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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.10086","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2019-01-29T03:34:48Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"53bac68fd34e73680a96b038477a7a903f5fa3de467e8e0a18e420c30d5a711f","abstract_canon_sha256":"38700a3354c0725f6b1c39e276da2698bcd86c76117acbf5f79e8cf617507452"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:16.202436Z","signature_b64":"xOVrIIzT4BshrwHu6GBCcyj/EEji9ffItbX+wmq6HtKjx+8UVVPgm0QuRyZtTBaEXRTYn+3BVb6dFpSh6FZHDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"712dbce07f1496703c4aab17ec3344b07c970326eb616afaebd41be32a18e0d7","last_reissued_at":"2026-05-17T23:55:16.201946Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:16.201946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $p$. 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