{"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/explicit-isogenies-in-quadratic-time-in-any","title":"Explicit isogenies in quadratic time in any characteristic","arxiv_id":"1603.00711","date":"2016-03-02","proceeding":null,"authors":["Luca De Feo","Cyril Hugounenq","Jérôme Plût","Éric Schost"],"abstract":"Consider two elliptic curves $E,E'$ defined over the finite field $\\mathbb{F}_q$, and suppose that there exists an isogeny $\\psi$ between $E$ and $E'$. We propose an algorithm that determines $\\psi$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $\\ell$-torsion of the curves (where $\\ell$ is a prime different from the characteristic $p$ of the base field). Our approach is inspired by a previous algorithm due to Couveignes, that involved computations using the $p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\\tilde{O}(r^2) p^{O(1)}$ base field operations. On the other hand, the cost of our algorithm is $\\tilde{O}(r^2 + \\sqrt{r} \\log(q))$; this makes it an interesting alternative for the medium- and large-characteristic cases.","url_abs":"http://arxiv.org/abs/1603.00711v2","url_pdf":"http://arxiv.org/pdf/1603.00711v2.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":"explicit-isogenies-in-quadratic-time-in-any","repo_url":"https://github.com/Hugounenq-Cyril/Two_curves_on_a_volcano","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}