ECC smart's attack

先记录着 有时间好好学学再理解

from sage.all import *

p = 75206427479775622966537995406541077245842499523456803092204668034148875719001
a = 40399280641537685263236367744605671534251002649301968428998107181223348036480
b = 34830673418515139976377184302022321848201537906033092355749226925568830384464

E = EllipticCurve(GF(p), [a, b])
def SmartAttack(P,Q,p):
    E = P.curve()
    Eqp = EllipticCurve(Qp(p, 2), [ ZZ(t) + randint(0,p)*p for t in E.a_invariants() ])

    P_Qps = Eqp.lift_x(ZZ(P.xy()[0]), all=True)
    for P_Qp in P_Qps:
        if GF(p)(P_Qp.xy()[1]) == P.xy()[1]:
            break

    Q_Qps = Eqp.lift_x(ZZ(Q.xy()[0]), all=True)
    for Q_Qp in Q_Qps:
        if GF(p)(Q_Qp.xy()[1]) == Q.xy()[1]:
            break

    p_times_P = p*P_Qp
    p_times_Q = p*Q_Qp

    x_P,y_P = p_times_P.xy()
    x_Q,y_Q = p_times_Q.xy()

    phi_P = -(x_P/y_P)
    phi_Q = -(x_Q/y_Q)
    k = phi_Q/phi_P
    return ZZ(k)
G = E(63199291976729017585116731422181573663076311513240158412108878460234764025898 ,11977959928854309700611217102917186587242105343137383979364679606977824228558 )
P = E(75017275378438543246214954287362349176908042127439117734318700769768512624429 , 39521483276009738115474714281626894361123804837783117725653243818498259351984 )
flag = SmartAttack(G,P,p)
print(flag)