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In this paper, we introduce a novel approach for generating random elementsof a finite group given a set of generators of that. Our method draws uponcombinatorial group theory and automata theory to achieve this objective.Furthermore, we explore the application of this method in generating randomelements of a particularly significant group, namely the symmetric group (orgroup of permutations on a set). Through rigorous analysis, we demonstrate thatour proposed method requires fewer average swaps to generate permutationscompared to existing approaches. However, recognizing the need for practicalapplications, we propose a hardware-based implementation based on ourtheoretical approach, and provide a comprehensive comparison with previousmethods. Our evaluation reveals that our method outperforms existing approachesin certain scenarios. Although our primary proposed method only aims to speedup the shuffling and does not decrease its time complexity, we also extend ourmethod to improve the time complexity.

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