"""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))"""
	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 getConstantCoef(*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