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from sage.all import *
import random
from xonly import xPoint
def Elkies(E, ell, Fp, lam = None, prev_j = None):
"""
Elkies algorithm for computing isogenies as detailed in [BSS - Elliptic
Curves in Cryptography, Chapter VII].
Input:
- E: starting curve
- ell: a split prime
- Fp: base field
- lam: if looking for a specific Elkies isogeny, optionally provide the
corresponding eigenvalue
- prev_j: if the curve is the codomain of another ell-isogeny,
optionally provide the j invariant of the previous curve to avoid
backtracking
Output:
- the kernel polynomial hc
"""
#TODO maybe we want montomery curves later
p = Fp.characteristic()
assert ell > 2
d = (ell-1)/2
j = Fp(E.j_invariant())
_, _, _, a, b = [Fp(c) for c in E.a_invariants()]
R = Fp['X,Y']
X, Y = R._first_ngens(2)
E4 = -48*a
E6 = 864*b
jp = -E6*j/E4
phi_ell = R(classical_modular_polynomial(ell))
X, Y = phi_ell.variables()
phi_ellx = derivative(phi_ell, X)
phi_ellxx = derivative(phi_ellx, X)
phi_ellxy = derivative(phi_ellx, Y)
phi_elly = derivative(phi_ell, Y)
phi_ellyy = derivative(phi_elly, Y)
phi_ell_eval = phi_ell(X, j).univariate_polynomial()
if prev_j:
x = phi_ell_eval.variables()[0]
phi_ell_eval = phi_ell_eval // (x - prev_j)
# print(f"Finding j: {time()-tstart}")
if prev_j:
j_ells = [phi_ell_eval.any_root()] #There is only one root in this case!
#assert len(j_ells) == 1, f"j_ells was {j_ells}, and prev_j was {prev_j}"
else:
j_ells = [r for r,e in phi_ell_eval.roots()]
assert len(j_ells) == 2, f"j_ells was {j_ells}"
#print(j_ells)
jt = j_ells[0]
assert jt != 0 and jt != 1728, "encountered E0"
hc = _derive_hc(Fp, a, b, ell, d, j, jt, jp, E4, E6, phi_ellx, phi_ellxx,
phi_ellxy, phi_elly, phi_ellyy)
if lam and not prev_j:
if not CheckElkies(E, ell, hc, lam):
#Do the other one
assert j_ells[1] != 0 and j_ells[1] != 1728, "encountered E0"
hc = _derive_hc(Fp, a, b, ell, d, j, j_ells[1], jp, E4, E6, phi_ellx, phi_ellxx, phi_ellxy, phi_elly, phi_ellyy)
return hc
def WeierstrassP(a,b,d):
# Weierstrass p up to degree d in w=z^2
coefs = [-a/5, -b/7]
for k in range(2, d):
ck = 0
for j in range (k-1):
ck = ck + coefs[j]*coefs[k-2-j]
ck = ck*3/((k-1)*(2*(k+1)+3))
coefs.append(ck)
return coefs
def _derive_hc(Fp, a, b, ell, d, j, jt, jp, E4, E6, phi_ellx, phi_ellxx, phi_ellxy, phi_elly, phi_ellyy):
jtp = -jp*phi_ellx(j, jt)/(ell*phi_elly(j, jt))
at = -jtp**2/(48*jt*(jt-1728))
bt = -jtp**3/(864*jt**2*(jt-1728))
E4t = -48*at
E6t = 864*bt
fracjs = -(jp**2*phi_ellxx(j, jt) + 2*ell*jp*jtp*phi_ellxy(j, jt) + (ell*jtp)**2*phi_ellyy(j, jt))/ (jp*phi_ellx(j, jt))
p1 = ell*fracjs/2 + (Fp(ell)/4)*(E4**2/E6 - ell*E4t**2/E6t) + (Fp(ell)/3)*(E6/E4 - ell*E6t/E4t)
c = WeierstrassP(a,b,d)
ckst = WeierstrassP(ell**4*at, ell**6*bt, d)
ct = [ckst[i] - ell*c[i] for i in range(len(c))] # difference in formula VII.23
# Computing the coefficients of VII.23 and store as A[i]
Fpw, w = PowerSeriesRing(Fp, 'w', default_prec=d+1).objgen()
evp = -(p1/2)*w - sum(w**(i+1)*ct[i-1] / ((2*i+1)*(2*i+2)) for i in range(1,d))
exp_evp = evp.exp(d+1)
A = exp_evp.coefficients()
C = sum(c[i-1]*w**i for i in range(1, d+1))
# Computing all powers of C starting with zeroth power
Cpow = [Fpw(1), C];
for i in range(2, d+1):
Cpow.append(C*Cpow[-1])
# Now doing recurrence relation VII.24
hc = [1, -p1/2]
for i in range(2, d+1):
newcoeff = A[i]
for k in range(1, i+1):
insum = 0
for j in range(k+1):
insum += Fp(binomial(d-i+k, k-j))*Cpow[k-j][j]
newcoeff -= insum*hc[i-k]
hc.append(newcoeff)
Rx, x = PolynomialRing(Fp, 'x').objgen()
hc = Rx(hc[::-1])
return hc
def CheckElkies(E, ell, h, lam):
p = E.base_field().characteristic()
if kronecker_symbol(-p, ell) != 1:
assert False, "not Elkies"
_, _, _, a, b = E.a_invariants()
f = h.parent()([b, a, 0, 1])
B = pow(f, (p-1)/2, h)
check_wrong = False
if lam > ell//2:
lam = ell-lam
check_wrong = True
if lam == 1:
if check_wrong:
return B != 1
else:
return B == 1
# Stupid way for now, no sage function for directly computing mod h
RR = h.parent().quotient_ring(h)
y_coord = E.multiplication_by_m(lam)[1]
y_coord_num = y_coord.numerator()
x, y = y_coord_num.variables()
y_coord_den = y_coord.denominator()
if check_wrong:
return RR(B)*RR(y_coord_den) != RR(y_coord_num(x, 1).univariate_polynomial())
else:
return RR(B)*RR(y_coord_den) == RR(y_coord_num(x, 1).univariate_polynomial())
def NonElkiesIsogeny(E, ell, Fp2):
# For u, v, when not elkies prime.
# Make that either ell divides p+1, the other way works...
E = E.change_ring(Fp2)
#print(factor(E.division_polynomial(ell)))
# Make sure these are the only cases...
h_ell = factor(E.division_polynomial(ell))[0][0]
m = (ell-1)//(2*h_ell.degree())
if m == 1:
return h_ell
#Deuring for the people <3
from sage.schemes.elliptic_curves.isogeny_small_degree import _least_semi_primitive
a = _least_semi_primitive(ell)
fs = [h_ell]
Fbig = Fp2.extension(h_ell)
x = Fbig.gens()[0]
xi = xPoint(x, E.change_ring(Fbig))
for _ in range(1, m):
xi = xi.mul(a)
fs.append(xi.X.minpoly())
h_ell = prod(fs)
assert h_ell.degree() == (ell-1)/2, f"Degree : {h_ell.degree()}, shouldve been {(ell-1)/2}"
return h_ell
### This is way slower for now :((
def ElkiesDirect(E, ell, lam = None):
psi_x = E.division_polynomial(ell)
Fp = E.base_field()
p = Fp.characteristic()
if not lam:
F_ell = GF(ell)
lam = F_ell(-p).sqrt()
R = psi_x.parent()
x = psi_x.variables()[0]
RR = R.quotient_ring(psi_x)
X = RR._first_ngens(1)[0]
check_wrong = False
if lam > ell//2:
lam = ell-lam
check_wrong = True
if lam == 1:
h_sq_bar = X**p - X
else:
mult_by_lam = E.multiplication_by_m(lam)[0]
mult_by_lam_num = mult_by_lam.numerator().univariate_polynomial()
mult_by_lam_denom = mult_by_lam.denominator().univariate_polynomial()
h_sq_bar = X**p*mult_by_lam_denom(X) - mult_by_lam_num(X)
h_sq = 0
x_pow = x**0
for ai in list(h_sq_bar):
h_sq += ai*x_pow
x_pow *= x
h_sq = gcd(psi_x, h_sq)
#print(h_sq)
#print(factor(h_sq))
#print(Elkies(E, ell, Fp))
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