# GNU GPLv3, see LICENSE

"""Arithmetic operations on Galois Field 2**8. See https://en.wikipedia.org/wiki/Finite_field_arithmetic"""

import operator

def _gfmul(a, b):

	"""Basic multiplication. Russian peasant algorithm."""

	res = 0

	while a and b:

		if b&1: res ^= a

		if a&0x80: a = 0xff&(a<<1)^0x1b

		else: a <<= 1

		b >>= 1

	return res

g = 3  # generator

E = [None]*256  # exponentials

L = [None]*256  # logarithms

acc = 1

for i in range(256):

	E[i] = acc

	L[acc] = i

	acc = _gfmul(acc, g)

L[1] = 0

INV = [E[255-L[i]] if i!=0 else None for i in range(256)]  # multiplicative inverse

def gfmul(a, b):

	"""Fast multiplication. Basic multiplication is expensive. a*b==g**(log(a)+log(b))"""

	assert 0<=a<=255, 0<=b<=255

	if a==0 or b==0: return 0

	t = L[a]+L[b]

	if t>255: t -= 255

	return E[t]

def evaluate(coefs, x):

	"""Evaluate polynomial's value at x.

	:param coefs: [an, ..., a1, a0]."""

	res = 0

	for a in coefs:  # Horner's rule

		res = gfmul(res, x)

		res ^= a

	return res

def get_constant_coef(weights, y_coords):

	"""Compute constant polynomial coefficient given the points.

	See https://en.wikipedia.org/wiki/Shamir's_Secret_Sharing#Computationally_Efficient_Approach"""

	return reduce(

		operator.xor,

		map(lambda ab: gfmul(*ab), zip(weights, y_coords))

	)

def compute_weights(x_coords):

	assert x_coords

	res = [

			reduce(

				gfmul,

				(gfmul(xj, INV[xj^xi]) for xj in x_coords if xi!=xj),

				1

			) for xi in x_coords

	]

	return res