A General Introduction to Shor's Algorithm and Its Applications

Abstract

Shor's algorithm, a quantum algorithm for integer factorization, poses a fundamental threat to modern cryptographic systems, particularly RSA encryption. Classical factorization methods operate in sub-exponential time, making them impractical for large numbers. In contrast, Shor's algorithm leverages quantum principles-superposition, entanglement, and the Quantum Fourier Transform-to achieve polynomial-time factorization. This breakthrough significantly reduces the security of encryption protocols that underpin global financial, governmental, and cybersecurity infrastructures. While practical quantum computers capable of executing Shor's algorithm at scale are not yet available, rapid advancements in quantum computing necessitate the development of quantum-resistant cryptographic methods. This paper briefly outlines the theoretical foundations of Shor's algorithm, its computational advantages over classical factorization, and the broader implications for cybersecurity in the quantum era.

Keywords: Shor's Algorithm, Quantum Computing, RSA Encryption, Integer Factorization, Cryptography, Post-Quantum Security