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Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography

Tianshuo Yang

TL;DR

We extend scalar Kloosterman sums to matrix settings in $\mathrm{GL}_n(\mathbb{F}_q)$, develop two computational algorithms based on Jordan decomposition and Green polynomials, and relate these sums to generating functions and symmetric-power $L$-functions. Using Deligne–Katz equidistribution and Sato–Tate theory, we show normalized matrix Kloosterman sums exhibit random-matrix statistics tied to monodromy groups such as $\mathrm{SU}(2)$ and $\mathrm{Sp}(2k)$, with precise distributions given by complete symmetric polynomials in Frobenius eigenvalues. We validate the theory with extensive numerical experiments and demonstrate cryptographic applications by proposing a spectral test that distinguishes truly random sequences from algebraic biases, supported by empirical results. This framework integrates algebraic geometry, representation theory, and cryptography to provide a principled tool for randomness assessment and security analysis.

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.

Matrix Kloosterman Sums, Random Matrix Statistics, and Cryptography

TL;DR

We extend scalar Kloosterman sums to matrix settings in , develop two computational algorithms based on Jordan decomposition and Green polynomials, and relate these sums to generating functions and symmetric-power -functions. Using Deligne–Katz equidistribution and Sato–Tate theory, we show normalized matrix Kloosterman sums exhibit random-matrix statistics tied to monodromy groups such as and , with precise distributions given by complete symmetric polynomials in Frobenius eigenvalues. We validate the theory with extensive numerical experiments and demonstrate cryptographic applications by proposing a spectral test that distinguishes truly random sequences from algebraic biases, supported by empirical results. This framework integrates algebraic geometry, representation theory, and cryptography to provide a principled tool for randomness assessment and security analysis.

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 -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 and , 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.
Paper Structure (35 sections, 13 theorems, 29 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 35 sections, 13 theorems, 29 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conjugacy Classes of $\mathrm{GL}_n(\mathbb{F}_q)$
  4. Representations of $\mathrm{GL}_n(\mathbb{F}_q)$
  5. Symmetric Functions and Green Polynomials
  6. Matrix Kloosterman Sums
  7. Definition
  8. Reduction Formulas
  9. General Formula
  10. Algorithms for Computation
  11. Normalization
  12. Algorithm I: Jordan Block Computation
  13. Algorithm II: General Matrix Computation
  14. Complexity Analysis
  15. $L$-Functions and Symmetric Functions
  16. ...and 20 more sections

Key Result

Theorem 2.2

green1955characters For any $y \in \mathrm{GL}_n(\mathbb{F}_q)$, there exists an integer $s \geq 1$, $d_1, \ldots, d_s \geq 1$, regular elliptic elements $y_j \in \mathrm{GL}_{d_j}(\mathbb{F}_q)$ (with mutually disjoint eigenvalues over $\overline{\mathbb{F}_q}$), and non-empty partitions $\bm{\mu}_ where $J_{\bm{\mu}_j}(y_j)$ denotes the Jordan block matrix associated with partition $\bm{\mu}_j$

Figures (4)

  • Figure 1: Graphs 1-1, 1-2, 1-3, 1-4: Distribution of $K_p$ and $\theta_p$ for small sample sizes.
  • Figure 2: Graphs 2-1, 2-2, 2-3, 2-4: Histograms for $p=11$ (top) and $p=53$ (bottom).
  • Figure 3: Graphs 3-1, 3-2: Trace and Angle distribution for $p=2$ over varying extensions. Despite the small field, the statistics align with SU(2).
  • Figure 4: Graphs 4-1 to 4-4: Top: Random Sequence $C_1$. Bottom: Patterned Sequence $C_2$. The patterned sequence shows significant bias and deviation from the semi-circle law.

Theorems & Definitions (20)

  • Definition 2.1: Regular Elliptic Element
  • Theorem 2.2: Conjugacy Classification
  • Theorem 2.3: Green's Parameterization
  • Definition 2.4: Partitions
  • Definition 2.5: Green Polynomials
  • Definition 2.6: Hyper-Kloosterman Sum
  • Definition 2.7: Matrix Kloosterman Sums
  • Theorem 2.8: Direct Sum Decomposition
  • Theorem 2.9: Regular Elliptic Reduction
  • Theorem 2.10: Jordan Block Formula
  • ...and 10 more