{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/multiplication-polynomials-for-elliptic","title":"Multiplication polynomials for elliptic curves over finite local rings","arxiv_id":"2302.03650","date":"2023-02-07","proceeding":null,"authors":["Riccardo Invernizzi","Daniele Taufer"],"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.","url_abs":"https://arxiv.org/abs/2302.03650v2","url_pdf":"https://arxiv.org/pdf/2302.03650v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"multiplication-polynomials-for-elliptic","repo_url":"https://github.com/r98inver/ec-local-rings","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}