{"paper":{"title":"Near-Optimal Time-Sparsity Trade-Offs for Solving Noisy Linear Equations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CR","cs.DM","math.ST","stat.TH"],"primary_cat":"cs.CC","authors_text":"Guy Bresler, Kiril Bangachev, Stefan Tiegel, Vinod Vaikuntanathan","submitted_at":"2024-11-19T13:53:43Z","abstract_excerpt":"We present a polynomial-time reduction from solving noisy linear equations over $\\mathbb{Z}/q\\mathbb{Z}$ in dimension $\\Theta(k\\log n/\\mathsf{poly}(\\log k,\\log q,\\log\\log n))$ with a uniformly random coefficient matrix to noisy linear equations over $\\mathbb{Z}/q\\mathbb{Z}$ in dimension $n$ where each row of the coefficient matrix has uniformly random support of size $k$. This allows us to deduce the hardness of sparse problems from their dense counterparts. In particular, we derive hardness results in the following canonical settings. 1) Assuming the $\\ell$-dimensional (dense) LWE over a poly"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.12512","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/2411.12512/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"}