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partially.py
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partially.py
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'''
This is an implemention of the partially blind signature scheme from the paper:
Provably Secure Partially Blind Signatures
Masayuki ABE and Tatsuaki OKAMOTO
'''
import os
from collections import namedtuple
import hashlib
Parameters = namedtuple('Parameters', ['L', 'N', 'p', 'q', 'g'])
Keypair = namedtuple('Keypair', ['x', 'y', 'parameters'])
### Math helper functions ###
def rand_int(nbits):
if nbits % 8 != 0:
raise ValueError("nbits must be divisible by 8 so it can be broken"
" into bytes.")
return int.from_bytes(os.urandom(nbits//8), byteorder='little')
def rand_less_than(upper_bound, nbits):
'''This could be smarter.'''
while True:
r = rand_int(nbits)
if r < upper_bound:
return r
def fermat_test(p, nbits):
'''Fermat primality test'''
for _ in range(5):
a = rand_less_than(p, nbits)
if not pow(a, p - 1, p) == 1:
return False
return True
def miller_rabin_test(p, nbits):
'''Miller-Rabin primality test'''
k = 5 # accuracy parameter, this should be turned up in practice
r = 1
while (pow(2, r) & p) != pow(2, r):
r += 1
d = p // pow(2, r)
for _ in range(k):
a = rand_less_than(p - 2, nbits)
x = pow(a, d, p)
if x == 1 or x == p - 1:
continue
for _ in range(r - 1):
x = pow(x, 2, p)
if x == 1:
return False
if x == p - 1:
break
else:
return False
return True
def prime_test(p, nbits):
return miller_rabin_test(p, nbits)
def rand_prime(nbits):
is_prime = False
while not is_prime:
p = rand_int(nbits)
is_prime = prime_test(p, nbits)
return p
### DSA functions ###
def choose_q(N):
return rand_prime(N)
def choose_p(L, N, q):
k = L - N
is_prime = False
while not is_prime:
p = (q*rand_int(k)) + 1
is_prime = prime_test(p, L)
return p
def choose_g(L, N, p, q):
h = 2
while True:
g = pow(h, (p - 1)//q, p)
if pow(g, q, p) == 1:
return g
h = rand_less_than(p, L)
def choose_parameters(L, N):
'''Returns DSA parameters p, q, g'''
q = choose_q(N)
p = choose_p(L, N, q)
g = choose_g(L, N, p, q)
return Parameters(L, N, p, q, g)
def choose_keypair(parameters):
x = rand_less_than(parameters.q, parameters.N)
return Keypair(x, pow(parameters.g, x, parameters.p), parameters)
def int_to_bytes(in_int):
i = in_int
byte_length = ((i).bit_length() + 7) // 8
return i.to_bytes(byte_length, 'little')
### Hashing functions ###
def do_hash(data):
'''hash helper'''
h = hashlib.sha256()
h.update(data)
return h.digest()
def full_domain_hash(data, target_length):
tl_bytes = target_length // 8
digest_size = hashlib.sha256().digest_size
ncycles = (tl_bytes // digest_size) + 1
out = bytearray()
for i in range(ncycles):
out.extend(do_hash(data + int_to_bytes(i)))
return bytes(out[:tl_bytes])
def digest(data, parameters):
'''F hash function from paper'''
hashed = full_domain_hash(data, parameters.L)
i = int.from_bytes(hashed, byteorder='little') % parameters.p
return pow(i, (parameters.p - 1)//parameters.q, parameters.p)
### Protocol stuff ###
class Signer:
'''Signer S from the paper'''
def __init__(self, parameters):
self.parameters = parameters
self.L, self.N, self.p, self.q, self.g = tuple(parameters)
self.keypair = choose_keypair(self.parameters)
def start(self, info):
self.z = digest(info, self.parameters)
def one(self):
usd = [rand_less_than(self.q, self.N) for _ in range(3)]
self.u, self.s, self.d = usd
self.a = pow(self.g, self.u, self.p)
self.b = (pow(self.g, self.s, self.p) *
pow(self.z, self.d, self.p)) % self.p
return self.a, self.b
def three(self, e):
self.c = (e - self.d) % self.q
self.r = (self.u - (self.c * self.keypair.x)) % self.q
return self.r, self.c, self.s, self.d
class User:
'''User U from the paper'''
def __init__(self, parameters, pubkey):
self.parameters = parameters
self.L, self.N, self.p, self.q, self.g = tuple(parameters)
self.y = pubkey
def start(self, info, msg):
ts = [rand_less_than(self.q, self.N) for _ in range(4)]
self.t1, self.t2, self.t3, self.t4 = ts
self.z = digest(info, self.parameters)
self.msg = msg
def two(self, a, b):
alpha = (a * pow(self.g, self.t1, self.p) *
pow(self.y, self.t2, self.p)) % self.p
beta = (b * pow(self.g, self.t3, self.p) *
pow(self.z, self.t4, self.p)) % self.p
e_bytes = bytearray()
for v in (alpha, beta, self.z):
e_bytes.extend(int_to_bytes(v))
e_bytes.extend(msg)
epsilon = int.from_bytes(full_domain_hash(e_bytes, self.N), 'little')
return (epsilon - self.t2 - self.t4) % self.q
def four(self, r, c, s, d):
rho = (r + self.t1) % self.q
omega = (c + self.t2) % self.q
delta = (s + self.t3) % self.q
sigma = (d + self.t4) % self.q
return rho, omega, delta, sigma
def check(rho, omega, delta, sigma, z, msg, y, parameters):
'''Signatuer verification'''
lhs = int_to_bytes((omega + sigma) % parameters.q)
rhs_one = (pow(parameters.g, rho, parameters.p) *
pow(y, omega, parameters.p)) % parameters.p
rhs_two = (pow(parameters.g, delta, parameters.p) *
pow(z, sigma, parameters.p)) % parameters.p
rhs_hash = full_domain_hash(int_to_bytes(rhs_one) + int_to_bytes(rhs_two) +
int_to_bytes(z) + msg, parameters.N)
rhs = int_to_bytes(int.from_bytes(rhs_hash, 'little') % parameters.q)
return rhs == lhs
if __name__ == '__main__':
L, N = 1024, 160
info = b'info'
msg = b'my msg'
params = choose_parameters(L, N)
signer = Signer(params)
signer.start(info)
user = User(params, signer.keypair.y)
user.start(info, msg)
a, b = signer.one()
e = user.two(a, b)
r, c, s, d = signer.three(e)
rho, omega, delta, sigma = user.four(r, c, s, d)
# p has proper form
assert (params.p - 1) % params.q == 0
# requirement to use this F
assert ((params.p - 1) % params.q**2) != 0
# test params are prime
assert prime_test(params.p, params.L)
assert prime_test(params.q, params.N)
# g has proper form
assert pow(params.g, params.q, params.p) == 1
# z is in g
assert pow(user.z, params.q, params.p) == 1
# signature works
assert check(rho, omega, delta, sigma, user.z, msg, user.y, params)