{"paper":{"title":"Finding normal bases over finite fields with prescribed trace self-orthogonal relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Qunying Liao, Rongquan Feng, Xiyong Zhang, Xuhong Gao","submitted_at":"2013-03-10T02:14:01Z","abstract_excerpt":"Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of $\\mathbb{F}_{2^n}$ over $\\mathbb{F}_2$ if and only if $4\\nmid n$. In this paper, we prove there exists a normal element $\\alpha$ of $\\mathbb{F}_{2^n}$ over $\\mathbb{F}_{2}$ corresponding to a prescribed vector $a=(a_0,a_1,...,a_{n-1})\\in \\mathbb{F}_2^n$ such that $a_i={Tr}_{2^n|2}(\\alpha^{1+2^i})$ for $0\\leq i\\leq n-1$, where $n$ is a 2-power or odd, if and only if the given vector $a$ is symmetric ($a_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2283","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":""},"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"}