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