Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912801751040000 |
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| author | Yang, Tianshuo |
| author_facet | Yang, Tianshuo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography Yang, Tianshuo Number Theory Cryptography and Security 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. |
| title | Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography |
| topic | Number Theory Cryptography and Security |
| url | https://arxiv.org/abs/2601.01603 |