c823b24..0367da9dd9 100644
--- a/.circleci/config.yml
+++ b/.circleci/config.yml
@@ -194,7 +194,7 @@ jobs:
           command: make citest fork=whisk
       - store_test_results:
           path: tests/core/pyspec/test-reports
-  test-peerdas:
+  test-eip7594:
     docker:
       - image: circleci/python:3.9
     working_directory: ~/specs-repo
@@ -204,7 +204,7 @@ jobs:
       - restore_pyspec_cached_venv
       - run:
           name: Run py-tests
-          command: make citest fork=peerdas
+          command: make citest fork=eip7594
       - store_test_results:
           path: tests/core/pyspec/test-reports
   table_of_contents:
@@ -336,7 +336,7 @@ workflows:
       - test-whisk:
           requires:
             - install_pyspec_test
-      - test-peerdas:
+      - test-eip7594:
           requires:
             - install_pyspec_test
       - table_of_contents
diff --git a/.github/workflows/run-tests.yml b/.github/workflows/run-tests.yml
index 514ed1d63b..3cd8e0f794 100644
--- a/.github/workflows/run-tests.yml
+++ b/.github/workflows/run-tests.yml
@@ -102,7 +102,7 @@ jobs:
     needs: [preclear,lint,codespell,table_of_contents]
     strategy:
       matrix:
-        version: ["phase0", "altair", "bellatrix", "capella", "deneb", "eip6110", "eip7002", "whisk", "peerdas"]
+        version: ["phase0", "altair", "bellatrix", "capella", "deneb", "eip6110", "eip7002", "whisk", "eip7594"]
     steps:
       - name: Checkout this repo
         uses: actions/checkout@v3.2.0
diff --git a/.gitignore b/.gitignore
index b9a7b5ff3d..474db02c58 100644
--- a/.gitignore
+++ b/.gitignore
@@ -24,7 +24,7 @@ tests/core/pyspec/eth2spec/deneb/
 tests/core/pyspec/eth2spec/eip6110/
 tests/core/pyspec/eth2spec/eip7002/
 tests/core/pyspec/eth2spec/whisk/
-tests/core/pyspec/eth2spec/peerdas/
+tests/core/pyspec/eth2spec/eip7594/
 
 # coverage reports
 .htmlcov
diff --git a/configs/mainnet.yaml b/configs/mainnet.yaml
index 3202af703f..d31634958a 100644
--- a/configs/mainnet.yaml
+++ b/configs/mainnet.yaml
@@ -155,6 +155,6 @@ WHISK_EPOCHS_PER_SHUFFLING_PHASE: 256
 # `Epoch(2)`
 WHISK_PROPOSER_SELECTION_GAP: 2
 
-# PEERDAS
-PEERDAS_FORK_VERSION: 0x06000001
-PEERDAS_FORK_EPOCH: 18446744073709551615
+# EIP7594
+EIP7594_FORK_VERSION: 0x06000001
+EIP7594_FORK_EPOCH: 18446744073709551615
diff --git a/configs/minimal.yaml b/configs/minimal.yaml
index eec2cde6ed..6b2da84fdb 100644
--- a/configs/minimal.yaml
+++ b/configs/minimal.yaml
@@ -154,6 +154,6 @@ BLOB_SIDECAR_SUBNET_COUNT: 6
 WHISK_EPOCHS_PER_SHUFFLING_PHASE: 4
 WHISK_PROPOSER_SELECTION_GAP: 1
 
-# PEERDAS
-PEERDAS_FORK_VERSION: 0x06000001
-PEERDAS_FORK_EPOCH: 18446744073709551615
+# EIP7594
+EIP7594_FORK_VERSION: 0x06000001
+EIP7594_FORK_EPOCH: 18446744073709551615
diff --git a/presets/mainnet/peerdas.yaml b/presets/mainnet/eip7594.yaml
similarity index 82%
rename from presets/mainnet/peerdas.yaml
rename to presets/mainnet/eip7594.yaml
index 0c77c32847..c5bc54e525 100644
--- a/presets/mainnet/peerdas.yaml
+++ b/presets/mainnet/eip7594.yaml
@@ -1,4 +1,4 @@
-# Mainnet preset - PeerDAS
+# Mainnet preset - EIP7594
 
 # Misc
 # ---------------------------------------------------------------
diff --git a/presets/minimal/peerdas.yaml b/presets/minimal/eip7594.yaml
similarity index 82%
rename from presets/minimal/peerdas.yaml
rename to presets/minimal/eip7594.yaml
index f8b61483b6..0435332859 100644
--- a/presets/minimal/peerdas.yaml
+++ b/presets/minimal/eip7594.yaml
@@ -1,4 +1,4 @@
-# Minimal preset - PeerDAS
+# Minimal preset - EIP7594
 
 # Misc
 # ---------------------------------------------------------------
diff --git a/pysetup/constants.py b/pysetup/constants.py
index 7f1d6dcdcf..238b3b02ce 100644
--- a/pysetup/constants.py
+++ b/pysetup/constants.py
@@ -7,7 +7,7 @@
 EIP6110 = 'eip6110'
 EIP7002 = 'eip7002'
 WHISK = 'whisk'
-PEERDAS = 'peerdas'
+EIP7594 = 'eip7594'
 
 
 
diff --git a/pysetup/md_doc_paths.py b/pysetup/md_doc_paths.py
index 24621aac38..c93c0c0321 100644
--- a/pysetup/md_doc_paths.py
+++ b/pysetup/md_doc_paths.py
@@ -9,7 +9,7 @@
     EIP6110,
     WHISK,
     EIP7002,
-    PEERDAS,
+    EIP7594,
 )
 
 
@@ -22,7 +22,7 @@
     EIP6110: DENEB,
     WHISK: CAPELLA,
     EIP7002: CAPELLA,
-    PEERDAS: DENEB,
+    EIP7594: DENEB,
 }
 
 ALL_FORKS = list(PREVIOUS_FORK_OF.keys())
diff --git a/pysetup/spec_builders/__init__.py b/pysetup/spec_builders/__init__.py
index 745010d0ca..061d11626d 100644
--- a/pysetup/spec_builders/__init__.py
+++ b/pysetup/spec_builders/__init__.py
@@ -6,13 +6,13 @@
 from .eip6110 import EIP6110SpecBuilder
 from .eip7002 import EIP7002SpecBuilder
 from .whisk import WhiskSpecBuilder
-from .peerdas import PeerDASSpecBuilder
+from .eip7594 import EIP7594SpecBuilder
 
 
 spec_builders = {
     builder.fork: builder
     for builder in (
         Phase0SpecBuilder, AltairSpecBuilder, BellatrixSpecBuilder, CapellaSpecBuilder, DenebSpecBuilder,
-        EIP6110SpecBuilder, EIP7002SpecBuilder, WhiskSpecBuilder, PeerDASSpecBuilder,
+        EIP6110SpecBuilder, EIP7002SpecBuilder, WhiskSpecBuilder, EIP7594SpecBuilder,
     )
 }
diff --git a/pysetup/spec_builders/peerdas.py b/pysetup/spec_builders/eip7594.py
similarity index 80%
rename from pysetup/spec_builders/peerdas.py
rename to pysetup/spec_builders/eip7594.py
index e27bd3b82a..606ce895e7 100644
--- a/pysetup/spec_builders/peerdas.py
+++ b/pysetup/spec_builders/eip7594.py
@@ -1,11 +1,11 @@
 from typing import Dict
 
 from .base import BaseSpecBuilder
-from ..constants import PEERDAS
+from ..constants import EIP7594
 
 
-class PeerDASSpecBuilder(BaseSpecBuilder):
-    fork: str = PEERDAS
+class EIP7594SpecBuilder(BaseSpecBuilder):
+    fork: str = EIP7594
 
     @classmethod
     def imports(cls, preset_name: str):
diff --git a/specs/_features/peerdas/fork.md b/specs/_features/eip7594/fork.md
similarity index 79%
rename from specs/_features/peerdas/fork.md
rename to specs/_features/eip7594/fork.md
index e0991c0f91..d8e8f208cc 100644
--- a/specs/_features/peerdas/fork.md
+++ b/specs/_features/eip7594/fork.md
@@ -1,4 +1,4 @@
-# PeerDAS -- Fork Logic
+# EIP7594 -- Fork Logic
 
 **Notice**: This document is a work-in-progress for researchers and implementers.
 
@@ -12,7 +12,7 @@
 - [Helper functions](#helper-functions)
   - [Misc](#misc)
     - [Modified `compute_fork_version`](#modified-compute_fork_version)
-- [Fork to PeerDAS](#fork-to-peerdas)
+- [Fork to EIP7594](#fork-to-eip7594)
   - [Fork trigger](#fork-trigger)
   - [Upgrading the state](#upgrading-the-state)
 
@@ -20,7 +20,7 @@
 
 ## Introduction
 
-This document describes the process of PeerDAS upgrade.
+This document describes the process of EIP7594 upgrade.
 
 ## Configuration
 
@@ -28,8 +28,8 @@ Warning: this configuration is not definitive.
 
 | Name | Value |
 | - | - |
-| `PEERDAS_FORK_VERSION` | `Version('0x05000000')` |
-| `PEERDAS_FORK_EPOCH` | `Epoch(18446744073709551615)` **TBD** |
+| `EIP7594_FORK_VERSION` | `Version('0x05000000')` |
+| `EIP7594_FORK_EPOCH` | `Epoch(18446744073709551615)` **TBD** |
 
 ## Helper functions
 
@@ -42,8 +42,8 @@ def compute_fork_version(epoch: Epoch) -> Version:
     """
     Return the fork version at the given ``epoch``.
     """
-    if epoch >= PEERDAS_FORK_EPOCH:
-        return PEERDAS_FORK_VERSION
+    if epoch >= EIP7594_FORK_EPOCH:
+        return EIP7594_FORK_VERSION
     if epoch >= DENEB_FORK_EPOCH:
         return DENEB_FORK_VERSION
     if epoch >= CAPELLA_FORK_EPOCH:
@@ -55,23 +55,23 @@ def compute_fork_version(epoch: Epoch) -> Version:
     return GENESIS_FORK_VERSION
 ```
 
-## Fork to PeerDAS
+## Fork to EIP7594
 
 ### Fork trigger
 
-PeerDAS does not need a hard fork. We only add this fork doc for compiling this new feature in pyspec.
+EIP7594 does not need a hard fork. We only add this fork doc for compiling this new feature in pyspec.
 
-For now, we assume the condition will be triggered at epoch `PEERDAS_FORK_EPOCH`.
+For now, we assume the condition will be triggered at epoch `EIP7594_FORK_EPOCH`.
 
-Note that for the pure PeerDAS networks, we don't apply `upgrade_to_peerdas` since it starts with PeerDAS version logic.
+Note that for the pure EIP7594 networks, we don't apply `upgrade_to_eip7594` since it starts with EIP7594 version logic.
 
 ### Upgrading the state
 
-If `state.slot % SLOTS_PER_EPOCH == 0` and `compute_epoch_at_slot(state.slot) == PEERDAS_FORK_EPOCH`,
-an irregular state change is made to upgrade to PeerDAS.
+If `state.slot % SLOTS_PER_EPOCH == 0` and `compute_epoch_at_slot(state.slot) == EIP7594_FORK_EPOCH`,
+an irregular state change is made to upgrade to EIP7594.
 
 ```python
-def upgrade_to_peerdas(pre: deneb.BeaconState) -> BeaconState:
+def upgrade_to_eip7594(pre: deneb.BeaconState) -> BeaconState:
     epoch = deneb.get_current_epoch(pre)
     post = BeaconState(
         # Versioning
@@ -80,7 +80,7 @@ def upgrade_to_peerdas(pre: deneb.BeaconState) -> BeaconState:
         slot=pre.slot,
         fork=Fork(
             previous_version=pre.fork.current_version,
-            current_version=PEERDAS_FORK_VERSION,  # [Modified in PeerDAS]
+            current_version=EIP7594_FORK_VERSION,  # [Modified in EIP7594]
             epoch=epoch,
         ),
         # History
diff --git a/specs/_features/peerdas/polynomial-commitments-sampling.md b/specs/_features/eip7594/polynomial-commitments-sampling.md
similarity index 100%
rename from specs/_features/peerdas/polynomial-commitments-sampling.md
rename to specs/_features/eip7594/polynomial-commitments-sampling.md
diff --git a/tests/core/pyspec/eth2spec/test/context.py b/tests/core/pyspec/eth2spec/test/context.py
index 5d71453c58..e940e24e72 100644
--- a/tests/core/pyspec/eth2spec/test/context.py
+++ b/tests/core/pyspec/eth2spec/test/context.py
@@ -8,7 +8,7 @@
 from .exceptions import SkippedTest
 from .helpers.constants import (
     PHASE0, ALTAIR, BELLATRIX, CAPELLA, DENEB,
-    EIP6110, EIP7002, PEERDAS,
+    EIP6110, EIP7002, EIP7594,
     WHISK,
     MINIMAL,
     ALL_PHASES,
@@ -510,7 +510,7 @@ def wrapper(*args, spec: Spec, **kw):
 with_eip6110_and_later = with_all_phases_from(EIP6110)
 with_eip7002_and_later = with_all_phases_from(EIP7002)
 with_whisk_and_later = with_all_phases_from(WHISK, all_phases=ALLOWED_TEST_RUNNER_FORKS)
-with_peerdas_and_later = with_all_phases_from(PEERDAS, all_phases=ALLOWED_TEST_RUNNER_FORKS)
+with_eip7594_and_later = with_all_phases_from(EIP7594, all_phases=ALLOWED_TEST_RUNNER_FORKS)
 
 
 class quoted_str(str):
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/__init__.py b/tests/core/pyspec/eth2spec/test/eip7594/__init__.py
similarity index 100%
rename from tests/core/pyspec/eth2spec/test/peerdas/__init__.py
rename to tests/core/pyspec/eth2spec/test/eip7594/__init__.py
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/__init__.py b/tests/core/pyspec/eth2spec/test/eip7594/unittests/__init__.py
similarity index 100%
rename from tests/core/pyspec/eth2spec/test/peerdas/unittests/__init__.py
rename to tests/core/pyspec/eth2spec/test/eip7594/unittests/__init__.py
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/__init__.py b/tests/core/pyspec/eth2spec/test/eip7594/unittests/polynomial_commitments/__init__.py
similarity index 100%
rename from tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/__init__.py
rename to tests/core/pyspec/eth2spec/test/eip7594/unittests/polynomial_commitments/__init__.py
diff --git a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py b/tests/core/pyspec/eth2spec/test/eip7594/unittests/polynomial_commitments/test_polynomial_commitments.py
similarity index 95%
rename from tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
rename to tests/core/pyspec/eth2spec/test/eip7594/unittests/polynomial_commitments/test_polynomial_commitments.py
index cff774114a..d3e848772a 100644
--- a/tests/core/pyspec/eth2spec/test/peerdas/unittests/polynomial_commitments/test_polynomial_commitments.py
+++ b/tests/core/pyspec/eth2spec/test/eip7594/unittests/polynomial_commitments/test_polynomial_commitments.py
@@ -2,7 +2,7 @@
 from eth2spec.test.context import (
     spec_test,
     single_phase,
-    with_peerdas_and_later,
+    with_eip7594_and_later,
 )
 from eth2spec.test.helpers.sharding import (
     get_sample_blob,
@@ -10,7 +10,7 @@
 from eth2spec.utils.bls import BLS_MODULUS
 
 
-@with_peerdas_and_later
+@with_eip7594_and_later
 @spec_test
 @single_phase
 def test_fft(spec):
@@ -27,7 +27,7 @@ def test_fft(spec):
     assert poly_coeff_inversed == poly_coeff
 
 
-@with_peerdas_and_later
+@with_eip7594_and_later
 @spec_test
 @single_phase
 def test_verify_cell_proof(spec):
@@ -40,7 +40,7 @@ def test_verify_cell_proof(spec):
     assert spec.verify_cell_proof(commitment, cell_id, cells[cell_id], proofs[cell_id])
 
 
-@with_peerdas_and_later
+@with_eip7594_and_later
 @spec_test
 @single_phase
 def test_verify_cell_proof_batch(spec):
@@ -57,7 +57,7 @@ def test_verify_cell_proof_batch(spec):
     )
 
 
-@with_peerdas_and_later
+@with_eip7594_and_later
 @spec_test
 @single_phase
 def test_recover_polynomial(spec):
diff --git a/tests/core/pyspec/eth2spec/test/helpers/constants.py b/tests/core/pyspec/eth2spec/test/helpers/constants.py
index a79feb96e8..fc939ce097 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/constants.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/constants.py
@@ -19,7 +19,7 @@
 EIP6110 = SpecForkName('eip6110')
 EIP7002 = SpecForkName('eip7002')
 WHISK = SpecForkName('whisk')
-PEERDAS = SpecForkName('peerdas')
+EIP7594 = SpecForkName('eip7594')
 
 #
 # SpecFork settings
@@ -38,7 +38,7 @@
     # Experimental patches
     EIP6110,
     EIP7002,
-    PEERDAS,
+    EIP7594,
 )
 # The forks that have light client specs
 LIGHT_CLIENT_TESTING_FORKS = (*[item for item in MAINNET_FORKS if item != PHASE0], DENEB)
@@ -59,7 +59,7 @@
     EIP6110: DENEB,
     WHISK: CAPELLA,
     EIP7002: CAPELLA,
-    PEERDAS: DENEB,
+    EIP7594: DENEB,
 }
 
 # For fork transition tests