Multiplication polynomials for elliptic curves over finite local rings

TL;DR

This paper introduces multiplication polynomials for elliptic curves over finite local rings, providing explicit formulas and implications for the elliptic curve discrete logarithm problem (ECDLP) in these settings.

Contribution

It establishes the structure of multiplication polynomials over finite local rings and demonstrates their use in solving the ECDLP efficiently in these contexts.

Findings

Multiplication polynomials are degree-i rational polynomials without constant terms.

No primes greater than i appear in denominators of the polynomial terms.

ECDLP can be efficiently solved on elliptic curves over certain finite local rings.

Abstract

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ 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 $\mathbb{F}_q$ and any $k\in\mathbb{N}^*$, we prescribe the group structure of a generic elliptic curve defined over $\mathbb{F}_q[X]/(X^k)$, and we show that their ECDLP on $E^{\infty}$ may be efficiently solved.