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VSS2_main.py
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VSS2_main.py
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import random
import math
# from decimal import Decimal
import time
def isprime(n):
if n == 2:
return True
if n == 1 or n % 2 == 0:
return False
i = 3
while i <= math.sqrt(n):
if n % i == 0:
return False
i = i + 2
return True
def initial(Z_lower=100):
start = time.time()
# generate q bigger than z_lower
q = Z_lower
while True:
if isprime(q):
break
else:
q = q + 1
print("q = " + str(q))
print("\nq is prime\n")
# Find p and r
r = 1
while True:
p = r * q + 1
if isprime(p):
print("r = " + str(r))
print("p = " + str(p))
print("\np is prime\n")
break
r = r + 1
# Compute elements of Z_p*
Z_p_star = []
for i in range(0, p):
if (math.gcd(i, p) == 1):
Z_p_star.append(i)
# if len(Z_p_star) > 10:
# break
# print("Z_p* = ")
# print(Z_p_star) # , len(Z_p_star) same length, i.e. range(p)
# Compute elements of G = {h^r mod p | h in Z_p*}
G = []
for i in Z_p_star:
G.append((i ** r) % p)
G = list(set(G))
G.sort()
# print("\nG = ")
# print(G)
print("Order of G is " + str(len(G)) + ". This must be equal to q:",q)
# Since the order of G is prime, any element of G except 1 is a generator
g = random.choice(list(filter(lambda g: g != 1, G)))
print("\ng = " + str(g) + "\n")
print("initialization time ", time.time()-start)
return p, q, r, g
def generate_shares(n, t, secret, p, q, r, g, idxs):
if secret==0: secret=1 # one exception after quantization
assert secret >= 1 and secret <= q, "secret not in range "+str(secret)
FIELD_SIZE = q
coefficients = coeff(t, secret, FIELD_SIZE)
# print("coefficients",coefficients,type(coefficients[0]))
users = list(idxs) # users are to recieve the shares
assert n==len(users), "these two number should be identical"
shares = []
for i in users:
f_i = f(i, coefficients, q)
shares.append((i, f_i))
start = time.time()
commitments = commitment(coefficients, g, p)
print("commitments:", commitments)
print("commitments take ", time.time()-start,"seconds")
verifications = []
startv = time.time()
for i in users:
# start = time.time()
check1 = quick_pow(g, share_ith(shares, i), p)
# print("check1 time ", time.time()-start)
# start = time.time()
# check1 = g ** share_ith(shares, i) % p
check2 = verification(g, commitments, i, p, q)
# print("check2 time ", time.time()-start)
verifications.append(check2)
if (check1 % p) == (check2 % p) :
pass
else:
# print(g, share_ith(shares, i), p , q, i, commitments, shares)
print("checking fails with:", check1, check2)
# print(i, "-th user ============= tag at time ", time.time()-start,"seconds =============")
print("verification time ", time.time()-startv)
# commitments, verifications = [0,], [0,]
return shares, commitments, verifications
def share_ith(shares, i):
for share in shares:
if share[0] == i:
return int(share[1])
return None
def coeff(t, secret, FIELD_SIZE):
coeff = [random.randrange(0, FIELD_SIZE) for _ in range(t - 1)]
coeff.append(secret) # a0 is secret
return coeff
def f(x, coefficients, q):
# y = Decimal('0')
# for coefficient_index, coefficient_value in enumerate(coefficients[::-1]):
# y += (Decimal(str(x)) ** Decimal(str(coefficient_index)) * Decimal(str(coefficient_value)))
# y = Decimal(int(y)%q)
# return int(y)
y = 0
for coefficient_index, coefficient_value in enumerate(coefficients[::-1]):
y += x ** coefficient_index * coefficient_value
y = int(y) % q
return y
def commitment(coefficients, g, p):
commitments = []
for coefficient_index, coefficient_value in enumerate(coefficients[::-1]):
# c = g ** coefficient_value % p
c = quick_pow(g, coefficient_value, p)
commitments.append(c)
return commitments
def verification(g, commitments, i, p, q):
v = 1
for k, c in enumerate(commitments):
# start = time.time()
# v = ( v * (c ** ((i ** k) % q)) ) % p
v = (v * quick_pow(c, (i ** k) % q , p)) % p
# print(c,i,k)
# print("verification once takes:", time.time() - start)
return v
def quick_pow(a, b, q): # compute a^b mod q, in a faster way?
temp = 1
for i in range(1, b + 1):
temp = (temp * a) % q
return temp % q
def reconstruct_secret(pool, q):
start = time.time()
x_s,y_s = [],[]
for share in pool:
x_s.append(int(share[0]))
y_s.append(int(share[1]))
out = _lagrange_interpolate(0, x_s, y_s, q)
print("reconstruct_secret time takes:", time.time() - start)
return out
def _lagrange_interpolate(x, x_s, y_s, p):
"""
Find the y-value for the given x, given n (x, y) points;
k points will define a polynomial of up to kth order.
"""
k = len(x_s)
assert k == len(set(x_s)), "points must be distinct"
def PI(vals): # upper-case PI -- product of inputs
accum = 1
for v in vals:
accum *= v
return accum
nums = [] # avoid inexact division
dens = []
for i in range(k):
others = list(x_s)
cur = others.pop(i)
nums.append(PI(int(x - o) for o in others))
dens.append(PI(int(cur - o) for o in others))
den = PI(dens)
num = sum([_divmod(int(int(nums[i]) * int(den) * int(y_s[i]) % p), int(dens[i]), p) for i in range(k)])
# debug: overflow, ^ here : cast to int
# num = 0
# for i in range(k):
# temp = int(int(nums[i]) * int(den) * int(y_s[i]) % p)
# num += _divmod(temp, int(dens[i]), p)
return (_divmod(num, den, p) + p) % p
def _extended_gcd(a, b):
"""
Division in integers modulus p means finding the inverse of the
denominator modulo p and then multiplying the numerator by this
inverse (Note: inverse of A is B such that A*B % p == 1) this can
be computed via extended Euclidean algorithm
https://en.wikipedia.org/wiki/Modular_multiplicative_inverse#Computation
"""
x = 0
last_x = 1
y = 1
last_y = 0
while b != 0:
quot = a // b
a, b = b, a % b
x, last_x = last_x - quot * x, x
y, last_y = last_y - quot * y, y
return int(last_x), int(last_y)
def _divmod(num, den, p):
"""Compute num / den modulo prime p
To explain what this means, the return value will be such that
the following is true: den * _divmod(num, den, p) % p == num
"""
inv, _ = _extended_gcd(den, p)
return num * inv
# Driver code
if __name__ == '__main__':
# initialization
time_start = time.time()
print("========Main VSS Starts==========")
p,q,r,g = initial(2**18) # 2 * 10**7)
# Secret taken from the group Z_q*
t, n = 7, 40
secret = q - 123
print(f'Original Secret: {secret}')
# Phase I: Generation of shares
users = [i+1 for i in range(n)]
# print(n, t, secret, p, q, r, g, users)
shares, commitments, verifications= generate_shares(n, t, secret, p, q, r, g, users)
# print(f'Shares: {", ".join(str(share) for share in shares)}')
# print(f'Commitments: {", ".join(str(commitment) for commitment in commitments)}')
# print(f'verifications: {", ".join(str(verification) for verification in verifications)}')
# Phase II: Secret Reconstruction
# Picking t shares randomly for reconstruction
pool = random.sample(shares, t)
# print(f'Combining shares: {", ".join(str(share) for share in pool)}')
secret_reconstructed = reconstruct_secret(pool, q)
print("reconstruct_secret:",secret_reconstructed)
time_end = time.time()
print('time cost in second:', time_end-time_start)