{"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/finding-orientations-of-supersingular","title":"Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders","arxiv_id":"2308.11539","date":"2023-08-22","proceeding":null,"authors":["Sarah Arpin","James Clements","Pierrick Dartois","Jonathan Komada Eriksen","Péter Kutas","Benjamin Wesolowski"],"abstract":"Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $\\mathfrak{O}$-orientable for a fixed imaginary quadratic order $\\mathfrak{O}$ provides non-trivial information towards computing an endomorphism corresponding to the $\\mathfrak{O}$-orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at $p$ and $\\infty$. We provide code implementations in Sagemath which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to $O(p)$, even for cryptographically sized $p$.","url_abs":"https://arxiv.org/abs/2308.11539v1","url_pdf":"https://arxiv.org/pdf/2308.11539v1.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":"finding-orientations-of-supersingular","repo_url":"https://github.com/jtcc2/finding-orientations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}