Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913098257924096 |
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| author | Luo, Ming-Xing |
| author_facet | Luo, Ming-Xing |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_15858 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$ Luo, Ming-Xing Cryptography and Security Quantum Physics 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. |
| title | Module Lattice Security (Part I): Unconditional Verification of Weber's Conjecture for $k \le 12$ |
| topic | Cryptography and Security Quantum Physics |
| url | https://arxiv.org/abs/2604.15858 |