def _factor_order(m, u, era=None):
"""
Return triple (k, q, primes) if m is factored as m = kq,
where k > 1 and q is a probable prime > u.
u is expected to be (n^(1/4)+1)^2 for Atkin-Morain ECPP.
If m is not successfully factored into desired form,
return (False, False, primes).
The third component of both cases is a list of primes
to do trial division.
Algorithm 27 (Atkin-morain ECPP) Step3
"""
if era is None:
v = min(500000, int(log(m)))
era = factor.mpqs.eratosthenes(v)
k = 1
q = m
for p in era:
if p > u:
break
e, q = arith1.vp(q, p)
k *= p**e
if q < u or k == 1:
return False, False, era
if not prime.millerRabin(q):
return False, False, era
return k, q, era
def factor_orders(m, n):
"""
m = kq
Args:
m: for factorizations
n: the integer for prime proving. Used to compute bound
Returns:
"""
k = 1
q = m
bound = (floorpowerroot(n, 4) + 1) ** 2
for p in small_primes:
'''
Check again.
'''
# if q becomes too small, fails anyway
if q <= bound:
break
# if p is over the bound, and p is a factor of q. A solution is found.
if p > bound and q % p == 0:
k *= q/p
q = p
break
# When q is > bound. Try to factorize through small primes.
while q % p == 0:
q = q/p
k *= p
if q <= bound or k == 1:
return None
if not prime.millerRabin(q):
return None
return k, q
def _factor_order(m, u, era=None):
"""
Return triple (k, q, primes) if m is factored as m = kq,
where k > 1 and q is a probable prime > u.
u is expected to be (n^(1/4)+1)^2 for Atkin-Morain ECPP.
If m is not successfully factored into desired form,
return (False, False, primes).
The third component of both cases is a list of primes
to do trial division.
Algorithm 27 (Atkin-morain ECPP) Step3
"""
if era is None:
v = min(500000, int(log(m)))
era = factor.mpqs.eratosthenes(v)
k = 1
q = m
for p in era:
if p > u:
break
e, q = arith1.vp(q, p)
k *= p ** e
if q < u or k == 1:
return False, False, era
if not prime.millerRabin(q):
return False, False, era
return k, q, era
