def ecpp(n, era=None):
"""
Algorithm 27 (Atkin-Morain ECPP)
Input : a natural number n
Output : If n is prime, retrun True. Else, return False
"""
_log.info('n = %d' % n)
disc_and_order_info = _ecpp_find_disc(n, era)
if not disc_and_order_info:
return False
# disc, m = k*q, era
disc, m, k, q, era = disc_and_order_info
# non(quadratic|cubic) residue
g = quasi_primitive(n, disc == -3)
try:
# find param corresponding to order m
for param in param_gen(disc, n, g):
E = Elliptic(param, n)
P = E.choose_point()
if E.mul(m, P) == [0]:
break
# find a point [k]P is not the unit.
U = E.mul(k, P)
while U == [0]:
P = E.choose_point()
U = E.mul(k, P)
# compute [q]([k]P)
V = E.mul(q, U)
except ZeroDivisionError:
return False
if V == [0]:
if q > 1000000000:
return ecpp(q, era)
elif prime.trialDivision(q):
return True
else:
return False
else:
return False
def ecpp(n, era=None):
"""
Algorithm 27 (Atkin-Morain ECPP)
Input : a natural number n
Output : If n is prime, retrun True. Else, return False
"""
_log.info("n = %d" % n)
disc_and_order_info = _ecpp_find_disc(n, era)
if not disc_and_order_info:
return False
# disc, m = k*q, era
disc, m, k, q, era = disc_and_order_info
# non(quadratic|cubic) residue
g = quasi_primitive(n, disc == -3)
try:
# find param corresponding to order m
for param in param_gen(disc, n, g):
E = Elliptic(param, n)
P = E.choose_point()
if E.mul(m, P) == [0]:
break
# find a point [k]P is not the unit.
U = E.mul(k, P)
while U == [0]:
P = E.choose_point()
U = E.mul(k, P)
# compute [q]([k]P)
V = E.mul(q, U)
except ZeroDivisionError:
return False
if V == [0]:
if q > 1000000000:
return ecpp(q, era)
elif prime.trialDivision(q):
return True
else:
return False
else:
return False
def testTrialDivision(self):
self.assertTrue(prime.trialDivision(3))
self.assertTrue(prime.trialDivision(23))
self.assertFalse(prime.trialDivision(7 * 13))
self.assertTrue(prime.trialDivision(97))
self.assertFalse(prime.trialDivision(11 * 13))
def atkin_morain(n):
"""
Atkin-Morain ECPP Algorithm.
Args:
n: Probable Prime
Returns:
Certificate of primality, or False.
"""
if n < arbitrary_bound:
if prime.trialDivision(n):
return [n]
else:
return False
d = 0
m_found = False
while m_found is False:
try:
d, ms = choose_discriminant(n, d)
except ValueError:
return False
for m in ms:
factors = factor_orders(m, n)
if factors is not None:
k, q = factors
params = curve_parameters(d, n)
try:
# Test to see if the order of the curve is really m
a, b = params.pop()
ec = EllipticCurve(a, b, n)
while not test_order(ec, m):
a, b = params.pop()
ec = EllipticCurve(a, b, n)
#print n, a, b
m_found = True
break
except IndexError:
pass
# if no proper m can be found. Go back to choose_discriminant()
'''
If this step fails need to return false.
'''
try:
# operate on point
while True:
P = choose_point(ec)
U = ec.mul(k, P) # U = [m/q]P
if U != 0:
break
V = ec.mul(q, U)
except (ZeroDivisionError, ValueError):
return False
if V != 0:
return False
else:
if q > arbitrary_bound:
cert = atkin_morain(q)
if cert:
cert.insert(0, (q, m, a, b))
return cert
else:
if prime.trialDivision(q):
return [q]
else:
return False
