Cryptographic methods today are a crucial tool for constructing information security systems. At the same time, to solve the problem of encrypting large amounts of information, block or stream symmetric ciphers are mainly preferred because of their efficiency and proven cryptographic strength, including against perspective quantum cryptanalysis. The effectiveness of modern symmetric ciphers largely depends on the cryptographic S-boxes applied in their construction, the quality of which largely determines the degree of implementation of the concepts of diffusion and confusion by the cryptographic algorithm, while the presence of large sets of cryptographically high-quality S-boxes is also important, in the terms of their application as a long-term key. Today, the Nyberg construction is well-known and widely applied in ciphers, including widespread AES block symmetric cipher. This construction allows you to synthesize high-quality S-boxes that harmoniously satisfy the main criteria for cryptographic quality, however, the set of S-boxes synthesized using this construction is small, which makes the task of developing new methods for synthesizing large sets of cryptographically high-quality S-boxes highly relevant. At the same time, as research shows, the constructions of extended Galois fields are a promising raw material for solving this problem. In this paper, the Galois field transform matrices of order N=256 are constructed for all isomorphic representations of the extended Galois field GF(256) which are analogous to the Reed-Muller transform but for the case of many-valued logic functions. As part of the research, the isomorphism invariant row numbers of the Galois field transform matrices are identified, which allows to obtain bijective S-boxes, as well as bijective S-boxes that correspond to the main criteria for cryptographic quality of component Boolean functions such as algebraic degree of nonlinearity, distance of nonlinearity, error propagation criterion, and criterion of minimization of correlation of output and input vectors of the S-box. At the same time, the cardinality of the set of synthesized S-boxes is ~23 times higher than the cardinality of the set of S-boxes of the Nyberg construction, which allows them to be used as a long-term key. The proposed S-boxes can become the basis for improving the effectiveness of existing symmetric cryptographic algorithms and developing new ciphers.
1. Shannon, C.E. (1949). Communication theory of secrecy systems. The Bell system technical journal, 28(4), 656-715.
https://doi.org/10.1002/j.1538-7305.1949.tb00928.x
2. Mazurkov, M.I., & Sokolov, A.V. (2013). Nonlinear transformations based on complete classes of isomorphic and automorphic representations of field GF (256). Radioelectronics and Communications Systems, 56(11), 513-521.
https://doi.org/10.3103/S0735272713110022
3. Sokolov, A.V., & Djiofack, T.V.N. (2019). Nonlinear Properties of Rijndael S-boxes Represented by the Many-Valued Logic Functions. Proceedings of the International Workshop on Cyber Hygiene, Kyiv, Ukraine, 96-106.
4. Rostovcev, A.G. (2002). Cryptography and Data Protection, SPb.: Mir i Sem'ja.
5. Logachev, O.A., Sal'nikov, A.A., & Jashhenko, V.V. (2012). Boolean functions in coding theory and cryptology, USA: American Mathematical Society.
6. Mazurkov, M.I., & Sokolov, A.V. (2013). Constructive method for synthesis of complete classes of multilevel de Bruijn sequences. Radioelectronics and Communications Systems, 56(1), 36-41.
https://doi.org/10.3103/S0735272713010044
7. Wang, J., Zhu, Y., Zhou, C., & Qi, Z. (2020). Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm. Symmetry, 12, 2115.
https://doi.org/10.3390/sym12122115
8. Lambić, D. (2018). S-box design method based on improved one-dimensional discrete chaotic map. Journal of Information and Telecommunication, 2(2), 181-191.
https://doi.org/10.1080/24751839.2018.1434723
9. Lu, Q., Zhu, C., & Wang, G. (2019). A Novel S-Box Design Algorithm Based on a New Compound Chaotic System. Entropy, 21(10), 1004.
https://doi.org/10.3390/e21101004
10. Lai, Q., Akgul, A., Li, C., Xu, G., Çavuşoğlu, U. (2018). A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design, Entropy, 20(1), 12.
https://doi.org/10.3390/e20010012
11. Hussain, I., Anees, A., Al-Maadeed, T., & Mustafa M. (2019). Construction of S-Box Based on Chaotic Map and Algebraic Structures. Symmetry, 11(3), 351.
https://doi.org/10.3390/sym11030351
12. FIPS 197. (2001) Advanced encryption standard. Retrieved from: http://csrc.nist.gov/publications/
13. Waheed, A., Subhan, F., Suud, M.M., Alam, M., & Ahmad, S. (2023). An analytical review of current S-box design methodologies, performance evaluation criteria, and major challenges. Multimedia Tools and Applications, 82(19), 1-24.
https://doi.org/10.1007/s11042-023-14910-3
14. Stankovic, R.S., Astola, J.T., & Moraga, C. (2012). Representations of Multiple-Valued Logic Functions. Morgan and Claypool Publishers, 2012.
https://doi.org/10.1007/978-3-031-79852-8
15. Sokolov, A.V., & Overchuk, Yu.S. (2018). On the possibility of synthesizing an algebraic normal form of quaternary functions over the field GF (4). Proceedings of the first international scientific and practical conference "Problems of cyber security of information and telecommunication systems", 384-388.
16. Grošek, O., & Fabšič, T. (2018). Computing multiplicative inverses in finite fields by long division, Journal of Electrical Engineering, 69(5), 400-402.
https://doi.org/10.2478/jee-2018-0059
17. Berlekamp, E.R. (2015) Algebraic coding theory: Revised Edition. World Scientific.
https://doi.org/10.1142/9407