import random
import math

n = 8
k = 3
x = "11110101"




def findS():
	for s_candidate in range(k, n+1):
		binom = math.factorial(s_candidate) / ( math.factorial(k-1) * (math.factorial(s_candidate - (k-1))) )
		if binom >= n:
			return s_candidate


s = findS()


def makeOneSeq(seq):
	ones_remainding_counter = k - 1
	while ones_remainding_counter != 0:
		rand_idx = random.randint(0, s - 1)
		if seq[rand_idx] == "0":
			seq_temp = list(seq)
			seq_temp[rand_idx] = "1"
			seq = "".join(seq_temp)
			ones_remainding_counter -= 1
	return seq

def makeSequences():
	sequnces = [ "0"*s for j in range(n) ]
	for i, seq in enumerate(sequnces):
		candidate = makeOneSeq(seq)
		while( candidate in sequnces):
			candidate = makeOneSeq(seq)
		sequnces[i] = candidate
	sequnces.sort()
	return sequnces


sequnces = makeSequences()

#print(sequnces)


def user_send(rands, i):
	gs = [[rands[l] * z + int(i[l]) for l in range(0, s)] for z in range(1, k+1)]
	return gs


def servers_comp(gs):
	Fs = []
	for g in gs:
		F = 0
		for j in range(0, n):
			f = 1
			j_bitstring = sequnces[j]
			for l in range(s):
				if int(j_bitstring[l]) == 1:
					f *= g[l]
			F += f * int(x[j])
		Fs.append(F)
	return Fs

def poly_interpolation(x, Fs):
	add_product = 0
	for i in range(0, k):
		mult_product = 1
		for j in range(0, k):
			if j != i:
				mult_product *= (x - (j+1))/((i+1) - (j+1))
		add_product += mult_product * Fs[i]
	return int(add_product)


def protocol(i):
	rands = [random.randint(1,20) for r in range(s)]

	gs = user_send(rands, i)
	Fs = servers_comp(gs)
	F0 = poly_interpolation(0, Fs)

	# print(F0)

	return F0


for i in range(n):
	i_bitstring = sequnces[i]
	F0 = protocol(i_bitstring)
	assert (F0 == int(x[i]))