{"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-computation-of-2-isogenies-in-dimension","title":"Fast computation of 2-isogenies in dimension 4 and cryptographic applications","arxiv_id":"2407.15492","date":"2024-07-22","proceeding":null,"authors":["Pierrick Dartois"],"abstract":"Dimension 4 isogenies have first been introduced in cryptography for the cryptanalysis of Supersingular Isogeny Diffie-Hellman (SIDH) and have been used constructively in several schemes, including SQIsignHD, a derivative of SQIsign isogeny based signature scheme. Unlike in dimensions 2 and 3, we can no longer rely on the Jacobian model and its derivatives to compute isogenies. In dimension 4 (and higher), we can only use theta-models. Previous works by Romain Cosset, David Lubicz and Damien Robert have focused on the computation of $\\ell$-isogenies in theta-models of level $n$ coprime to $\\ell$ (which requires to use $n^g$ coordinates in dimension $g$). For cryptographic applications, we need to compute chains of $2$-isogenies, requiring to use $\\geq 3^g$ coordinates in dimension $g$ with state of the art algorithms. In this paper, we present algorithms to compute chains of $2$-isogenies between abelian varieties of dimension $g\\geq 1$ with theta-coordinates of level $n=2$, generalizing a previous work by Pierrick Dartois, Luciano Maino, Giacomo Pope and Damien Robert in dimension $g=2$. We propose an implementation of these algorithms in dimension $g=4$ to compute endomorphisms of elliptic curve products derived from Kani's lemma with applications to SQIsignHD and SIDH cryptanalysis. We are now able to run a complete key recovery attack on SIDH when the endomorphism ring of the starting curve is unknown within a few seconds on a laptop for all NIST SIKE parameters.","url_abs":"https://arxiv.org/abs/2407.15492v1","url_pdf":"https://arxiv.org/pdf/2407.15492v1.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-computation-of-2-isogenies-in-dimension","repo_url":"https://github.com/Pierrick-Dartois/SQISignHD-lib","is_official":1,"mentioned_in_paper":0,"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}