A Systematic Review of Number-Theoretic Foundations of Post-Quantum Cryptographic Protocols: Methods, Architectures, and Future Research Directions

Abstract

The advent of quantum computing poses a fundamental threat to classical public-key cryptographic systems grounded in integer factorization and discrete logarithm problems. This paradigm shift has catalyzed the development of post-quantum cryptographic protocols rooted in advanced number-theoretic constructs such as lattices, error-correcting codes, multivariate polynomials, and isogeny-based systems. This paper presents a systematic review of number-theoretic foundations underpinning post-quantum cryptography, emphasizing their algorithmic structures, architectural implementations, and integration into modern software engineering ecosystems. The methodology involves a structured analysis of recent literature spanning 2018 to 2025, focusing on cryptographic primitives, security assumptions, performance trade-offs, and practical deployment considerations. The findings reveal a dominant shift toward lattice-based constructions due to their efficiency and strong worst-case hardness guarantees, alongside emerging hybrid models incorporating artificial intelligence for parameter tuning and security evaluation. The paper contributes a comprehensive synthesis of existing approaches, identifies critical research gaps in scalability and standardization, and outlines future directions for integrating post-quantum protocols into secure software pipelines.