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- load("div.sage")
- # Initialize an elliptic curve
- p = 115792089237316195423570985008687907853269984665640564039457584007908834671663
- r = 115792089237316195423570985008687907852837564279074904382605163141518161494337
- Fp = GF(p) # Base Field
- Fr = GF(r) # Scalar Field
- A = 0
- B = 7
- E = EllipticCurve(GF(p), [A, B])
- assert(E.cardinality() == r)
- K.<x> = PolynomialRing(Fp, implementation="generic")
- L.<y> = PolynomialRing(K, implementation="generic")
- M.<z> = L[]
- eqn = y^2 - x^3 - A * x - B
- P0 = LabelPoint(E.random_element(), {"P₀": 1})
- P1 = LabelPoint(E.random_element(), {"P₁": 1})
- P2 = LabelPoint(E.random_element(), {"P₂": 1})
- Q = -int(Fr(5)^-1) * (P0.P + 2*P1.P + 3*P2.P)
- Q = LabelPoint(Q, {"Q": 1})
- A0 = LabelPoint(E.random_element(), {"A₀": 1})
- A1 = LabelPoint(E.random_element(), {"A₁": 1})
- X1 = div_line(A0, A1)
- # i = 0
- L0 = div_line(P0, -P0)
- L1 = div_line(P1, -P1)
- L2 = div_line(P2, -P2)
- L3 = div_line(Q, -Q)
- # P₀ + 2P₁ + 3P₂ + 5Q
- # i = 1
- D1 = div_line(P1, P1)
- R1 = P1 + P1
- D2 = div_line(P2, P2)
- R2 = P2 + P2
- D3 = div_line(P2, Q)
- R3 = P2 + Q
- D4 = div_line(Q, Q)
- R4 = Q + Q
- D5 = D4
- R5 = R4
- D6 = L0
- R6 = P0
- # i = 2
- D1 = D1 + D2 + div_line(R1, R2) - (div_line(R1, -R1) + div_line(R2, -R2))
- R1 = R1 + R2
- D2 = D3 + D4 + div_line(R3, R4) - (div_line(R3, -R3) + div_line(R4, -R4))
- R2 = R3 + R4
- D3 = D5 + D6 + div_line(R5, R6) - (div_line(R5, -R5) + div_line(R6, -R6))
- R3 = R5 + R6
- # i = 3
- Dx = D1
- Rx = R1
- D1 = D2 + D3 + div_line(R2, R3) - (div_line(R2, -R2) + div_line(R3, -R3))
- R1 = R2 + R3
- D2, R2 = Dx, Rx
- # i = 4
- D = D1 + D2 + div_line(R1, R2) - (div_line(R1, -R1) + div_line(R2, -R2))
- assert D.is_equiv({
- "P₀": 1,
- "P₁": 2,
- "P₂": 3,
- "Q": 5,
- "∞": -11
- })
- assert X1.eval(D) == (-1)^D.effective_degree() * D.eval(X1)
- f_numer = D.func.numerator().mod(eqn)
- f_denom = D.func.denominator().mod(eqn)
- f = f_numer / f_denom
- assert f.denominator() == 1
- f = f.numerator()
- print(f)
- save([P0.P, P1.P, P2.P, Q.P, f], "div.sobj")
- print("Saved (P₀, P₁, P₂, Q, f)")
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