# GNU GPLv3, see LICENSE

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


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: [a0, a1, ...]."""
	res = 0
	xk = 1
	for a in coefs:
		res ^= gfmul(a, xk)
		xk = gfmul(xk, x)
	return res


def get_constant_coef(*points):
	"""Compute constant polynomial coefficient given the points.

	See https://en.wikipedia.org/wiki/Shamir's_Secret_Sharing#Computationally_Efficient_Approach"""
	k = len(points)
	res = 0
	for i in range(k):
		(x, y) = points[i]
		prod = 1
		for j in range(k):
			if i==j: continue
			(xj, yj) = points[j]
			prod = gfmul(prod, (gfmul(xj, INV[xj^x])))
		res ^= gfmul(y, prod)
	return res