from . import fft

def compute_x(n):

	return fft.precompute_x(fft.ceil_size(n))[:n]

	# we might be concerned with zero coefficients degenerating our polynomial,

	# but there's no reason - we still need k shares to determine it is the case

	coefs = [int(secret_b)]+[int(b) for b in os.urandom(k-1)]

	return fft.evaluate(coefs, n)

	xs = compute_x(n)

	return [(xi, bytes([s[i] for s in shares])) for (i, xi) in enumerate(xs)]

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]

def gfpow(x, k):

	"""Compute x**k."""

	i = 1

	res = 1

	while i <= k:

		if k&i:

			res = gfmul(res, x)

		x = gfmul(x, x)

		i <<= 1

	return res

# GNU GPLv3, see LICENSE

import random

import functools

import operator

from unittest import TestCase

from .. import gf256

from ..fft import *

def batch_evaluate(coefs, xs):

	return [gf256.evaluate(coefs, x) for x in xs]

class TestFFT(TestCase):

	def test_complex_dft(self):

		self.assertEqual(complex_dft([0]), [0+0j])

		self.assertEqual(complex_dft([1]), [1+0j])

		self.assertEqual(complex_dft([2]), [2+0j])

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1]), [4+0j, 2+0j]))

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1, 4]), [8+0j, 0.5+2.59807621j, 0.5-2.59807621j]))

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1, 4, 1]), [9+0j, -1+0j, 5+0j, -1+0j]))

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(

			complex_dft([3, 1, 4, 1, 5]),

			[14+0j, 0.80901699+2.04087031j, -0.30901699+5.20431056j, -0.30901699-5.20431056j, 0.80901699-2.04087031j]

		))

	def test_complex_prime_fft(self):

		random.seed(1918)

		for divisors in [[3], [2, 3], [3, 5], [3, 5, 17], [2, 3, 5, 7, 11]]:

			n = functools.reduce(operator.mul, divisors)

			coefficients = [random.randint(-128, 127) for i in range(n)]

			a = prime_fft(coefficients, divisors, complex_dft)

			b = complex_dft(coefficients)

			all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))

	def test_finite_dft(self):

		random.seed(1918)

		x = {i: precompute_x(i) for i in [3, 5, 15, 17]}  # all sets of xs

		for n in [3, 5, 15, 17]:

			coefficients = [random.randint(0, 255) for i in range(n)]

			self.assertEqual(

				dft(coefficients),

				batch_evaluate(coefficients[::-1], x[n])

			)

	def test_finite_prime_fft(self):

		random.seed(1918)

		for divisors in [[3], [3, 5], [3, 17], [5, 17], [3, 5, 17]]:

			n = functools.reduce(operator.mul, divisors)

			coefficients = [random.randint(0, 255) for i in range(n)]

			a = prime_fft(coefficients, divisors)

			b = dft(coefficients)

			all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))

	def test_gfpow(self):

		self.assertEqual(gfpow(0, 0), 1)

		for i in range(1, 256):

			self.assertEqual(gfpow(i, 0), 1)

			self.assertEqual(gfpow(i, 1), i)

			self.assertEqual(gfpow(0, i), 0)

			self.assertEqual(gfpow(1, i), 1)

			self.assertEqual(gfpow(i, 256), i)

			self.assertEqual(gfpow(i, 2), gfmul(i, i))

		random.seed(1918)

		for i in range(256):

			j = random.randint(2, 255)

			k = random.randint(3, 255)

			y = 1

			for m in range(k):

				y = gfmul(y, j)

			self.assertEqual(gfpow(j, k), y)

from ..core import encode, decode, detect_encoding, _share_byte, compute_x

		xs = compute_x(3)