{"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/fast-and-frobenius-rational-isogeny","title":"Fast and Frobenius: Rational Isogeny Evaluation over Finite Fields","arxiv_id":"2306.16072","date":"2023-06-28","proceeding":null,"authors":["Gustavo Banegas","Valerie Gilchrist","Anaëlle Le Dévéhat","Benjamin Smith"],"abstract":"Consider the problem of efficiently evaluating isogenies $\\phi: E \\to E/H$ of elliptic curves over a finite field $\\mathbb{F}_q$, where the kernel $H = \\langle G\\rangle$ is a cyclic group of odd (prime) order: given $E$, $G$, and a point (or several points) $P$ on $E$, we want to compute $\\phi(P)$. This problem is at the heart of efficient implementations of group-action- and isogeny-based post-quantum cryptosystems such as CSIDH. Algorithms based on V{\\'e}lu's formulae give an efficient solution to this problem when the kernel generator $G$ is defined over $\\mathbb{F}_q$. However, for general isogenies, $G$ is only defined over some extension $\\mathbb{F}_{q^k}$, even though $\\langle G\\rangle$ as a whole (and thus $\\phi$) is defined over the base field $\\mathbb{F}_q$; and the performance of V{\\'e}lu-style algorithms degrades rapidly as $k$ grows. In this article we revisit the isogeny-evaluation problem with a special focus on the case where $1 \\le k \\le 12$. We improve V{\\'e}lu-style isogeny evaluation for many cases where $k = 1$ using special addition chains, and combine this with the action of Galois to give greater improvements when $k > 1$.","url_abs":"https://arxiv.org/abs/2306.16072v1","url_pdf":"https://arxiv.org/pdf/2306.16072v1.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":"fast-and-frobenius-rational-isogeny","repo_url":"https://github.com/vgilchri/k-velu","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}