Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$

Computer Science > Cryptography and Security [Submitted on 17 Apr 2026] Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for k \le 12 Ming-Xing Luo Weber's conjecture (1886) governs three aspects of lattice-based cryptography: the solvability of the Principal Ideal Problem, the freeness of modules over rings of integers, and the tightness of worst-case-to-average-case reductions in Ring-LWE (R-LWE) and Module-LWE (MLWE). Existing verifications for k \ge 9 rely on Generalized Riemann Hypothesis (GRH). In this paper, we present the first unconditional proof for k \le 12. Our method combines the Fukuda-Komatsu computational sieve, inductive structure of the cyclotomic \mathbb{Z}_2-tower, and Herbrand's theorem. Comments: 24 pages Subjects: Cryptography and Security (cs.CR); Quantum Physics (quant-ph) Cite as: arXiv:2604.15858 [cs.CR]   (or arXiv:2604.15858v1 [cs.CR] for this version)   https://doi.org/10.48550/arXiv.2604.15858 Focus to learn more Submission history From: Mingxing Luo [view email] [v1] Fri, 17 Apr 2026 09:05:50 UTC (28 KB) Access Paper: HTML (experimental) view license Current browse context: cs.CR < prev   |   next > new | recent | 2026-04 Change to browse by: cs quant-ph References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)