def velu_walk(E, l, lam, k):
q = E.base_field().order()
lam = GF(l)(lam)
r = lam.multiplicative_order()
mu = GF(l)((lam ** (-1)) * q)
r_mu = mu.multiplicative_order()
if k < 0:
return velu_walk(E.quadratic_twist(), l, -mu, -k)
if r_mu == r:
print("Invalid parameters")
return
if r_mu < r:
return velu_walk(E.quadratic_twist(), l, -lam, k)
t = E.trace_of_frobenius()
order = rec_order(q, t, r)
cur = extend_field(E, r)
for i in range(k):
cur = velu_step(cur, l, lam, order, q)
cur = EllipticCurve_from_j(GF(q)(cur.j_invariant()))
return cur
def walk_the_crater(E,l,lam):
k = E.base_field()
q = k.order()
lam = GF(l)(lam)
r = lam.multiplicative_order()
t = E.trace_of_frobenius()
order = rec_order(q, t, r)
cur = extend_field(E, r)
crater = [E.j_invariant()]
cur = velu_step(cur, l, lam, order, q)
while k(cur.j_invariant())!=E.j_invariant():
crater.append(k(cur.j_invariant()))
cur = velu_step(cur, l, lam, order, q)
return crater
def elkies_first_step(E, l, lam):
q = E.base_field().order()
lam = GF(l)(lam)
Phi = ClassicalModularPolynomialDatabase()[l]
x = PolynomialRing(E.base_field(), 'x').gen()
f = Phi(x, E.j_invariant())
j_1, j_2 = f.roots()[0][0], f.roots()[1][0]
E1 = elkies_mod_poly(E, j_1, l)
try:
I = EllipticCurveIsogeny(E, None, E1, l)
except:
I = l_isogeny(E, E1, l)
r = lam.multiplicative_order()
k = GF(q ** r)
ext = extend_field(E, r)
try:
P = ext.lift_x(I.kernel_polynomial().any_root(k))
except:
return j_2
if ext(P[0] ** q, P[1] ** q) == Integer(lam) * P:
return j_1
else:
return j_2