Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$
TLDR
This paper provides the first unconditional proof of Weber's Conjecture for k ≤ 12, crucial for lattice-based cryptography.
Key contributions
- Presents the first unconditional proof for Weber's Conjecture for k ≤ 12.
- Removes reliance on the Generalized Riemann Hypothesis for existing verifications.
- Impacts the Principal Ideal Problem, module freeness, and R-LWE/MLWE reductions.
- Combines Fukuda-Komatsu sieve, cyclotomic ℤ₂-tower, and Herbrand's theorem.
Why it matters
This paper provides an unconditional proof for Weber's Conjecture, a cornerstone of lattice-based cryptography. Removing reliance on the unproven GRH strengthens the theoretical foundations of Ring-LWE and Module-LWE. This work is critical for the security and efficiency of future cryptographic systems.
Original 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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