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some punctuation

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b-wagn 2024-07-31 17:30:36 +02:00
parent 85a42cd7bf
commit 225c486183
1 changed files with 10 additions and 10 deletions
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20
specs/_features/eip7594/polynomial-commitments-sampling.md
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@ -685,13 +685,13 @@ def recover_polynomialcoeff(cell_indices: Sequence[CellIndex],
"""
Recover the polynomial in coefficient form that when evaluated at the roots of unity will give the extended blob.
"""
# Get the extended domain. This will be referred to as the FFT domain
# Get the extended domain. This will be referred to as the FFT domain.
roots_of_unity_extended = compute_roots_of_unity(FIELD_ELEMENTS_PER_EXT_BLOB)
# Flatten the cells into evaluations
# If a cell is missing, then its evaluation is zero
# If a cell is missing, then its evaluation is zero.
# We let E(x) be a polynomial of degree FIELD_ELEMENTS_PER_EXT_BLOB - 1
# that interpolates the evaluations including the zeros for missing ones
# that interpolates the evaluations including the zeros for missing ones.
extended_evaluation_rbo = [0] * FIELD_ELEMENTS_PER_EXT_BLOB
for cell_index, cell in zip(cell_indices, cells):
start = cell_index * FIELD_ELEMENTS_PER_CELL
@ -699,8 +699,8 @@ def recover_polynomialcoeff(cell_indices: Sequence[CellIndex],
extended_evaluation_rbo[start:end] = cell
extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
# Compute the vanishing polynomial Z(x) in coefficient form
# Z(x) is the polynomial which vanishes on all of the evaluations which are missing
# Compute the vanishing polynomial Z(x) in coefficient form.
# Z(x) is the polynomial which vanishes on all of the evaluations which are missing.
missing_cell_indices = [CellIndex(cell_index) for cell_index in range(CELLS_PER_EXT_BLOB)
if cell_index not in cell_indices]
zero_poly_coeff = construct_vanishing_polynomial(missing_cell_indices)
@ -710,18 +710,18 @@ def recover_polynomialcoeff(cell_indices: Sequence[CellIndex],
# Compute (E*Z)(x) = E(x) * Z(x) in evaluation form over the FFT domain
# Note: over the FFT domain, the polynomials (E*Z)(x) and (P*Z)(x) agree, where
# P(x) is the polynomial we want to reconstruct (degree FIELD_ELEMENTS_PER_BLOB - 1)
# P(x) is the polynomial we want to reconstruct (degree FIELD_ELEMENTS_PER_BLOB - 1).
extended_evaluation_times_zero = [BLSFieldElement(int(a) * int(b) % BLS_MODULUS)
for a, b in zip(zero_poly_eval, extended_evaluation)]
# We know that (E*Z)(x) and (P*Z)(x) agree over the FFT domain
# and we know that (P*Z)(x) has degree at most FIELD_ELEMENTS_PER_EXT_BLOB - 1
# We know that (E*Z)(x) and (P*Z)(x) agree over the FFT domain,
# and we know that (P*Z)(x) has degree at most FIELD_ELEMENTS_PER_EXT_BLOB - 1.
# Thus, an inverse FFT of the evaluations of (E*Z)(x) (= evaluations of (P*Z)(x))
# yields the coefficient form of (P*Z)(x).
extended_evaluation_times_zero_coeffs = fft_field(extended_evaluation_times_zero, roots_of_unity_extended, inv=True)
# Next step is to divide the polynomial (P*Z)(x) by polynomial Z(x) to get P(x)
# Do this in evaluation form over a coset of the FFT domain to avoid division by 0
# Next step is to divide the polynomial (P*Z)(x) by polynomial Z(x) to get P(x).
# We do this in evaluation form over a coset of the FFT domain to avoid division by 0.
# Convert (P*Z)(x) to evaluation form over a coset of the FFT domain
extended_evaluations_over_coset = coset_fft_field(extended_evaluation_times_zero_coeffs, roots_of_unity_extended)