Quantum Threats to Classical Cryptography: A Mathematical and Algorithmic Perspective

Abstract

Quantum computing offers transformative computational power by harnessing phenomena such as superposition and entanglement. Yet, this potential simultaneously poses severe risks to classical cryptographic systems. In this paper, we explore how quantum algorithms, most notably Shor’s algorithm for integer factorization and Grover’s algorithm for unstructured search, could break widely deployed cryptographic schemes. We present a rigorous mathematical analysis of these quantum algorithms, compare their complexity with classical methods, and examine their impact on public-key and symmetric cryptography. Finally, we discuss potential countermeasures, including post-quantum cryptographic solutions, to safeguard data in the quantum era.