Abstract This work introduces a new post-quantum key exchange protocol based on the computational hardness of the Nilpotent Commutator Inversion Problem (NCIP) in Heisenberg Lie algebras over GF(p), where p ≈ 2256. Compared to traditional methods based on integer factorization or discrete logarithms, our scheme relies on the non-linear algebraic properties of nilpotent matrices for quantum security. The protocol performs extremely well in sparse environments with 64-byte public keys requiring 0.024ms for 3 × 3 matrices and 128-bit post-quantum security. Through comprehensive comparison with NIST-standardized schemes (Kyber, SPHINCS+), we show superior performance in memory footprint (0.01KB) and computational overhead, particularly for IoT applications. Experimental evaluation confirms 95.1% success rates for 8 × 8 matrices (256-bit security) with sub-millisecond latency, while security analysis establishes resistance to both classical linear algebra attacks and quantum algorithms (Shor’s, Grover’s) through the NCIP hardness assumption. The work provides concrete parameter recommendations aligned with NIST security levels and demonstrates practical implementation in Python using optimized matrix operations over GF(pn). This Lie algebra-based approach offers a mathematically distinct alternative to existing post-quantum primitives, with particular advantages for scalable, lightweight deployments in edge computing and large-scale networks. Future work directions include hardware acceleration, formal security reductions, and integration with low-power network protocols.
Abstract This work introduces a new post-quantum key exchange protocol based on the computational hardness of the Nilpotent Commutator Inversion Problem (NCIP) in Heisenberg Lie algebras over GF(p), where p ≈ 2256. Compared to traditional methods based on integer factorization or discrete logarithms, our scheme relies on the non-linear algebraic properties of nilpotent matrices for quantum security. The protocol performs extremely well in sparse environments with 64-byte public keys requiring 0.024ms for 3 × 3 matrices and 128-bit post-quantum security. Through comprehensive comparison with NIST-standardized schemes (Kyber, SPHINCS+), we show superior performance in memory footprint (0.01KB) and computational overhead, particularly for IoT applications. Experimental evaluation confirms 95.1% success rates for 8 × 8 matrices (256-bit security) with sub-millisecond latency, while security analysis establishes resistance to both classical linear algebra attacks and quantum algorithms (Shor’s, Grover’s) through the NCIP hardness assumption. The work provides concrete parameter recommendations aligned with NIST security levels and demonstrates practical implementation in Python using optimized matrix operations over GF(pn). This Lie algebra-based approach offers a mathematically distinct alternative to existing post-quantum primitives, with particular advantages for scalable, lightweight deployments in edge computing and large-scale networks. Future work directions include hardware acceleration, formal security reductions, and integration with low-power network protocols.
The advent of quantum computing has precipitated an estimated attack resistant cryptography system necessitating quantum attacks. The classical asymmetric algorithms, i.e., RSA and ECC are susceptible to Shor algorithm and a transition should be made to Post-Quantum Cryptography (PQC) anymore. The novelty of this work lies in providing a unified comparative framework that analyzes PQC families through security assumptions, concrete key/ciphertext sizes, and implementation complexity an aspect not addressed collectively in prior surveys. This survey gives a general picture of the status of the design process and general evaluation of PQC accounting its application within the paradigm of the diversified networks including the IoT and 5G/6G and in the cyber-physical environment. Furthermore, this review uniquely synthesizes practical hybrid‑deployment strategies, including protocol layering, coexistence models, and backward‑compatibility mechanisms, offering actionable guidance for real‑world migration. Additionally, the manuscript contributes a prioritized list of open problems across IoT, 5G/6G, blockchain, and cyber‑physical systems, highlighting domains facing the most urgent quantum‑resilience challenges. In this paper, the authors will evaluate the most popular PQC schemes, including schemes based on lattice, code, hash, and isogeny, depending on their performance, scalability, and interoperability (with reference to the recent research). It also examines the problems of integration as implemented with respect to the computational overhead, latency and adoption of hybrid with classical cryptography. This is expounded more to the standardization of PQC by NISTs and with a focus on such algorithms as CRYSTALS-Kyber and Dilithium. The findings suggest that, hybrid PQC protocols are able to control quantum resistance and actual performance in numerous infrastructures in computing systems.
The emergence of quantum computer will threaten the security of existing public-key cryptosystems, including the Diffie Hellman key exchange protocol, encryption scheme and etc, and it makes the study of resistant quantum cryptography very urgent. This motivate us to design a new key exchange protocol and encryption scheme in this paper. Firstly, some acknowledged mathematical problems was introduced, such as ergodic matrix problem and tensor decomposition problem, the two problems have been proved to NPC hard. From the computational complexity prospective, NPC problems have been considered that there is no polynomial-time quantum algorithm to solve them. From the algebraic structures prospective, non-commutative cryptography has been considered to resist quantum. The matrix and tensor operator we adopted also satisfied with this non-commutative algebraic structures, so they can be used as candidate problems for resisting quantum from perspective of computational complexity theory and algebraic structures. Secondly, a new problem was constructed based on the introduced problems in this paper, then a key exchange protocol and a public key encryption scheme were proposed based on it. Finally the security analysis, efficiency, recommended parameters, performance evaluation and etc. were also been given. The two schemes has the following characteristics, provable security, security bits can be scalable, to achieve high efficiency, quantum resistance, and etc.
Modern public-key cryptography relies on the hardness of mathematical problems such as integer factorization and discrete logarithms. However, the development of quantum computing poses an imminent threat to these assumptions. Shor’s algorithm, in particular, can factor large semiprimes exponentially faster than classical algorithms, compromising systems like RSA, DSA, and ECC. This paper explores the mathematical foundations of pre-quantum cryptography, discusses the limitations of classical security models when confronted with quantum capabilities, and then pays attention to post-quantum cryptography (PQC), a field dedicated to developing cryptographic schemes resilient against both classical and quantum attacks. Among the proposed families, this paper focuses specifically on hash function–based cryptography for its simplicity and minimal reliance on algebraic structure. This study focuses in particular on SPHINCS+, a stateless hash-based digital signature scheme currently under consideration by NIST. Through detailed mathematical explanation and a visual example, we analyze its construction using Winternitz One-Time Signatures and Merkle trees. The results highlight SPHINCS+ as a robust candidate for post-quantum security due to its reliance on well-understood hash primitives and its resistance to known quantum algorithms such as Grover’s. Finally, this paper discusses ongoing challenges such as performance trade-offs, standardization, and real-world deployment. This research underscores the urgency of adopting quantum-resistant cryptographic systems before large-scale quantum computers become a reality.
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.
The advance of quantum computing poses significant risks to traditional cryptographic systems, including RSA and ECC, due to algorithms such as Shor's, prompting an urgent need for robust post-quantum cryptography (PQC) solutions. This paper presents a systematic review of PQC algorithms, encompassing lattice-based, code-based, multivariate polynomial, hash-based, and supersingular isogeny-based cryptography. By evaluating recent publications and NIST standardization efforts, we assess these approaches in terms of quantum resistance, computational efficiency, scalability, and implementation challenges. Lattice-based schemes, notably CRYSTALS-Kyber, excel in quantum-secure encryption and performance, whereas codebased systems like McEliece are hindered by large key sizes. Hash-based signatures, such as SPHINCS+, and isogeny-based protocols, including SIDH, provide promising alternatives but require further optimization to reduce computational overhead. Our analysis highlights the potential of hybrid PQC methods and lightweight hardware implementations for securing emerging domains like IoT, blockchain, and financial systems. The review serves as a foundational resource for researchers and practitioners, advocating the adoption of quantum-resistant cryptography to preserve data confidentiality, integrity, and authenticity in the quantum era.
The advent of quantum computing poses an existential threat to contemporary cryptographic standards, particularly those securing decentralized blockchain networks and cloud infrastructures.Classical public-key cryptosystems such as RSA, ECC, and DH, which rely on factorization and discrete logarithm problems, are rendered obsolete by Shor's algorithm, necessitating the transition toward post-quantum cryptographic (PQC) solutions.This study explores the integration of PQC algorithms, including lattice-based, hash-based, code-based, multivariate, and isogeny-based cryptographic mechanisms, within blockchain-ledger technologies and cloud architectures to ensure long-term security against quantum adversaries.A comparative analysis is conducted to evaluate computational efficiency, key size implications, communication overhead, and security resilience under quantum attack models.The research highlights the adaptation of PQC within blockchain consensus mechanisms, smart contract execution, and cryptographic primitives such as digital signatures, zero-knowledge proofs, and secure multi-party computation (MPC).Additionally, it examines the impact of PQC on cloud security, addressing challenges in quantum-safe key exchange protocols, homomorphic encryption for secure computations, and cross-platform interoperability within hybrid quantum-classical cloud ecosystems.Real-world implementations and benchmarking data provide insights into the feasibility of large-scale adoption, shedding light on standardization efforts by NIST and industry consortia.The study concludes with future directions, emphasizing the need for efficient PQC algorithm optimization, lightweight cryptographic frameworks for IoT-driven blockchain applications, and scalable post-quantum identity management systems.By establishing quantum-resistant security frameworks, this research underscores the imperative need for early adoption to mitigate cryptographic vulnerabilities in the impending post-quantum era.