Author:

Keywords:

12ZZC23N#57206766, G023721N#57210626, G0F5921N#57206508, Science & Technology, Technology, Physical Sciences, Computer Science, Theory & Methods, Mathematics, Applied, Computer Science, Mathematics, elliptic curve, local finite ring, points at infinity, addition law, multiplication polynomials

Abstract:

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 Px, which can be therefore used to parameterize all its multiples nP. We refer to the coefficient of (Px)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 𝔽q and any k∈ℕ∗, we prescribe the group structure of a generic elliptic curve defined over 𝔽q[X]/(Xk), and we show that their ECDLP on E∞ may be efficiently solved.