Since the launch of the post-quantum cryptography standardization project by NIST, post-quantum cryptography has become a prominent research area. Non-commutative cryptography constructed using NP-complete problems is widely regarded resistant to quantum computing attacks, so it has become an important branch of post-quantum cryptography. Nonetheless, the existing non-commutative cryptographic protocol still exhibits certain shortcomings. In this paper, a non-commutative combinatorial semigroup with matrix power function is constructed from a modified medial semigroup by the method of semidirect product, and then a key exchange protocol is developed on it to provide a kind of novel non-commutative cryptographic protocol. Due to the non-commutativity of cryptographic platform, the proposed protocol could perform well in respect of antiquantum computing attack, which is superior to traditional cryptographic protocol. In addition, the security analysis shows that the protocol has significant advantages in resisting algebraic and brute force attacks, as well as aganist quantum cryptanalysis; the complexity analysis demonstrates that computation and storage complexities are of polynomial order, ensuring efficient operation even for large matrix sizes.
Since the launch of the post-quantum cryptography standardization project by NIST, post-quantum cryptography has become a prominent research area. Non-commutative cryptography constructed using NP-complete problems is widely regarded resistant to quantum computing attacks, so it has become an important branch of post-quantum cryptography. Nonetheless, the existing non-commutative cryptographic protocol still exhibits certain shortcomings. In this paper, a non-commutative combinatorial semigroup with matrix power function is constructed from a modified medial semigroup by the method of semidirect product, and then a key exchange protocol is developed on it to provide a kind of novel non-commutative cryptographic protocol. Due to the non-commutativity of cryptographic platform, the proposed protocol could perform well in respect of antiquantum computing attack, which is superior to traditional cryptographic protocol. In addition, the security analysis shows that the protocol has significant advantages in resisting algebraic and brute force attacks, as well as aganist quantum cryptanalysis; the complexity analysis demonstrates that computation and storage complexities are of polynomial order, ensuring efficient operation even for large matrix sizes.
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.
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 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.
We present in this paper an algorithm for exchanging session keys, coupled with a hashing encryption module. We show schemes designed for their potential invulnerability to classical and quantum attacks. In turn, if the parameters included were appropriate, brute-force attacks exceed the (five) security levels used in the NIST competition of new post-quantum standards. The original idea consists of products of rectangular matrices in Zp as public values and whose factorization is proved to be an NP-complete problem. We present running times as a function of the explored parameters and their link with operational safety. To our knowledge there are no classical and quantum attacks of polynomial complexity available at hand, remaining only the systematic exploration of the private-key space.
The development of quantum computing poses a very real threat to traditional cryptography, especially these, which are number-theoretic like RSA and ECC. With the more and more development of quantum algorithms such as Shor and Grover, there is an immediate necessity to come up with cryptographic protocols that are resistant to quantum attacks. This paper reviews and discusses different quantum-resistant (post-quantum) cryptographic schemes, such as lattice-based, code-based, multivariate, hash-based, and isogeny-based cryptography. It compares these protocols on the assumptions of their security and their efficiency in their computation and the size of their keys as well as their possible integrations into the existing communication infrastructure. Moreover, we also evaluate the standardization process by NIST and the difficulties of moving towards post-quantum security in practical use. This paper proposes a scalable comparison system, which correlates the hypothetical security basis of postquantum cryptography with the ability to effectively operate in the real world. It has emphasized protocol interoperability, migration compatibility and trade-offs in performance that gives a viable roadmap towards quantum-resistant cryptographic protocols implementation into current communication infrastructures whilst also providing scalable and future secure communication systems. This study, by emphasizing the potential and the drawbacks of the existing post-quantum cryptography schemes, helps in a more precise idea of the directions to follow on the way to a secure, future-proof communications scheme.
The rapid development of quantum computing poses a direct threat to RSA, DSA, and ECC modern cryptographic systems due to the Shor's algorithm potential application. In response to this threat, the NIST is conducting post-quantum cryptography standardization, having selected lattice-based and hash-function-based algorithms in 2022. Cryptosystems based on noncommutative groups, despite their potential resistance to quantum attacks owing to natural properties of noncommutative algebraic structures, were not included in the first set of standards due to the complexity of security analysis. This research conducts a comprehensive analysis of the cryptographic competitiveness of noncommutative group-based systems, evaluates their advantages and disadvantages compared to existing post-quantum solutions, and determines prospects for practical application as an alternative or complementary solution to ensure cryptographic diversification under quantum threats.