Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$
For cryptographers relying on Ring-LWE and Module-LWE, this removes a conditional assumption for small parameters, though the result is incremental as it extends verification to only k≤12.
Weber's conjecture, which underpins lattice-based cryptography, was previously verified only under GRH for k≥9. This paper provides the first unconditional proof for k≤12, eliminating the reliance on unproven assumptions.
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.