{"paper":{"title":"On the $k$-error linear complexity of binary sequences derived from the discrete logarithm in finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"cs.CR","authors_text":"Qiuyan Wang, Zhixiong Chen","submitted_at":"2019-01-29T03:34:48Z","abstract_excerpt":"Let $q=p^r$ be a power of an odd prime $p$. We study binary sequences $\\sigma=(\\sigma_0,\\sigma_1,\\ldots)$ with entries in $\\{0,1\\}$ defined by using the quadratic character $\\chi$ of the finite field $\\mathbb{F}_q$: $$ \\sigma_n=\\left\\{ \\begin{array}{ll} 0,& \\mathrm{if}\\quad n= 0,\\\\ (1-\\chi(\\xi_n))/2,&\\mathrm{if}\\quad 1\\leq n< q, \\end{array} \\right. $$ for the ordered elements $\\xi_0,\\xi_1,\\ldots,\\xi_{q-1}\\in \\mathbb{F}_q$. The $\\sigma$ is Legendre sequence if $r=1$.\n  Our first contribution is to prove a lower bound on the linear complexity of $\\sigma$ for $r\\geq 2$.\n  The bound improves some "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.10086","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"}