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

TL;DR

This paper provides the first unconditional proof for Weber's conjecture for modules of rank up to 12, advancing lattice-based cryptography without relying on GRH.

Contribution

It introduces a novel proof method combining computational sieves, cyclotomic tower structures, and Herbrand's theorem to verify Weber's conjecture unconditionally for k ≤ 12.

Findings

Unconditional verification of Weber's conjecture for k ≤ 12

First proof not relying on Generalized Riemann Hypothesis

Method combines computational sieve, cyclotomic tower, and Herbrand's theorem

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.