def galois_action(self, t, N):
from sage.modular.modsym.p1list import lift_to_sl2z
if self.is_infinity():
return self
if not isinstance(t, Integer):
t = Integer(t)
a = self._Cusp__a
b = self._Cusp__b * t.inverse_mod(N)
if b.gcd(a) != ZZ(1):
_,_,a,b = lift_to_sl2z(a,b,N)
a = Integer(a); b = Integer(b)
# Now that we've computed the Galois action, we efficiently
# construct the corresponding cusp as a Cusp object.
return Cusp(a,b,check=False)
def galois_action(self, t, N):
from sage.modular.modsym.p1list import lift_to_sl2z
if self.is_infinity():
return self
if not isinstance(t, Integer):
t = Integer(t)
a = self._Cusp__a
b = self._Cusp__b * t.inverse_mod(N)
if b.gcd(a) != ZZ(1):
_,_,a,b = lift_to_sl2z(a,b,N)
a = Integer(a); b = Integer(b)
# Now that we've computed the Galois action, we efficiently
# construct the corresponding cusp as a Cusp object.
return Cusp(a,b,check=False)
def method(r,k,D,max_trials=10000, g=0):
"""
Description:
Run the Cocks-Pinch method to find an elliptic curve
Input:
r - prime
k - embedding degree, r % k == 1
D - (negative) fundamental discriminant where D is a square mod r
max_trials - the number of integers q to test for primality in the CP method
g - an element of order k in Z_r^*
Output:
(q,t) - tuple where q is a prime and t is chosen such that there exists
an elliptic curve E over F_q with trace t, and r | q+1-t;
if the algorithm fails to find (q,t), it will return (0,0)
"""
assert test_promise(r,k,D), 'Invalid inputs'
if g != 0:
assert power_mod(g,k,r) == 1, 'Invalid inputs'
else:
g = find_element_of_order(k,r)
D = Integer(D)
t = Integer(g) + 1
root_d = Integer(Mod(D, r).sqrt())
u = Integer(Mod((t-2)*root_d.inverse_mod(r) ,r))
q = 1
j = Integer(0)
i = Integer(0)
count = 0
while (count < max_trials):
q = Integer( (t+i*r)**2 - D*(u + j*r)**2)
if q % 4 ==0:
q = q//4
if utils.is_suitable_q(q):
return (q, t+i*r)
q = 1
if random() < 0.5:
j+=1
else:
i+=1
count+=1
return (0, 0) # no prime found, so end
def method(r, k, D, max_trials=10000, g=0):
"""
Description:
Run the Cocks-Pinch method to find an elliptic curve
Input:
r - prime
k - embedding degree, r % k == 1
D - (negative) fundamental discriminant where D is a square mod r
max_trials - the number of integers q to test for primality in the CP method
g - an element of order k in Z_r^*
Output:
(q,t) - tuple where q is a prime and t is chosen such that there exists
an elliptic curve E over F_q with trace t, and r | q+1-t;
if the algorithm fails to find (q,t), it will return (0,0)
"""
assert test_promise(r, k, D), 'Invalid inputs'
if g != 0:
assert power_mod(g, k, r) == 1, 'Invalid inputs'
else:
g = find_element_of_order(k, r)
D = Integer(D)
t = Integer(g) + 1
root_d = Integer(Mod(D, r).sqrt())
u = Integer(Mod((t - 2) * root_d.inverse_mod(r), r))
q = 1
j = Integer(0)
i = Integer(0)
count = 0
while (count < max_trials):
q = Integer((t + i * r)**2 - D * (u + j * r)**2)
if q % 4 == 0:
q = q // 4
if utils.is_suitable_q(q):
return (q, t + i * r)
q = 1
if random() < 0.5:
j += 1
else:
i += 1
count += 1
return (0, 0) # no prime found, so end
