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Advances in Applied Mathematics

Mathematical Analysis of Public Key Encryption Algorithm and Security of Anti Quantum Cryptography

彪高泽 唐

Published Jan 1, 2018• 0 citations
Publisher
DOI
10.12677/aam.2018.710147

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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.
Newer variants of the Fujisaki–Okamoto transformation are used in most candidates of the third round of the NIST Post-Quantum Cryptography standardization call in the category of public key encryption schemes. These transformations are applied to obtain a highly secure key encapsulation mechanism from a less secure public key encryption scheme. Furthermore, there are five candidates (three finalists and two alternatives) that passed to the third round of the process and whose security is based in lattice problems. This work analyzes the different ways in which the lattice-based candidates of the NIST call apply the Fujisaki–Okamoto transformation and the particularities of each application. The study of such differences and their repercussion in the design of the proposals will allow a better understanding of the algorithms. Moreover, we propose a modification of the Kyber algorithm—the only public key encryption candidate established as a PQC standard by NIST in its more recent publication—in order to avoid the re-encryption in the decapsulation algorithm and, in this way, to reduce the side channel attacks vulnerability.
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.