POST-QUANTUM CRYPTOGRAPHY AND ITS MATHEMATICAL FOUNDATIONS

Abstract

The rapid advancement of quantum computing technologies has posed a significant challenge to the security and reliability of classical cryptographic systems, particularly those based on integer factorization, discrete logarithms, and elliptic curve cryptography. Post-quantum cryptography (PQC) has therefore emerged as one of the most crucial research directions in modern information security, focusing on the design and analysis of cryptographic algorithms that remain secure even in the presence of powerful quantum adversaries. This article provides a comprehensive overview of post-quantum cryptography and its mathematical foundations, highlighting the theoretical underpinnings, algorithmic approaches, and current trends in this evolving field. The discussion begins with an examination of the fundamental mathematical problems upon which post-quantum cryptographic constructions are based, including lattice-based problems, error-correcting codes, multivariate polynomial equations, isogenies of elliptic curves, and hash-based cryptographic techniques. These problems are believed to be resistant to both classical and quantum attacks, making them suitable candidates for the next generation of secure communication protocols. The article also explores the complexity assumptions, reductions, and hardness proofs that establish the security of post-quantum schemes, thereby emphasizing the importance of rigorous mathematical reasoning in cryptographic design. Furthermore, the article analyzes the advantages and limitations of different PQC families, considering aspects such as key sizes, computational efficiency, implementation challenges, and resistance to side-channel attacks. Special attention is given to the ongoing standardization process led by the National Institute of Standards and Technology (NIST), which aims to select algorithms for widespread adoption across governmental and commercial applications. The paper also reflects on the future prospects of post-quantum cryptography in the context of global cybersecurity, digital infrastructure, and secure data transmission in the post-quantum era. The annotation highlights that the significance of post-quantum cryptography extends far beyond academic interest, as it directly addresses the urgent need for cryptographic resilience in finance, healthcare, defense, e-governance, and critical digital services. The mathematical foundations outlined in this article provide a bridge between abstract theory and practical cryptographic systems, ensuring that security in the digital age can withstand the transformative impact of quantum technologies.