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Our goal here is to minimize the exponent of $q$ in the case where every equation contains at most three unknowns. %Applying hypergraph-theoretic methods we prove The main result states that if t"},"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":"1309.5292","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2013-09-20T15:06:19Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"cc17570bdda9b2b2908ef1d8ac529a95fa5904439e83673a9b62f2c51b44c34b","abstract_canon_sha256":"8f2cb5a2ec9d531037e13a3bd119fbb3bde26e8936a5e4c98c8f0a0c4f9f2498"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:47.075477Z","signature_b64":"8HrGnGXK/GtKrd2TpyEAM+IRYlJGJOd+FwHOOv+DZP/SuCqMtA0XZ8qGr+LKOYQc9ip0ZpA8Vo7cGlGeUqoKBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3420a80b7ca87de518cefaac44ecbc09b9930926957249bc0b1aa5fe93fad3be","last_reissued_at":"2026-05-18T03:12:47.074868Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:47.074868Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Speeding up Deciphering by Hypergraph Ordering","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CR","authors_text":"Peter Horak, Zsolt Tuza","submitted_at":"2013-09-20T15:06:19Z","abstract_excerpt":"The \"Gluing Algorithm\" of Semaev [Des.\\ Codes Cryptogr.\\ 49 (2008), 47--60] --- that finds all solutions of a sparse system of linear equations over the Galois field $GF(q)$ --- has average running time $O(mq^{\\max \\left\\vert \\cup_{1}^{k}X_{j}\\right\\vert -k}), $ where $m$ is the total number of equations, and $\\cup_{1}^{k}X_{j}$ is the set of all unknowns actively occurring in the first $k$ equations. 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