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Module Lattice Security (Part III): Structured CVP Distance on the Log-Unit Lattice

arXiv
Authors: Ming-Xing Luo

Year

2026

Paper ID

63846

Status

Preprint

Abstract Read

~2 min

Abstract Words

144

Citations

0

Abstract

We prove that the L2 CVP distance from a random short ring element to the log-unit lattice of Q\(ζ2k\) converges to fracπ{2sqrt{6}}sqrt{n} as n=2k-1→infty. We then show that this target lies inside the Voronoi cell of the origin for kge 4. For the Linfty norm, the maximum over n sub-Gaussian coordinates yields O\(sqrt{log n}\) which translates into a sub-polynomial approximation factor for the Short Generator Problem. We show a Coarse Lattice Theorem that Babai's algorithm returns zero for all structured targets, yet exactly recovers unit perturbations of arbitrary size. For module determinant ideals, we further prove the Trigamma Theorem that proves an intrinsic imbalance σg0=O(1) independent of the modulus q. Finally, combined with Parts I and II, we reduce the CDPR factor for ML-KEM from exp\(tO(sqrt{n}\)) to a sub-polynomial value.

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  • This paper contributes to the Quantum Machine Learning research area in the Quantum Articles archive.
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  • We prove that the L^2 CVP distance from a random short ring element to the log-unit lattice of Q(ζ2^k) converges to fracπ2sqrt6sqrtn as n=2^k-1 -> infty.

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2605.17404 [2] arXiv https://arxiv.org/abs/2605.17404 [3] Publisher https://arxiv.org/abs/2605.17404

Local Citation Graph (Related-Paper Links)

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External citation index: OpenAlex citation signal • updated 2026-06-15 00:29:27

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