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@@ -0,0 +1,53 @@
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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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+
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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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+
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+B0 = E.random_element()
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+B1 = E.random_element()
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+
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+# Base 3 representation
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+d0 = [ 1, -1, 0, 0, 0]
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+d1 = [-1, -1, 0, -1, 1]
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+
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+e0 = sum(d0_j*(-3)^j for j, d0_j in enumerate(d0))
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+assert e0 == 4
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+
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+e1 = sum(d1_j*(-3)^j for j, d1_j in enumerate(d1))
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+assert e1 == 110
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+
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+# We will prove this statement
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+Q = 4*B0 + 110*B1
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+assert Q == (
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+ (-3)^0 * ( B0 - B1) +
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+ (-3)^1 * (-B0 - B1) +
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+
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+ (-3)^3 * (-B1) +
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+ (-3)^4 * (B1)
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+)
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+
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+Q5 = E(0, 1, 0)
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+Q4 = -3*Q5 + B1
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+Q3 = -3*Q4 - B1
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+Q2 = -3*Q3
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+Q1 = -3*Q2 - B0 - B1
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+Q0 = -3*Q1 + B0 - B1
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+assert Q0 == Q
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+
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+a0 = (-3)^0
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+b0 = (-3)^1
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+assert e0 == a0 - b0
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+
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+a1 = (-3)^4
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+b1 = (-3)^0 + (-3)^1 + (-3)^3
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+assert e1 == a1 - b1
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+
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