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

• URL: http://arxiv.org/abs/2604.15858v1
• Date: Fri, 17 Apr 2026 09:05:50 GMT
• Title: Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$
• Abstract summary: Weber's conjecture governs three aspects of lattice-based cryptography.<n>We present the first unconditional proof for $k le 12$.
• Score: 1.0152838128195467
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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