def pollard_brent_search(f, n: int, x0=2, size=100):
"""Pollard's-rho algorithm to find a non-trivial
factor of `n` using Brent's cycle detection algorithm,
and including the product speed improvment."""
power = 1
lam = 0
tortoise = x0
hare = x0
d = 1
while d == 1:
prod = 1
last_pos = hare
for c in range(size):
if power == lam:
tortoise = hare
power *= 2
lam = 0
hare = f(hare)
lam += 1
prod = (prod * abs(tortoise - hare)) % n
d = gcd(prod, n)
if not (is_prime(d)):
return pollard_rho_search_brent(f, n, last_pos)
else:
return d
def pollard_rho_search_floyd(f, n: int, x0=2):
"""Pollard's-rho algorithm to find a non-trivial
factor of `n` using Floyd's cycle detection algorithm."""
d = 1
tortoise = x0
hare = x0
while d == 1:
tortoise = f(tortoise)
hare = f(f(hare))
d = gcd(abs(tortoise - hare), n)
return d
def frobenius_trace_mod_2(curve):
"""Computes the trace of the Frobenius endomorphism mod 2"""
R = Polynomials(curve.field)
x = R.x()
A, B = curve.parameters()
curve_polynomial = x ** 3 + A * x + B
field_polynomial = R([0] * curve.field.size() + [1]) - x
if gcd(curve_polynomial, field_polynomial).degree() == 0:
return FiniteField(2)(1)
else:
return FiniteField(2)(0)
def ECMFactor(n, sieve, limit, times):
"""Finds a non-trivial factor of `n` using
the elliptic-curve factorization method"""
for _ in range(times):
g = n
while g == n:
curve, point = randomCurve(n)
g = gcd(n, curve.discriminant().remainder())
if g > 1:
return g
for prime in sieve:
prod = prime
i = 0
while prod < limit and i < 10:
i += 1
try:
point = prime * point
except ZeroDivisionError as e:
return gcd(e.args[1], n)
prod *= prime
return n
def pollard_rho_search_brent(f, n: int, x0=2):
"""Pollard's-rho algorithm to find a non-trivial
factor of `n` using Brent's cycle detection algorithm."""
power = 1
lam = 0
tortoise = x0
hare = x0
d = 1
while d == 1:
if power == lam:
tortoise = hare
power *= 2
lam = 0
hare = f(hare)
d = gcd(abs(tortoise - hare), n)
lam += 1
return d