{"paper":{"title":"On Ideal Lattices and Gr\\\"obner Bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"Ambedkar Dukkipati, Maria Francis","submitted_at":"2014-09-27T10:01:19Z","abstract_excerpt":"In this paper, we draw a connection between ideal lattices and Gr\\\"{o}bner bases in the multivariate polynomial rings over integers. We study extension of ideal lattices in $\\mathbb{Z}[x]/\\langle f \\rangle$ (Lyubashevsky \\& Micciancio, 2006) to ideal lattices in $\\mathbb{Z}[x_1,\\ldots,x_n]/\\mathfrak{a}$, the multivariate case, where $f$ is a polynomial in $\\mathbb{Z}[X]$ and $\\mathfrak{a}$ is an ideal in $\\mathbb{Z}[x_1,\\ldots,x_n]$. Ideal lattices in univariate case are interpreted as generalizations of cyclic lattices. We introduce a notion of multivariate cyclic lattices and we show that mu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7788","kind":"arxiv","version":2},"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"}