Twelve math cases are officially safe from quantum computers without relying on a 160-year-old unproven theory.
April 20, 2026
Original Paper
Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$
arXiv · 2604.15858
The Takeaway
Weber's conjecture proof provides the first unconditional verification of certain lattice-based cryptographic structures. Previous security guarantees for this technology relied on the Generalized Riemann Hypothesis, which remains unproven. Lattice-based cryptography is the primary defense planned against future quantum computer attacks. Removing the reliance on an unproven hypothesis turns theoretical security into a mathematical certainty. This work ensures that our future encryption methods are built on a bedrock of proven logic rather than hopeful assumptions.
From the 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, inductiv