Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography

TL;DR

This paper explores matrix Kloosterman sums, their statistical properties, and applications in cryptography, revealing their distributional behavior aligns with random matrix theory and enabling new cryptographic security tests.

Contribution

We develop algorithms for evaluating matrix Kloosterman sums, analyze their distributional behavior, and introduce a spectral cryptographic test based on their statistical signatures.

Findings

Normalized sums follow Sato-Tate distribution

Algorithms for efficient evaluation of sums

New cryptographic security test based on spectral signatures

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.