def blum_blum_shub(x,p,q):
"""
Blum Blum Shub
Args:
x -- seed value, coprime to pq, and greater than 1
p -- prime congruent to 3 mod 4
q -- prime congruent to 3 mod 4
"""
m = p*q
require_integers(["x","p","q"], [x,p,q])
require_prime( ["p","q"], [p,q])
require_true(["p","q"], [p,q], lambda x: x % 4 == 3, "must be congruent to 3 mod 4")
require_geq(["x"], [x], 2)
K = legendre_symbol(2,(p-1)//2) + legendre_symbol(2,(q-1)//2)
if K == 2:
raise Exception("both (p-1)/2 and (q-1)/2 have 2 as a quadratic residue")
if gcd(m,x) != 1:
raise Exception("x must be coprime to pq")
while True:
yield x
x = (x*x)%m
def kronecker_symbol(a,n):
"""Extend the Legendre Symbol to all naturals"""
fac = prime_factorization(n)
out = 1
for f in fac:
out *= legendre_symbol(a,f)
return out
def MsgToPoint(m: int):
for i in range(0, N):
x = 100 * m + i
z = x**3
z %= MODULO_P
z += A * x + B
if sympy.legendre_symbol(z, MODULO_P) == 1:
y = sympy.sqrt_mod(z, MODULO_P)
return Point(x, y)
def jacobi_symbols(flatten=False):
"""
Irreguar Array of Jacobi Symbols: 1 at quadratic residues, -1 at nonresideus, 0 at zero
One row for each odd natural, n, of length n\n
OEIS
"""
for p in odds():
pfacs = canonical_factorization(p)
T = []
for a in range(p):
out = 1
for f, e in pfacs.items():
out *= legendre_symbol(a, f)**e
T.append(out)
if flatten == True:
yield from T
else:
yield tuple(T)
def find_C(p, q):
c = 2
while (legendre_symbol(c, p)) % 4 != (-p) % 4 or (legendre_symbol(
c, q)) % 4 != (-q) % 4:
c += 1
return c
'''def find_d(e, w):
tmp = (w+1)//2
x = exgcd(e, w)
nod = gcd(e,w)
if tmp % nod == 0:
d = (tmp*x//nod) % w
return d'''
C = find_C(p, q)
print('C: ', C)
A = find_A(C, R)
print('A: ', A)
w = (p - legendre_symbol(C, p)) * (q - legendre_symbol(C, q)) * exgcd(4, R) % R
print('w: ', w)
e = find_e(w)
print('e: ', e)
d = ((w + 1) * exgcd(2, w) * exgcd(e, w)) % w
#d = (pow(e,-1,w)*(w+1)//2) % w
print('Секретный ключ d: ', d)
print('p, q, w, d держатся в секрете.')
M = int(input('Введите передаваемое сообщение: '))
M = M % R
print('Сообщение для шифрования: ', M)
def find_XY(a, b, R, e):
X = []