{"paper":{"title":"Complexity of Low-Degree Skew Polynomial Multiplication over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"Ke Ye, Ruichen Qiu, Yichuan Cao","submitted_at":"2026-07-01T05:50:36Z","abstract_excerpt":"In this note, we study the complexity of multiplication in skew polynomial rings over finite fields. We prove that the product of two elements in $\\mathbb{F}_{q^n}[x;\\sigma]$ of degree at most $d < n$ can be computed using $\\widetilde O(d^{\\omega_K-1}n)$ arithmetic operations over $\\mathbb{F}_q$, where $\\sigma$ is the $q$-Frobenius automorphism. This matches the conjectural upper bound of Caruso--Le Borgne~[ISSAC'17] and is quasi-optimal in view of the lower bound of Chen--Ye [ISSAC'24]. The proof reduces the finite-field case to the split algebra case using the equivariant multiplication theo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00476","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.00476/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}