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Human computer interaction.

Post-Quantum Cryptography: Mathematical Foundations of Secure Communication in the Quantum Computing Era

Ilhan Arda Al

Published Oct 2, 2026• 0 citations
Publisher
DOI
10.62802/jhci.50

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The advent of quantum computing poses a significant threat to the foundational cryptographic algorithms that secure modern digital communications. [27]. Protocols such as HTTPS, digital certificates, and public key infrastructures (PKIs) heavily rely on cryptographic primitives like RSA, ECC, and Diffie-Hellman, which are vulnerable to quantum attacks—most notably Shor’s algorithm. [18]. This paper presents a comprehensive comparative analysis between classical cryptographic algorithms currently in widespread use and emerging post-quantum cryptographic schemes designed to withstand quantum adversaries. [24]. We review the cryptographic mechanisms underpinning modern internet security, outline the mathematical foundations of quantum attacks, and evaluate the security, performance, and implementation feasibility of quantum-resistant alternatives such as Kyber, Dilithium, and Falcon. [25]. Additionally, we assess the hybrid approaches currently being explored by institutions and tech companies to enable a smooth transition to post-quantum cryptography. [23]. By providing an in-depth comparison, this study aims to guide researchers, developers, and policymakers in understanding the critical implications of quantum computing on cryptographic infrastructures and the necessary steps for securing communications in the quantum era. [16].
As quantum computing continues its rapid progression, traditional cryptographic schemes based on the hardness of integer factorization and discrete logarithms, such as RSA and ECC, face obsolescence. This chapter explores the number theoretic underpinnings of Post-Quantum Cryptography (PQC), surveying emerging quantum-resistant protocols including lattice-based, code-based, multivariate polynomial, and isogeny-based cryptography. We analyze the mathematical assumptions behind their security, the role of hard problems in number theory (e.g., shortest vector problem, syndrome decoding, supersingular isogeny graphs), and the computational implications of these approaches. Furthermore, the chapter discusses the transition from classical cryptography to PQC within the NIST standardization process and how these cryptographic primitives align with the modern number theoretic and algorithmic challenges. Designed for both researchers and educators, this chapter aims to bridge theory and practice while emphasizing the continuing centrality of number theory in shaping the future of secure communication in the quantum era.