Elliptic Curve Discrete Logarithm Problem (ECDLP) is the problem of finding an integer n such that Q = nP
Find point Q(x,y) such that P +Q = 0
#!/usr/bin/env sage
import sys
from sage.all import *
# Curve: E: Y2 = X3 + 497 X + 1768, p: 9739
# Problem: Find the point Q(x,y) such that P + Q = O.
F = GF(9739)
E = EllipticCurve(F, [497, 1768])
P = E(8045,6936)
print(P) # P
print(-P) # Q
print(P+(-P)) # 0 / Point at Infinity
Calculate order of composed curve by using "subcurves"
Given p,q,a,b,e we want to find g (requires a prime modulus).
E1 = EllipticCurve(GF(p),[a,b])
E2 = EllipticCurve(GF(q),[a,b])
o1=K1.order()
o2=K2.order()
d=inverse_mod(e,o1*o2)
Getting Curve Parameters from 2 Points
a=(A.y**X - A.x**Y - B.y**X + B.x**Y) * inverse_mod(A.x-B.x,n) % n
b=(A.y**X - A.x**Y - a*A.x ) % n
