{"paper":{"title":"Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR"],"primary_cat":"math.NT","authors_text":"INRIA Rocquencourt), Sorina Ionica (INRIA Nancy - Grand Est / LORIA","submitted_at":"2012-03-29T07:48:08Z","abstract_excerpt":"Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial self-pairings of the $\\ell$-Tate pairing in terms of the action of the Frobenius on the $\\ell$-torsion of the Jacobian of a genus 2 curve. We apply similar techniques to study the non-degeneracy of the $\\ell$-Tate pairing restrained to subgroups of the $\\ell$-torsion which are maximal isotropic with respect to the Weil pairing. First, we deduce a criterion to verify whether the jacobian of a genus 2 curve has maximal endomorphism ring. Secondly, we derive a method to construct horizontal $(\\ell,\\ell)$-isogenies starting "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0222","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}