Implementation and Analysis of Regev's Quantum Factorization Algorithm

Abstract: Quantum computing represents a significant advancement in computational capabilities. Of particular concern is its impact on asymmetric cryptography through, notably, Shor's algorithm and the more recently developed Regev's algorithm for factoring composite numbers. We present our implementation of the latter. Our analysis encompasses both quantum simulation results and classical component examples, with particular emphasis on comparative cases between Regev's and Shor's algorithms. Our experimental results reveal that Regev's algorithm indeed outperforms Shor's algorithm for certain composite numbers in practice. However, we observed significant performance variations across different input values. Despite Regev's algorithm's theoretical asymptotic efficiency advantage, our implementation exhibited execution times longer than Shor's algorithm for small integer factorization in both quantum and classical components. These findings offer insights into the practical challenges and performance characteristics of implementing Regev's algorithm in realistic quantum computing scenarios.