Multiplication polynomials for elliptic curves over finite local rings
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For a given elliptic curve E over a finite local ring, we denote by E^∞ its subgroup at infinity. Every point P ∈ E^∞ can be described solely in terms of its x-coordinate P_x, which can be therefore used to parameterize all its multiples nP. We refer to the coefficient of (P_x)^i in the parameterization of (nP)_x as the i-th multiplication polynomial. We show that this coefficient is a degree-i rational polynomial without a constant term in n. We also prove that no primes greater than i may appear in the denominators of its terms. As a consequence, for every finite field 𝔽_q and any k∈ℕ^*, we prescribe the group structure of a generic elliptic curve defined over 𝔽_q[X]/(X^k), and we show that their ECDLP on E^∞ may be efficiently solved.
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