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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
- P1 = LabelPoint(E.random_element(), {"P₁": 1})
- P2 = LabelPoint(E.random_element(), {"P₂": 1})
- P3 = LabelPoint(E.random_element(), {"P₃": 1})
- P4 = LabelPoint(E.random_element(), {"P₄": 1})
- Q = -(P1.P + P2.P + P3.P + P4.P)
- Q = LabelPoint(Q, {"Q": 1})
- assert P1.P + P2.P + P3.P + P4.P + Q.P == E(0)
- # Challenge line
- A0 = LabelPoint(E.random_element(), {"A₀": 1})
- A1 = LabelPoint(E.random_element(), {"A₁": 1})
- X1 = div_line(A0, A1)
- # First loop in construct
- L1 = div_line(P1, P2)
- Q1 = P1 + P2
- L2 = div_line(P3, P4)
- Q2 = P3 + P4
- L3 = div_line(Q, -Q)
- Q3 = Q
- #print(f"L₁ = {L1}")
- #print(f"L₂ = {L2}")
- #print(f"L₃ = {L3}")
- divs = [L1, L2, L3]
- # Now apply reduction algo
- # len(divs) == 3
- D1 = L1
- Q1 = Q1
- # i = 0
- ℓ = div_line(Q2, Q3)
- D2 = ℓ + L2 + L3 - div_line(Q2, -Q2) - div_line(Q3, -Q3)
- Q2 = Q2 + Q3
- divs = [D1, D2]
- # len(divs) == 2
- ℓ = div_line(Q1, Q2)
- D1 = ℓ + D1 + D2 - div_line(Q1, -Q1) - div_line(Q2, -Q2)
- Q1 = Q1 + Q2
- divs = [D1]
- D = D1
- assert D.is_equiv({
- "P₁": 1,
- "P₂": 1,
- "P₃": 1,
- "P₄": 1,
- "Q": 1,
- "∞": -5
- })
- assert X1.eval(D) == (-1)^D.effective_degree() * D.eval(X1)
- # We should get the same result here:
- load("construct.sage")
- points = [P1, P2, P3, P4, Q]
- D = construct(points)
- print(f"D = {D}")
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