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- q = 47
- Fq = GF(q)
- E = EllipticCurve(Fq, (0, 5))
- R.<x, y> = PolynomialRing(Fq)
- def add(P, Q):
- Px, Py = P[0], P[1]
- Qx, Qy = Q[0], Q[1]
- # We don't handle this case yet :p
- assert Px != Qx
- gradient = (Qy - Py) / (Qx - Px)
- intersect = Py - gradient * Px
- f = y - gradient * x - intersect
- assert f(Px, Py) == 0
- assert f(Qx, Qy) == 0
- R = P + Q
- Rx, Ry = R[0], R[1]
- assert f(Rx, -Ry) == 0
- g = x - Rx
- assert g(Rx, Ry) == g(Rx, -Ry) == 0
- return f / g
- P = E(33, 9)
- Q = E(34, 39)
- # div(f) = [P] + [Q] - [P + Q] - [∞] ∈ Pic(E)
- # = ([P] + [Q] - 2[∞]) - ([P + Q] - [∞])
- # => [P] + [Q] - 2[∞] ~ [P + Q] - [∞]
- f = add(P, Q)
- print(f)
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