This paper studies how post-quantum cryptographic (PQC) security assumptions can be represented and communicated through a structured, layered framework that is useful for technical interpretation but does not replace formal cryptographic proofs. We propose ``Explainable PQC,'' an interdisciplinary framework connecting three layers: (1) a complexity-based interpretive model that distinguishes classical security, quantum security, and reduction-backed hardness, drawing on computational complexity classes as supporting language; (2) an exploratory mathematical investigation applying combinatorial Hodge theory and polyhedral geometry to study structural aspects of lattice hardness; and (3)~an empirical experimentation platform, implemented in Julia, for measuring the behavior of lattice basis reduction algorithms (LLL, BKZ) in low-dimensional settings. The motivating case study throughout the paper is lattice-based PQC, including ML-KEM (FIPS 203) and ML-DSA (FIPS 204). The contribution of this paper is conceptual and organizational: it defines a layered interpretive framework, clarifies its scope relative to formal cryptographic proofs and reduction-based security arguments, and identifies mathematical and implementation-level directions through which PQC security claims may be more transparently communicated. This paper does not claim new cryptographic hardness results, new attacks, or concrete security parameter estimates.
This paper studies how post-quantum cryptographic (PQC) security assumptions can be represented and communicated through a structured, layered framework that is useful for technical interpretation but does not replace formal cryptographic proofs. We propose ``Explainable PQC,'' an interdisciplinary framework connecting three layers: (1) a complexity-based interpretive model that distinguishes classical security, quantum security, and reduction-backed hardness, drawing on computational complexity classes as supporting language; (2) an exploratory mathematical investigation applying combinatorial Hodge theory and polyhedral geometry to study structural aspects of lattice hardness; and (3)~an empirical experimentation platform, implemented in Julia, for measuring the behavior of lattice basis reduction algorithms (LLL, BKZ) in low-dimensional settings. The motivating case study throughout the paper is lattice-based PQC, including ML-KEM (FIPS 203) and ML-DSA (FIPS 204). The contribution of this paper is conceptual and organizational: it defines a layered interpretive framework, clarifies its scope relative to formal cryptographic proofs and reduction-based security arguments, and identifies mathematical and implementation-level directions through which PQC security claims may be more transparently communicated. This paper does not claim new cryptographic hardness results, new attacks, or concrete security parameter estimates.
Abstract: This study investigates Quantum-AI number-theoretic frameworks for post-quantum cryptography (PQC) and long-term information security. The paper positions lattice-based, hash-based and code-based cryptographic standards as formal security substrates for a future in which adversaries may combine classical cyber operations, artificial intelligence and quantum computation. The proposed framework treats cryptography not merely as an algorithmic primitive but as a socio-technical assurance layer that links formal number theory, cryptographic agility, AI-assisted verification, hardware deployment and governance-grade migration planning. The analysis is motivated by the harvest-now-decrypt-later threat, the standardisation of ML-KEM, ML-DSA and SLH-DSA, and the selection of HQC as a backup encryption mechanism for resilience against unexpected cryptanalytic progress. The methodological contribution is a Literature-Reasoning (LR) framework that identifies mathematical research strategies through a reasoning agent, converts tentative proofs and design claims into machine-checkable representations through a proof agent, and validates each step to reduce hallucination, informal overclaiming and migration error. The empirical component uses real standards-derived data from NIST PQC publications, including parameter sizes, key/ciphertext footprints and signature footprints, together with a structured migration scorecard for long-term information security. Results show that PQC migration is not a one-dimensional replacement problem: stronger security categories increase payload and storage costs, while governance maturity, software dependency visibility and formal verification capacity determine whether long-term confidentiality can be preserved across finance, telecom, health, industry, government and cloud infrastructure. Keywords: post-quantum cryptography; quantum artificial intelligence; number theory; ML-KEM; ML-DSA; SLH-DSA; HQC; crypto-agility; long-term security; formal verification
The emergence of quantum computing is a serious threat to traditional cryptographic algorithms, making it critical to transition to post-quantum cryptography (PQC). This work proposes a holistic conceptual and analytical framework for deploying PQC in multi-cloud environments where the complexity of different infrastructure, workloads and key management schemes pose significant security and efficiency problems. The proposed framework for crypto-agility follows a layered approach, with a cryptographic abstraction layer, a policy-based orchestration engine, a centralized hardware security module (HSM) management plane, and an ongoing monitoring module to facilitate a smooth transition from classical to hybrid and fully quantum-resistant cryptographic systems. He researches identifies and critically analyzes major challenges in deploying PQC technologies in multi-cloud environments, such as inter-cloud interoperability, computational complexity, latency, key management, and regulatory and compliance considerations. The framework tackles these challenges by incorporating hybrid cryptographic protocols that blend classic cryptographic algorithms with NIST-approved PQC algorithms, namely ML-KEM (FIPS 203) and ML-DSA (FIPS 204), to maintain backward compatibility and increase security against quantum-powered adversaries. Moreover, the framework proposes dynamic policy-driven algorithm selection to implement context-sensitive cryptographic enforcement based on sensitivity labels, security threats, and regulatory considerations. An experimental framework is proposed to assess the efficiency and scalability of PQC algorithms deployed in a multi-cloud environment, including metrics such as CPU usage, memory usage, latency and throughput with different workloads. Although this research is conceptual, it draws on previous benchmarking to discuss the securityperformance trade-offs associated with PQC. The results highlight the need for centralized trust anchors, crypto-agility and hybrid deployment models to address risks from cryptographic diversity and ensure the sustainability of security operations. This study advances the research on quantum-safe cloud security by offering a systematic, scalable and policy-oriented framework for PQC deployment. It provides practical guidance for businesses, cloud architects, and security professionals on how to safeguard against quantum vulnerabilities and ensure efficient operations in distributed systems by implementing quantum-safe cryptographic infrastructures.
The advent of quantum computing poses an existential threat to contemporary public-key cryptographic systems, which rely on the computational hardness of problems such as integer factorization and discrete logarithms. Shor's algorithm, capable of solving these problems efficiently on a sufficiently powerful quantum computer, necessitates a paradigm shift towards post-quantum cryptography (PQC). This paper proposes a unified security framework that bridges the conceptual gap between traditional number-theoretic hardness assumptions and the lattice-based problems central to PQC. We delve into the foundational mathematical problems, comparing the security arguments of factoring and discrete logarithm problems with the Shortest Vector Problem (SVP) and its variants in lattices. The framework emphasizes a consistent methodology for assessing quantum resistance, facilitating a more holistic understanding of cryptographic security in the post-quantum era. By analyzing the inherent complexities and interconnections of these hard problems, we aim to provide a structured approach to evaluate and design quantum-resistant cryptographic schemes, fostering robust and future-proof digital security infrastructure. This work contributes to the ongoing efforts in PQC standardization by offering a comparative lens for diverse cryptographic primitives.
Background: The imminent threat posed by cryptographically relevant quantum computers to classical public-key infrastructure specifically RSA and Elliptic-Curve Cryptography (ECC), whose security rests on the conjectured computational intractability of integer factorisation and the discrete logarithm problem respectively has propelled lattice-based cryptography to the centre of global post-quantum standardisation efforts. On 13 August 2024, the United States National Institute of Standards and Technology (NIST) published three landmark Federal Information Processing Standards (FIPS): FIPS 203 (ML-KEM, derived from CRYSTALS-Kyber), FIPS 204 (ML-DSA, derived from CRYSTALS-Dilithium), and FIPS 205 (SLH-DSA, derived from SPHINCS+), with FIPS 206 (FN-DSA, derived from FALCON) in advanced draft. The first three of these standards rest on algebraic structures intimately connected to classical number theory: polynomial rings over integer quotients, cyclotomic fields, and the algebraic geometry of ideal lattices in rings of integers of number fields. Objectives: This article provides: (1) a rigorous mathematical exposition of the lattice problems underpinning NIST’s post-quantum standards, specifically the Learning With Errors (LWE), Ring-LWE (RLWE), Module-LWE (MLWE), Short Integer Solution (SIS), and NTRU problems, and their connections to algebraic number theory; (2) a comparative performance analysis of NIST-standardised lattice-based algorithms against classical RSA and ECC benchmarks; (3) a security analysis examining known classical and quantum attacks on each hard problem; (4) a comparative evaluation of the four NIST-selected algorithms across key size, signature size, and computational benchmarks; and (5) a strategic assessment of the migration challenges facing deployed cryptographic infrastructure. Methods: The study synthesises peer-reviewed literature, NIST official standards documentation (FIPS 203, 204, 205, and draft FIPS 206), and independently published performance benchmarks published between June 2024 and June 2026. Mathematical security analysis is grounded in the worst-case to average-case reductions established by Regev (2005) for LWE and by Lyubashevsky, Peikert, and Regev (2013) [11, 18] for Ring-LWE. Performance data are drawn from Garani (2024) [8], the ARM Cortex-M0+ benchmarking study of Ratnayaka and Fischer (2026) [4], and the cross-platform evaluation by Alzahrani et al. (2025) [2]. Results: NIST’s three August 2024 standards establish a comprehensive lattice-based post-quantum portfolio. ML-KEM-768 produces a 1,184-byte public key and 1,088-byte ciphertext, achieving IND-CCA2 security under the MLWE assumption at NIST Level 3 (AES-192 equivalent). ML-DSA-65 produces 1,952-byte public keys and 3,309-byte signatures under the Module-LWE and Module-SIS assumptions at Level 3. FN-DSA-512 (FALCON, draft FIPS 206) produces 897-byte public keys and 666-byte signatures under the NTRU assumption, offering the most compact lattice-based signatures. Performance benchmarks on an Intel Core i7-12700K (3.6 GHz) confirm that ML-KEM key generation is orders of magnitude faster than RSA-3072 key generation. The algebraic structure of cyclotomic polynomial rings ∑₂ⁿ = Z[x]/(x^n + 1) is central to the efficiency of all three lattice-based NIST standards. Conclusion: Lattice-based cryptography, grounded in the algebraic number theory of ideal lattices over cyclotomic fields, provides the most comprehensive and mathematically well-founded response to the quantum computational threat. The August 2024 NIST standardisation of ML-KEM, ML-DSA, and SLH-DSA represents a landmark in cryptographic history, and the imminent finalisation of FN-DSA will complete the post-quantum toolbox. Organisations should prioritise immediate migration planning, leveraging hybrid classical-quantum schemes during transition, with full transition to NIST post-quantum standards recommended by 2030 per NSA and CISA guidance.
The rapid development of quantum computing technologies poses a serious threat to modern public-key cryptographic systems. Classical cryptographic algorithms such as RSA and elliptic-curve cryptography rely on mathematical problems that can be efficiently solved using quantum algorithms. Post-quantum cryptography (PQC) aims to develop cryptographic schemes that remain secure even in the presence of quantum adversaries. This paper investigates the mathematical foundations of PQC, focusing on lattice-based, code-based, and hash-based cryptographic constructions. Security models based on computational complexity theory are analyzed, and illustrative examples using Learning With Errors (LWE) and decoding problems are presented. Experimental analysis demonstrates that PQC schemes provide strong resistance against both classical and quantum attacks when appropriate parameters are selected. The results highlight the importance of adopting new cryptographic standards to ensure long-term data security in quantum computing environments.
Abstract It is viewed as revolutionary technology, offering incredible potential in terms of computing power within different scientific and industrial sectors. Quantum computers provide remarkable computing power; however, they also present significant security risks to existing public-key cryptography systems used to secure digital communication, online banking systems, government, medical applications, cloud computing systems, and IoT systems. Security of commonly used encryption schemes like RSA, DH, and ECC depends on mathematical problems that are impossible for classical computers to solve but can be easily resolved with the help of powerful enough quantum computers by using Shor’s algorithm. Furthermore, Grover’s algorithm makes exhaustive search attacks much faster, thus effectively lowering the security level of symmetric encryption schemes since it provides the speed-up for brute-force attacks. Post-Quantum Cryptography (PQC) turned out to be an effective and scalable method to protect digital data from attacks using quantum computing. In contrast to quantum cryptography, which requires a special quantum communication channel, PQC uses classical cryptographic algorithms that cannot be broken down by either classical or quantum computer due to certain mathematical problems. This research paper offers an extensive exploration of post-quantum cryptography through the analysis of the development of quantum computing, weaknesses of current cryptographic techniques, and mathematics behind quantum-resistant algorithms. Moreover, this paper explores the major classes of PQC techniques such as lattice-based, code-based, hash-based, multivariate, and isogeny-based cryptography and provides details about their security and application. Moreover, the report gives an overview of the NIST Post-Quantum Cryptography Standardization Project and analyzes recent standardizations of quantum-proof algorithms to be utilized in future digital infrastructures. Keywords: Post-Quantum Encryption, Quantum Security, Quantum-Proof Public-Key Cryptography, Lattice-Based Security Protocols, ML-KEM Key Encapsulation Mechanism, ML-DSA Digital Signature Algorithm, NIST Post-Quantum Cryptography, Contemporary Cybersecurity, Secure Communications, Cryptographic Algorithms.