def discrete_log(prime, public_key, primitive_root):
m = int(math.floor((prime - 1) ** .5))
base_exponents = {}
for j in xrange(0, m):
base_exponents[fast_exponentiation(primitive_root, j, prime)] = j
inverse_b = extended_euclidean_algorithm(primitive_root, prime) % prime
c = fast_exponentiation(inverse_b, m, prime)
x = public_key
for i in xrange(0, m):
if x in base_exponents:
return (m * i) + base_exponents[x]
else:
x = (x * c) % prime
def generate_keys():
'''Alice builds a dicsrete-log public-key cipher system.
'''
cyclic_group = large_random_prime()
generator = random_near_primitive_root(cyclic_group)
private_key = random.randint(1, cyclic_group - 1)
public_key = fast_exponentiation(generator, private_key, cyclic_group)
return (generator, public_key, cyclic_group), private_key
def miller_rabin(n, b):
'''Miller-Rabin test for primality.
'''
r = 0
m = n - 1
while m % 2 == 0:
r += 1
m = m / 2
if fast_exponentiation(b, m, n) in (n - 1, 1):
return True
power = 1
while power < r - 1:
b_m_squared_mod_n = fast_exponentiation(b, m * (2 ** power), n)
if b_m_squared_mod_n == 1:
return False
elif b_m_squared_mod_n == n - 1:
return True
else:
power = power * 2
return False
def test_attack_elgamal(self):
#Alice
p = 1009
b = 101
c = 482
l = discrete_log(prime=p,
primitive_root=b,
public_key=c)
#Bob
x = 559
r = 291
c_to_the_r = fast_exponentiation(c, r, p)
encrypted_message = x * c_to_the_r % p
header = fast_exponentiation(b, r, p)
decrypted_message = decrypt(encrypted_message, header, l, p)
print attack_elgamal(_primitive_root=b,
_public_key=c,
_prime=p,
message=encrypted_message,
header=header)
def primitive_root(group, divisors, prime):
for q in divisors:
if fast_exponentiation(group, (prime - 1) / q, prime) == 1:
return False
return True
def decrypt(message, header, l, p):
c_r = fast_exponentiation(header, l, p)
inverse = extended_euclidean_algorithm(c_r, p)
plaintext = (inverse * message) % p
return plaintext
def encrypt(message, b, c, p):
r = large_random_prime()
c_r = fast_exponentiation(c, r, p)
encrypted_message = message * c_r % p
header = fast_exponentiation(b, r, p)
return encrypted_message, header
def encrypt(n, encrypt_key, message):
return fast_exponentiation(message, encrypt_key, n)
def decrypt(encrypted_message, decrypt_key, n):
return fast_exponentiation(encrypted_message, decrypt_key, n)