Random generation of group elements using combinatorial group theory and automata theory, along with a hardware example

Abstract

In this paper, we introduce a novel approach for generating random elements of a finite group given a set of generators of that. Our method draws upon combinatorial group theory and automata theory to achieve this objective. Furthermore, we explore the application of this method in generating random elements of a particularly significant group, namely the symmetric group (or group of permutations on a set). Through rigorous analysis, we demonstrate that our proposed method requires fewer average swaps to generate permutations compared to existing approaches. However, recognizing the need for practical applications, we propose a hardware-based implementation based on our theoretical approach, and provide a comprehensive comparison with previous methods. Our evaluation reveals that our method outperforms existing approaches in certain scenarios. Although our primary proposed method only aims to speed up the shuffling and does not decrease its time complexity, we also extend our method to improve the time complexity.