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.

翻译：本文提出了一种利用给定生成元集生成有限群随机元素的新方法。该方法融合组合群论与自动机理论实现这一目标。我们进一步探讨了该方法在生成对称群（即集合上的置换群）这一重要群类随机元素中的应用。通过严格分析证明，与现有方法相比，本文提出的方法在生成置换时所需的平均交换次数更少。考虑到实际应用需求，我们基于理论方法提出了硬件实现方案，并与既有方法进行了全面比较。评估结果表明，在某些场景下本方法优于现有方案。虽然主要方法仅旨在加速置换过程而不降低时间复杂度，但本文也拓展了方法以改进时间复杂度。