def fk20_single_data_availability_optimized(polynomial, setup):
"""
Special version of the FK20 for the situation of data availability checks:
The upper half of the polynomial coefficients is always 0, so we do not need to extend to twice the size
for Toeplitz matrix multiplication
"""
assert is_power_of_two(len(polynomial))
n = len(polynomial) // 2
assert all(x == 0 for x in polynomial[n:])
reduced_polynomial = polynomial[:n]
# Preprocessing part -- this is independent from the polynomial coefficients and can be
# done before the polynomial is known, it only needs to be computed once
x = setup[0][n - 2::-1] + [b.Z1]
xext_fft = toeplitz_part1(x)
toeplitz_coefficients = reduced_polynomial[-1::] + [0] * (
n + 1) + reduced_polynomial[1:-1]
# Compute the vector h from the paper using a Toeplitz matric multiplication
h = toeplitz_part3(toeplitz_part2(toeplitz_coefficients, xext_fft))
h = h + [b.Z1] * n
# The proofs are the DFT of the h vector
return fft(h, MODULUS, get_root_of_unity(2 * n))
def fk20_multi_data_availability_optimized(polynomial, l, setup):
"""
FK20 multi-proof method, optimized for dava availability where the top half of polynomial
coefficients == 0
"""
assert is_power_of_two(len(polynomial))
n = len(polynomial) // 2
k = n // l
assert is_power_of_two(n)
assert is_power_of_two(l)
assert k >= 1
assert all(x == 0 for x in polynomial[n:])
reduced_polynomial = polynomial[:n]
# Preprocessing part -- this is independent from the polynomial coefficients and can be
# done before the polynomial is known, it only needs to be computed once
xext_fft = []
for i in range(l):
x = setup[0][n - l - 1 - i::-l] + [b.Z1]
xext_fft.append(toeplitz_part1(x))
add_instrumentation()
hext_fft = [b.Z1] * 2 * k
for i in range(l):
toeplitz_coefficients = reduced_polynomial[- i - 1::l] + [0] * (k + 1) \
+ reduced_polynomial[2 * l - i - 1: - l - i:l]
# Compute the vector h from the paper using a Toeplitz matric multiplication
hext_fft = [
b.add(v, w) for v, w in zip(
hext_fft, toeplitz_part2(toeplitz_coefficients, xext_fft[i]))
]
# Final FFT done after summing all h vectors
h = toeplitz_part3(hext_fft)
h = h + [b.Z1] * k
# The proofs are the DFT of the h vector
return fft(h, MODULUS, get_root_of_unity(2 * k))
def fk20_multi(polynomial, l, setup):
"""
For a polynomial of size n, let w be a n-th root of unity. Then this method will return
k=n/l KZG proofs for the points
proof[0]: w^(0*l + 0), w^(0*l + 1), ... w^(0*l + l - 1)
proof[0]: w^(0*l + 0), w^(0*l + 1), ... w^(0*l + l - 1)
...
proof[i]: w^(i*l + 0), w^(i*l + 1), ... w^(i*l + l - 1)
...
"""
n = len(polynomial)
k = n // l
assert is_power_of_two(n)
assert is_power_of_two(l)
assert k >= 1
# Preprocessing part -- this is independent from the polynomial coefficients and can be
# done before the polynomial is known, it only needs to be computed once
xext_fft = []
for i in range(l):
x = setup[0][n - l - 1 - i::-l] + [b.Z1]
xext_fft.append(toeplitz_part1(x))
hext_fft = [b.Z1] * 2 * k
for i in range(l):
toeplitz_coefficients = polynomial[-i - 1::l] + [0] * (
k + 1) + polynomial[2 * l - i - 1:-l - i:l]
# Compute the vector h from the paper using a Toeplitz matric multiplication
hext_fft = [
b.add(v, w) for v, w in zip(
hext_fft, toeplitz_part2(toeplitz_coefficients, xext_fft[i]))
]
h = toeplitz_part3(hext_fft)
# The proofs are the DFT of the h vector
return fft(h, MODULUS, get_root_of_unity(k))
def data_availabilty_using_fk20(polynomial, setup):
"""
Computes all the KZG proofs for data availability checks. This involves sampling on the double domain
and reordering according to reverse bit order
"""
assert is_power_of_two(len(polynomial))
n = len(polynomial)
extended_polynomial = polynomial + [0] * n
all_proofs = fk20_single_data_availability_optimized(
extended_polynomial, setup)
return list_to_reverse_bit_order(all_proofs)
def toeplitz_part1(x):
"""
Performs the first part of the Toeplitz matrix multiplication algorithm, which is a Fourier
transform of the vector x extended
"""
assert is_power_of_two(len(x))
root_of_unity = get_root_of_unity(len(x) * 2)
# Extend x with zeros (neutral element of G1)
xext = x + [b.Z1] * len(x)
xext_fft = fft(xext, MODULUS, root_of_unity, inv=False)
return xext_fft
def fk20_single(polynomial, setup):
"""
Compute all n (single) proofs according to FK20 method
"""
assert is_power_of_two(len(polynomial))
n = len(polynomial)
x = setup[0][n - 2::-1] + [b.Z1]
xext_fft = toeplitz_part1(x)
toeplitz_coefficients = polynomial[-1::] + [0] * (n + 1) + polynomial[1:-1]
# Compute the vector h from the paper using a Toeplitz matric multiplication
h = toeplitz_part3(toeplitz_part2(toeplitz_coefficients, xext_fft))
# The proofs are the DFT of the h vector
return fft(h, MODULUS, get_root_of_unity(n))
def toeplitz_part2(toeplitz_coefficients, xext_fft):
"""
Performs the second part of the Toeplitz matrix multiplication algorithm
"""
# Extend the toeplitz coefficients to get a circulant matrix into which the Toeplitz
# matrix is embedded
assert is_power_of_two(len(toeplitz_coefficients))
root_of_unity = get_root_of_unity(len(xext_fft))
toeplitz_coefficients_fft = fft(toeplitz_coefficients,
MODULUS,
root_of_unity,
inv=False)
hext_fft = [
b.multiply(v, w) for v, w in zip(xext_fft, toeplitz_coefficients_fft)
]
return hext_fft
