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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. 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) roots_of_unity_extended = compute_roots_of_unity(FIELD_ELEMENTS_PER_EXT_BLOB)
# Flatten the cells into evaluations # 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 # 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 extended_evaluation_rbo = [0] * FIELD_ELEMENTS_PER_EXT_BLOB
for cell_index, cell in zip(cell_indices, cells): for cell_index, cell in zip(cell_indices, cells):
start = cell_index * FIELD_ELEMENTS_PER_CELL 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_rbo[start:end] = cell
extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo) extended_evaluation = bit_reversal_permutation(extended_evaluation_rbo)
# Compute the vanishing polynomial Z(x) in coefficient form # Compute the vanishing polynomial Z(x) in coefficient form.
# Z(x) is the polynomial which vanishes on all of the evaluations which are missing # 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) missing_cell_indices = [CellIndex(cell_index) for cell_index in range(CELLS_PER_EXT_BLOB)
if cell_index not in cell_indices] if cell_index not in cell_indices]
zero_poly_coeff = construct_vanishing_polynomial(missing_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 # 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 # 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) extended_evaluation_times_zero = [BLSFieldElement(int(a) * int(b) % BLS_MODULUS)
for a, b in zip(zero_poly_eval, extended_evaluation)] 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 # 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 # 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)) # Thus, an inverse FFT of the evaluations of (E*Z)(x) (= evaluations of (P*Z)(x))
# yields the coefficient form 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) 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) # 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 # 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 # 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) extended_evaluations_over_coset = coset_fft_field(extended_evaluation_times_zero_coeffs, roots_of_unity_extended)