{"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/enumerating-hyperelliptic-curves-over-finite","title":"Enumerating hyperelliptic curves over finite fields in quasilinear time","arxiv_id":"2401.15255","date":"2024-01-27","proceeding":null,"authors":["Everett W. Howe"],"abstract":"We present an algorithm that, for every fixed genus $g$, will enumerate all hyperelliptic curves of genus $g$ over a finite field $k$ of odd characteristic in quasilinear time; that is, the time required for the algorithm is $\\widetilde{O}(q^{2g-1})$, where $q=\\#k$. Such an algorithm already exists in the case $g=2$, thanks to work of Mestre and Cardona and Quer, and in the case $g=3$, thanks to work of Lercier and Ritzenthaler. Experimentally, it appears that our new algorithm is about two orders of magnitude faster in practice than ones based on their work.","url_abs":"https://arxiv.org/abs/2401.15255v5","url_pdf":"https://arxiv.org/pdf/2401.15255v5.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":"enumerating-hyperelliptic-curves-over-finite","repo_url":"https://github.com/everetthowe/hyperelliptic","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}