def factorize(x):
r"""
>>> factorize(a)
[3, 41]
>>> factorize(b)
[2, 2, 2, 3, 19]
>>>
"""
savex = x
prime = 2
x = _g.mpz(x)
factors = []
while x >= prime:
newx, mult = _g.remove(x, prime)
if mult:
factors.extend([int(prime)] * mult)
x = newx
prime = _g.next_prime(prime)
for factor in factors:
assert _g.is_prime(factor)
from operator import mul
from functools import reduce
assert reduce(mul, factors) == savex
return factors
def square_modulo_root(a, p):
if (p - 3) % 4 == 0:
x = a**((p - 3) // 4 + 1) % p
return x, p - x
elif (p - 5) % 8 == 0:
k = (p - 5) // 8
if a**(2 * k + 1) % p == 1:
x = a**(k + 1) % p
return x, p - x
else:
x = a**(k + 1) * 2**(2 * k + 1) % p
return x, p - x
else:
k = (p - 1) // 8
d, s = gmpy2.remove(p - 1, 2)
for num in PRIMES:
if gmpy2.legendre(num, p) == -1:
b = num
break
ta, tb = (p - 1) // 2, 0
for r in range(s - 1):
ta //= 2
tb //= 2
if a**ta * b**tb % p == p - 1:
tb += (p - 1) // 2
x = a**((d + 1) // 2) * b**(tb // 2) % p
return x, p - x
def get_power(x):
"""Returns p | i ** p == x[i], or None"""
x = tuple(x)
if len(x) > 2 and x[0] == 0 and x[1] == 1:
f, power = gmpy2.remove(x[2], 2)
if f != 1:
return
for i, xi in enumerate(x):
if i**power != xi:
return
return power
def isqrt_modp(n, p):
"""
Compute an integer s such that s*s = n in Z/pZ
or return None if there is no such integer
Implements Tonelli-Shanks algorithm
```
assert isqrt_modp(5, 41) == 28
p = None
while p is None:
p = rand_prime(3, 1000000)
for x in range(p):
s = isqrt_modp(x, p)
if s is not None:
assert gmpy2.powmod(s, 2, p) == x
```
"""
if not is_quadratic_residue(n, p):
return None
Q, S = gmpy2.remove(p - 1, 2)
for z in range(3, p):
if not is_quadratic_residue(z, p):
break
M = S
c = gmpy2.powmod(z, Q, p)
t = gmpy2.powmod(n, Q, p)
R = gmpy2.powmod(n, (Q + 1) // 2, p)
for its in range(10000):
if t == 0:
return 0
if t == 1:
return R
for i in range(1, M):
if t == 1:
break
t = gmpy2.powmod(t, 2, p)
b = gmpy2.powmod(c, gmpy2.powmod(2, M - i - 1, p), p)
M = i
b2 = gmpy2.powmod(b, 2, p)
c = b2
t = gmpy2.f_mod(t * b2, p)
R = gmpy2.f_mod(R * b, p)
def factorize(x):
r'''
>>> factorize(a)
[3, 41]
>>> factorize(b)
[2, 2, 2, 3, 19]
>>>
'''
import gmpy2 as _g
savex=x
prime=2
x=_g.mpz(x)
factors=[]
while x>=prime:
newx,mult=_g.remove(x,prime)
if mult:
factors.extend([int(prime)]*mult)
x=newx
prime=_g.next_prime(prime)
for factor in factors: assert _g.is_prime(factor)
from operator import mul
assert reduce(mul, factors)==savex
return factors