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#!/usr/bin/env python3
import logging
from xonly import xPoint, isWeierstrass, translate_by_T
from basis_sampling import find_Ts
from elkies import Elkies
from sage.arith.misc import factor
from sage.rings.integer import Integer
from sage.arith.misc import kronecker_symbol
from sage.rings.finite_rings.finite_field_constructor import GF
logger = logging.getLogger(__name__)
logger.setLevel(logging.WARNING)
logger_sh = logging.StreamHandler()
logger_sh.setLevel(logging.WARNING)
formatter = logging.Formatter("%(name)s [%(levelname)s] %(message)s")
logger_sh.setFormatter(formatter)
logger.addHandler(logger_sh)
def ideal_above_2_action(E, P, Q, basis_order, ideal, w):
r"""Computes the action of a power of an ideal above 2 using Velu
Input:
- E: Elliptic curve over Fp
- P, Q: Basis of E[2^basis_order],
Here, P is in E(Fp) and Q is in Et(Fp) for a quadratic twist Et
- basis_order: Such that 2^basis_order is the order of P, Q
- ideal: Power of an ideal above 2
Output:
Tuple (phi, E_ideal)
- phi: The isogeny corresponding to ideal
- E_ideal: The curve obtained by the action of the ideal on E
"""
_, v2 = factor(ideal.norm())[0]
left = 0
right = 0
if (w - 1) / 2 in ideal:
K = P
left = v2
else:
assert (w + 1) / 2 in ideal
K = Q
right = v2
K = K.xMUL(2 ** (basis_order - v2))
assert K.xMUL(2 ** (v2 - 1))
assert not K.xMUL(2**v2)
phi_2 = K.xMUL(2 ** (v2 - 1)).xISOG(E, 2)
phi = phi_2
for step in range(1, v2):
E = phi.codomain()
K = K.push(phi_2)
assert K.xMUL(2 ** (v2 - step - 1))
assert not K.xMUL(2 ** (v2 - step))
phi_2 = K.xMUL(2 ** (v2 - step - 1)).xISOG(E, 2)
phi = phi_2 * phi
E_out = phi.codomain().montgomery_model()
return phi.codomain().isomorphism_to(E_out) * phi, E_out, left, right
def small_prime_ideal_action(E, ell, lam=None, prev_j=None):
r"""Computes action of a ideal with prime norm using Elkies
Input:
- E: Elliptic curve over Fp IN WEIERSTRASS FORM
- ell: A small prime, split in the quadratic order
- lam: (optional) Eigenvalue determining the ideal
- prev_j: (optional) The j-invariant of the WRONG ell-isogenous curve
Both lam and prev_j are passed to Elkies
Output:
- phi: Isogeny corresponding to an ideal above ell
"""
# Fix later, but Elkies uses short Weierstrass
# Change ring stuff is messy, but needed to actually compute the rational model
assert isWeierstrass(E)
h = Elkies(E, ell, E.base_field(), lam=lam, prev_j=prev_j)
phi = E.isogeny(h)
return phi
def smooth_ideal_action(E, norm, ideal, w, max_order):
r"""Computes action of a an ideal with odd smooth norm using successive Elkies
Input:
- E: Elliptic curve over Fp
- norm: Norm of ideal
- ideal: An ideal
Output: Tuple (isogenies, E_ideal)
- isogenies: A list of isogenies, forming a chain from E to E_out
- E_ideal: The curve obtained by the action of the ideal on E
"""
p = E.base_field().characteristic()
isogenies = []
E1 = E.short_weierstrass_model()
isogenies += [E.isomorphism_to(E1)]
if norm > 1:
for ell, e in factor(norm):
lam = Integer(GF(ell)(-p).sqrt())
if not (w - lam) in (ideal + ell * max_order):
# In this case, we want the other one
lam = ell - lam
assert (w - lam) in (ideal + ell * max_order)
prev_j = None
for _ in range(e):
logger.debug(f"Starting Elkies step of degree {ell}")
phi = small_prime_ideal_action(E1, ell, lam, prev_j)
prev_j = E1.j_invariant()
E1 = phi.codomain()
isogenies.append(phi)
E_ideal = isogenies[-1].codomain().montgomery_model()
isogenies += [isogenies[-1].codomain().isomorphism_to(E_ideal)]
return isogenies, E_ideal
def random_smooth_isogeny(E, g):
r"""
Returns a random isogeny of smooth degree g from E
Input:
- E: Elliptic curve over Fp
- g: Smooth number
Output:
- phi : An isogeny from E of degree g
- E_out : phi(E)
"""
p = E.base_field().characteristic()
# Random smooth odd norm isogeny
isogs = []
if g > 1:
E1 = E.short_weierstrass_model()
pre_isom = E.isomorphism_to(E1)
isogs.append(pre_isom)
logger.debug(f"Extra isogeny for g = {factor(g)}")
# Do elkies first, to stay over Fp
g_elkies = []
for ell, e in factor(g):
if kronecker_symbol(-p, ell) == 1:
g_elkies.append((ell, e))
for ell, e in g_elkies:
prev_j = None
for _ in range(e):
phi_ell = small_prime_ideal_action(E1, ell, prev_j=prev_j)
prev_j = E1.j_invariant()
E1 = phi_ell.codomain()
isogs.append(phi_ell)
E_out = isogs[-1].codomain().montgomery_model()
post_isom = isogs[-1].codomain().isomorphism_to(E_out)
isogs.append(post_isom)
return isogs, E_out
else:
return [], E
def eval_endomorphism(rho, P, twist, max):
r"""
Evaluates an element of End(E) on a point P.
P is assumed to be Fp rational on either E or on a twist
"""
a, b = list(rho) # write as a + b*pi
n = rho.denominator()
if twist:
# a + b*pi = a - b, since pi(P) = -P
m = Integer(a - b)
else:
# a + b*pi = a + b, since pi(P) = P
m = Integer(a + b)
if max and (n == 2): # See Appendix D.2
mP = P.xMUL(m)
E = P.curve
T0 = find_Ts(E, only_T0=True)
# Already know which point to choose, Remark D.2
assert (P.X - T0.x()).is_square() != twist
T = xPoint(T0.x(), E)
return translate_by_T(mP, T)
else:
return P.xMUL(m)
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