Implementation and Analysis of Regev's Quantum Factorization Algorithm

2025-02-13

AI Summary

This work systematically evaluates the practical performance of Regev’s quantum factoring algorithm against Shor’s algorithm. To address the lack of empirical validation, we implement the first end-to-end Regev pipeline—including LLL lattice basis reduction, modular exponentiation preprocessing, polynomial sampling, and Qiskit-based quantum circuit simulation—enabling cross-algorithm benchmarking. Our experiments reveal strong input sensitivity: Regev’s algorithm outperforms Shor’s on certain large composites but exhibits significant inefficiency for small integers. Quantitative analysis confirms that both classical and quantum runtime components exceed those of Shor’s algorithm under small-scale inputs, substantiating the conditional nature of Regev’s theoretical advantage. The contributions include: (1) the first open-source, reproducible implementation of Regev’s factoring algorithm; (2) the first empirically derived performance dataset; and (3) a rigorous characterization of its practical applicability boundaries—providing critical empirical foundations for post-Shor quantum cryptanalysis.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Algorithm Configuration
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Implementation of Regev's quantum factorization algorithm.
• Comparison between Regev's and Shor's algorithms.
• Performance analysis in quantum and classical components.

Innovation

• Implements Regev's quantum factorization
• Compares Regev's and Shor's algorithms
• Analyzes quantum simulation results