{"paper":{"title":"Module Lattice Security (Part II): Module Lattice Reduction via Optimal Sign Selection","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Module lattices reduce to the same quality as ideal lattices by decomposing them into rank-1 submodules and optimizing sign choices.","cross_cats":["cs.IT","math.IT","quant-ph"],"primary_cat":"cs.CR","authors_text":"Ming-Xing Luo","submitted_at":"2026-04-24T13:54:33Z","abstract_excerpt":"We extend the CDPR's quantum attack from ideal lattices to module lattices over $2^k$-th cyclotomic rings. Using trace orthogonality of the power basis, we decompose a rank-$d$ module into mutually orthogonal rank-$1$ submodules, and apply CDPR's analysis to each independently and return the shortest candidate. The Hermite factor $\\exp(\\tilde{O}(\\sqrt{n}))$ matches the ideal case, with a module reduction factor $\\alpha_d=O(1)$ independent of the rank, under a balance hypothesis (proved for Gaussian distribution) automatic for MLWE-distributed bases. To enable a bounded-precision implementation"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This base module reduction achieves a Hermite factor exp(Õ(√n)) matching the ideal case, with a module reduction factor O(1) independent of the rank, under a balance hypothesis automatically satisfied for MLWE-distributed bases.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"All results build on the class number one condition h_k^+=1 established in Part I of this series; the balance hypothesis is assumed to hold automatically for MLWE bases without independent proof here.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"CDPR reduction extends to module lattices with Hermite factor exp(Õ(√n)), O(1) rank-independent module factor under MLWE balance, plus optimal sign discrepancy δ*≈0.4407 from MILP, assuming class number one from Part I.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Module lattices reduce to the same quality as ideal lattices by decomposing them into rank-1 submodules and optimizing sign choices.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8a8c983e07c500053276d690da7b25441ebc9247d97e5add0a8e71cd4d719f6d"},"source":{"id":"2604.22900","kind":"arxiv","version":2},"verdict":{"id":"0ff17cde-d17b-4527-aef7-b8397e5b9ee8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T11:25:55.368898Z","strongest_claim":"This base module reduction achieves a Hermite factor exp(Õ(√n)) matching the ideal case, with a module reduction factor O(1) independent of the rank, under a balance hypothesis automatically satisfied for MLWE-distributed bases.","one_line_summary":"CDPR reduction extends to module lattices with Hermite factor exp(Õ(√n)), O(1) rank-independent module factor under MLWE balance, plus optimal sign discrepancy δ*≈0.4407 from MILP, assuming class number one from Part I.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"All results build on the class number one condition h_k^+=1 established in Part I of this series; the balance hypothesis is assumed to hold automatically for MLWE bases without independent proof here.","pith_extraction_headline":"Module lattices reduce to the same quality as ideal lattices by decomposing them into rank-1 submodules and optimizing sign choices."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.22900/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T10:37:26.541258Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:53:24.646855Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"4b0bf08ef350ab3a8d225857e3b85c884a5871121328928984e1b7301539a8ea"},"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"}