SIDH key exchange

sidh.sage

### SIDH 
### https://nbviewer.org/url/defeo.lu/docet/assets/jupyter/2018-03-22-PostScryptum.ipynb

p = 431
F.<i> = GF(p^2, modulus=x^2+1)

E = EllipticCurve(F, [1, 0])

E.order() == (p+1)^2

p + 1 == 2^4 * 3^3 

Pa, Qa = E.torsion_basis(2^4)

Pb, Qb = E.torsion_basis(3^3)

Sa = randint(1, 2^4)

Sb = randint(1, 3^3)

phi_a = E.isogeny(Pa + Sa * Qa)
phi_b = E.isogeny(Pb + Sb * Qb)


Ea = phi_a.codomain()
Eb = phi_b.codomain()

phi_a(Pb) + Sb * phi_a(Qb) == phi_a(Pb + Sb * Qb)
phi_b(Pa) + Sa * phi_b(Qa) == phi_b(Pa + Sa * Qa)


Eab = Ea.isogeny(phi_a(Pb + Sb * Qb)).codomain()
Eba = Eb.isogeny(phi_b(Pa + Sa * Qa)).codomain()


Eab == Eba == E.isogeny([Pa + Sa * Qa, Pb + Sb * Qb])