Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography

• URL: http://arxiv.org/abs/2601.01603v1
• Date: Sun, 04 Jan 2026 17:04:52 GMT
• Title: Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography
• Authors: Tianshuo Yang,
• Abstract summary: This paper presents a comprehensive study of matrix Kloosterman sums.<n>It includes their computational aspects, distributional behavior, and applications in cryptographic analysis.
• Score: 1.7250279414563907
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper presents a comprehensive study of matrix Kloosterman sums, including their computational aspects, distributional behavior, and applications in cryptographic analysis. Building on the work of [Zelingher, 2023], we develop algorithms for evaluating these sums via Green's polynomials and establish a general framework for analyzing their statistical distributions. We further investigate the associated $L$-functions and clarify their relationships with symmetric functions and random matrix theory. We show that, analogous to the eigenvalue statistics of random matrices in compact Lie groups such as $SU(n)$ and $Sp(2n)$, the normalized values of matrix Kloosterman sums exhibit Sato-Tate equidistribution. Finally, we apply this framework to distinguish truly random sequences from those exhibiting subtle algebraic biases, and we propose a novel spectral test for cryptographic security based on the distributional signatures of matrix Kloosterman sums.

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