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.

翻译：本文对矩阵Kloosterman和进行了系统研究，涵盖其计算特性、分布行为及其在密码分析中的应用。基于[Zelingher, 2023]的研究成果，我们开发了通过格林多项式计算这些和的算法，并建立了分析其统计分布的通用框架。我们进一步研究了相关的$L$函数，阐明了它们与对称函数及随机矩阵理论的联系。研究表明，类似于$SU(n)$和$Sp(2n)$等紧李群中随机矩阵的特征值统计特性，矩阵Kloosterman和的归一化值同样满足Sato-Tate等分布规律。最后，我们将该框架应用于区分真随机序列与具有细微代数偏差的序列，并提出了一种基于矩阵Kloosterman和分布特征的密码安全性新型谱检测方法。