Multiplication polynomials for elliptic curves over 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.03650
 arXiv:
 arXiv:2302.03650
 Bibcode:
 2023arXiv230203650I
 Keywords:

 Mathematics  Number Theory;
 Computer Science  Cryptography and Security;
 11G07;
 11T55;
 11C08;
 13B25
 EPrint:
 In International Symposium on Symbolic and Algebraic Computation 2023 (ISSAC 2023). ACM, New York, NY, USA