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- 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
- B0 = E.random_element()
- B1 = E.random_element()
- # Base 3 representation
- d0 = [ 1, -1, 0, 0, 0]
- d1 = [-1, -1, 0, -1, 1]
- e0 = sum(d0_j*(-3)^j for j, d0_j in enumerate(d0))
- assert e0 == 4
- e1 = sum(d1_j*(-3)^j for j, d1_j in enumerate(d1))
- assert e1 == 110
- # We will prove this statement
- Q = 4*B0 + 110*B1
- assert Q == (
- (-3)^0 * ( B0 - B1) +
- (-3)^1 * (-B0 - B1) +
- (-3)^3 * (-B1) +
- (-3)^4 * (B1)
- )
- Q5 = E(0, 1, 0)
- Q4 = -3*Q5 + B1
- Q3 = -3*Q4 - B1
- Q2 = -3*Q3
- Q1 = -3*Q2 - B0 - B1
- Q0 = -3*Q1 + B0 - B1
- assert Q0 == Q
- a0 = (-3)^0
- b0 = (-3)^1
- assert e0 == a0 - b0
- a1 = (-3)^4
- b1 = (-3)^0 + (-3)^1 + (-3)^3
- assert e1 == a1 - b1
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