import math import cmath import itertools from .gf256 import gfmul, gfpow # divisors of 255 and their factors in natural numbers DIVISORS = [3, 5, 15, 17, 51, 85, 255] FACTORS = {3: [3], 5: [5], 15: [3, 5], 17: [17], 51: [3, 17], 85: [5, 17], 255: [3, 5, 17]} # values of n-th square roots in GF256 SQUARE_ROOTS = {3: 189, 5: 12, 15: 225, 17: 53, 51: 51, 85: 15, 255: 3} def ceil_size(n): assert n <= DIVISORS[-1] for (i, ni) in enumerate(DIVISORS): if ni >= n: break return ni def precompute_x(n): """Return a geometric sequence [1, w, w**2, ..., w**(n-1)], where w**n==1. This can be done only for certain values of n.""" assert n in SQUARE_ROOTS, n w = SQUARE_ROOTS[n] # primitive N-th square root of 1 return list(itertools.accumulate([1]+[w]*(n-1), gfmul)) def complex_dft(p): """Quadratic formula from the definition. The basic case in complex numbers.""" N = len(p) w = cmath.exp(-2*math.pi*1j/N) # primitive N-th square root of 1 y = [0]*N for k in range(N): xk = w**k for n in range(N): y[k] += p[n] * xk**n return y def dft(p): """Quadratic formula from the definition. In GF256.""" N = len(p) x = precompute_x(N) y = [0]*N for k in range(N): for n in range(N): y[k] ^= gfmul(p[n], gfpow(x[k], n)) return y def compute_inverse(N1, N2): for i in range(N2): if N1*i % N2 == 1: return i raise ValueError("Failed to find an inverse to {0} mod {1}.".format(N1, N2)) def prime_fft(p, divisors, basic_dft=dft): """https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm""" if len(divisors) == 1: return basic_dft(p) N = len(p) N1 = divisors[0] N2 = N//N1 N1_inv = compute_inverse(N1, N2) N2_inv = compute_inverse(N2, N1) ys = [] for n1 in range(N1): # compute rows p_ = [p[(n2*N1+n1*N2) % N] for n2 in range(N2)] ys.append(prime_fft(p_, divisors[1:], basic_dft)) for k2 in range(N2): # compute cols p_ = [row[k2] for row in ys] y_ = basic_dft(p_) for (yi, row) in zip(y_, ys): # update col row[k2] = yi # remap and output res = [0]*N for k1 in range(N1): for k2 in range(N2): res[(k1*N2*N2_inv+k2*N1*N1_inv) % N] = ys[k1][k2] return res def evaluate(coefs, n): ni = ceil_size(n) extended_coefs = coefs + [0]*(ni-len(coefs)) ys = prime_fft(extended_coefs, FACTORS[ni]) return ys[:n]