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prng.py
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prng.py
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#!/usr/bin/env python
"""
Pseudorandom Number Generators and Probabilistic Encryption in Python (originally 2.4)
Graham Enos, Spring 2010
"""
def midsqr(x_0):
"""
Middle Square Generator; see Mills 2003
Input:
x_0 integer in Z_1e5
Example:
>>> my_midsqr = midsqr(1010)
>>> for i in range(3):
... print(my_mid_sqr.next(), end=" ")
2010 4010 8010
"""
x = x_0 # Initialize x as x_0
while True:
x = (x**2 / 10) % (10**4) # (x^2/10) (mod 10^4)
yield x
def fib(m, x_0, x_1):
"""
Fibonacci Generator; see Mills 2003
Input:
m integer >= 2
x_0, x_1 seeds, integers in Z_m
Example:
>>> my_fib = fib(10,1,1)
>>> for i in range(10):
... print(my_fib.next(), end=" ")
...
2 3 5 8 3 1 4 5 9 4
"""
y_0 = x_0
y_1 = x_1 # Initialize y_0 and y_1 as x_0 and x_1
while True:
temp_y = y_1
y_1 = (y_0 + y_1) % m
y_0 = temp_y
yield y_1
def lcg(m, a, b, x_0):
"""
Linear Congruential Generator, Algorithm 8.1 in Stinson 2006
Input:
m integer >= 2
a,b integers in Z_m
x_0 seed, integer in Z_m
Output:
ones and zeros
Example:
>>> my_lcg = lcg(31, 3, 5, 17)
>>> for i in range(10):
... print my_lcg.next(),
...
1 0 0 1 0 1 1 0 1 0
"""
x = x_0 # Initialize x as x_0
while True:
x = (a * x + b) % m # ax + b (mod m)
yield x % 2
def rsa(n, b, x_0):
"""
RSA Generator, Algorithm 8.2 in Stinson 2006
Input:
n composite n = p * q; p & q prime
b integer such that gcd(b, phi(n)) = 1
x_0 seed, any element of Z_n*
Output:
ones and zeros
Example:
>>> my_rsa = rsa(91261, 1547, 75634)
>>> for i in range(20):
... print my_rsa.next(),
...
1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 0 0 0
"""
x = x_0 # Intialize x as x_0
while True:
x = pow(x, b, n) # x^b (mod n)
yield x % 2
def dlg(p, a, x_0):
"""
Discrete Logarithm Generator, Algorithm 8.8 in Stinson 2006
Input:
p a prime
a primitive element mod p (has multiplicative order p-1)
x_0 seed, any nonzero element of Z_p
Output:
ones and zeros
Example:
>>> my_dlg = dlg(21383, 5, 15886)
>>> for i in range(50):
... print my_dlg.next(),
...
0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 1 1 1 0 0 1 0 0 1 1
1 0 0 0 1 0 1 0 0 0 0 1
"""
x = x_0 # Initialize x as x_0
half_p = p / 2.0
while True:
x = pow(a, x, p) # a^x (mod p)
if x > half_p:
yield 1
else:
yield 0
def bbs(n, x_0):
"""
Blum-Blum-Shub Generator, Algorithm 8.5 in Stinson 2006
Input:
n composite n = p*q; p & q are primes = 3 mod 4
x_0 seed, quadratic residue mod n
Output:
ones and zeros
Example:
>>> my_bbs = bbs(209, 3)
>>> for i in range(50):
... print my_bbs.next(),
...
1 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1
0 0 0 0 0 1 0 1 1 0 1 1
"""
x = x_0 # Initialize x as x_0
while True:
x = pow(x, 2, n) # x^2 (mod n)
yield x % 2
def bg_enc(n, r, x):
"""
Blum-Goldwasser Encryption
Requires:
bbs() BBS PRNG
Input:
n composite n = p*q; p & q are primes = 3 mod 4
r seed, quadratic residue mod n
x tuple of plaintext bits
Output:
ciphertext, a tuple of ones and zeros
Example
>>> bg_enc(272953, 159201, (1, 1, 0, 1, 0))
((0, 1, 1, 1, 0), 139680)
"""
my_bbs = bbs(n, r)
k = len(x)
z = ()
for i in range(k):
z += (my_bbs.next(),) # Grow keystream
s = pow(r, 2 ** (k + 1), n) # r^(2^(k+1)) mod n
y = tuple(a ^ b for (a, b) in zip(x, z)) # x XOR z
return (y, s)
def bg_dec(p, q, y, s):
"""
Blum-Goldwasser Decryption
Requires:
bbs() BBS PRNG
euclid Extended Euclidean Algorithm
Input:
p, q primes = 3 mod 4
y tuple of ciphertext bits
s hidden seed, quadratic residue mod p*q
Output:
plaintext, a tuple of ones and zeros
Example:
>>> bg_dec(499, 547, (0, 1, 1, 1, 0), 139680)
(0, 0, 1, 1, 0)
"""
n = p * q
k = len(y)
(a, b) = euclid(p, q)[:2] # Integers such that ap + bq = 1
d1 = pow((p + 1) / 4, k + 1, p - 1) # ((p+1)/4)^(k+1) mod (p-1)
d2 = pow((q + 1) / 4, k + 1, q - 1) # ((q+1)/4)^(k+1) mod (q-1)
u = pow(s, d1, p) # s^d1 mod p
v = pow(s, d2, q) # s^d2 mod q
r = (v * a * p + u * b * q) % n # vap + ubq mod n
my_bbs = bbs(n, r)
z = ()
for i in range(k):
z += (my_bbs.next(),) # Grow keystream
x = tuple(a ^ b for (a, b) in zip(y, z)) # y XOR z
return x
def euclid(a, b):
"""
Extended Euclidean Algorithm; finds (x,y,z) such that ax + by = gcd(a,b) = z
Written with help from
http://en.literateprograms.org/Extended_Euclidean_algorithm_%28Python%29
Input:
(a,b) two integers
Output:
(x, y, z) integers such that ax + by = gcd(a,b) = z
Example:
>>> euclid(12345,54321)
(3617, -822, 3)
"""
(x1, x2, x3) = (1, 0, a)
(y1, y2, y3) = (0, 1, b)
while y3 != 0:
quotient = x3 / y3
tmp1 = x1 - quotient * y1
tmp2 = x2 - quotient * y2
tmp3 = x3 - quotient * y3
(x1, x2, x3) = (y1, y2, y3)
(y1, y2, y3) = (tmp1, tmp2, tmp3)
return x1, x2, x3