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.