A software model to generate permutation keys through a square matrix
Abstract
Information security and data protection are among the key aspects, which should be intensively developing in the 21st century. A conventional approach to cryptographic algorithms offers to apply matrices to represent information. However, more recent approaches deploy other data structures, including permutations, thus necessitating accordance between differing data structures to integrate different methods into a wholistic system of processing and transmitting information. This study aims to generate permutations, which serve as a key for factorial data coding according to a known key matrix. The paper presents two algorithms for transforming a square matrix into a permutation. An example of matrix transformation following each of the proposed algorithms is given. A software model was created and described to investigate the transformation of square matrices into permutations with the Matlab software product. The authors have considered the built-in methods of statistical information processing in the Matlab program and their graphical representation by built-in functions, which are applied in the process of the software model. A matrix transformation has been performed according to the proposed algorithms. The paper investigates all possible combinations of a square matrix of order 2 with elements referring to the finite integer field modulo p = 17 and p = 23. According to each transforming algorithm, the results of a square matrix transforming into a permutation number are obtained in the lexicographic order. The statistical properties of the obtained results have been studied, and the most efficient algorithm for transforming matrices into permutations has been determined based on the distribution uniformity criterion for the generated permutation numbers. The study demonstrates that this algorithm can potentially be deployed in information exchange systems based on factorial data coding
Keywords
factorial data coding; data security; key agreement; statistics; uniform distribution
References
[1] Al-Shaarani, F., & Gutub, A. (2022). Securing matrix counting-based secret-sharing involving crypto steganography. Journal of King Saud University - Computer and Information Sciences, 34(9), 6909-6924. doi: 10.1016/j.jksuci.2021.09.009.
[2] Bleichenbacher Attack Explained. (2019). Retrieved from https://medium.com/@c0D3M/bleichenbacher-attack-explained-bc630f88ff25.
[3] Chernoff, H., & Lehmann, E.L. (1954). The use of maximum likelihood estimates in x2 tests for goodness of fit. The Annals of Mathematical Statistics, 25(3), 579-586. doi: 10.1214/aoms/1177728726.
[4] Gafsi, M., Abbassi, N., Hajjaji, M.A., Malek, J., & Mtibaa, A. (2020). Improved chaos-based cryptosystem for medical image encryption and decryption. Scientific Programming, 2020, article number 6612390. doi: 10.1155/2020/6612390.
[5] Greub, W. (1975). Linear algebra. New York: Springer. doi: 10.1007/978-1-4684-9446-4.
[6] Higham, D.J., & Higham, N.J. (2005). MATLAB guide. Philadelphia: Society for Industrial and Applied Mathematics.
[7] Huang, H., Li, C., & Deng, L. (2022). Public-key cryptography based on tropical circular matrices. Applied Sciences, 12(15), article number 7401. doi: 10.3390/app12157401.
[8] Issad, M., Anane, N., Bellemou, A.M., & Boudraa, B. (2020). Secure hybrid crypto-system AES/RSA on FPGA for data communication. Malaysian Journal of Computing and Applied Mathematics, 3(1), 1-10. doi: 10.37231/myjcam.2020.3.1.38.
[9] Joshi, S., Bairwa, A.K., Pljonkin, A.P., Garg, P., & Agrawal, K. (2023). From pre-quantum to post-quantum RSA. In Proceedings of the 6th international conference on networking, intelligent systems & security (pp. 1-8). doi: 10.1145/3607720.3607721.
[10] Kaptiol, Ye., & Horbenko, I. (2020). Analysis of the possibilities and peculiarities of programming cryptology problems on a quantum computer. Radiotekhnika, 3(202), 37-48. doi: 10.30837/rt.2020.3.202.03.
[11] Karatas, Z.Y., Luy, E., & Gonen, B. (2019). A public key cryptosystem based on matrices. International Journal of Computer Applications, 182(42), 47-50. doi: 10.5120/ijca2019918432.
[12] Kelesidis, E.-A. (2022). An optimization of Bleichenbacher’s oracle padding attack. In P.Y. Ryan & C. Toma (Eds.), Innovative security solutions for information technology and communications (pp. 145-155). Cham: Springer. doi: 10.1007/978-3-031-17510-7_10.
[13] Lavdanskyi, A., Faure, E., Skutskyi, A., & Bazilo, C. (2023). Accelerating operations on permutations using graphics processing units. In E. Faure, O. Danchenko, M. Bondarenko, Y. Tryus, C. Bazilo & G. Zaspa (Eds.), Information technology for education, science, and technics (pp. 3-12). Cham: Springer. doi: 10.1007/978-3-031-35467-0_1.
[14] Li, J., Yan, M., Peng, J., Huang, H., & El-Latif, A.A.A. (2024a). A lattice-based efficient certificateless public key encryption for big data security in clouds. Future Generation Computer Systems, 158, 255-266. doi: 10.1016/j.future.2024.04.039.
[15] Li, P., et al. (2024b). Scalable parallel ultrafast optical random bit generation based on a single chaotic microcomb. Light: Science & Applications, 13(1), article number 66. doi: 10.1038/s41377-024-01411-7.
[16] Mann, H.B., & Wald, A. (1942). On the choice of the number of class intervals in the application of the Chi square test. The Annals of Mathematical Statistics, 13(3), 306-317. doi: 10.1214/aoms/1177731569.
[17] Maturin, Yu., Komarnytska, L., & Hordiienko, I. (2023). Discrete math. Drohobych: Drohobych Ivan Franko State Pedagogical University.
[18] Maxrizal, M. (2022). Public key cryptosystem based on singular matrix. Trends in Sciences, 19(3), article number 2147. doi: 10.48048/tis.2022.2147.
[19] Naseri, A.R., Abbasi, A., & Atani, R.E. (2023). A new public key cryptography using Mq matrix. Journal of Mathematical Modeling, 11(4), 681-693. doi: 10.22124/jmm.2023.23982.2142.
[20] National Institute for Standards and Technology. (2001). Specification for the Advanced Encryption Standard (AES). Retrieved from https://nvlpubs.nist.gov/nistpubs/fips/nist.fips.197.pdf.
[21] Rani, S., Bhambri, P., Kataria, A., Khang, A., & Sivaraman, A.K. (2023). Big data, cloud computing and IoT: Tools and applications. Boca Raton: Chapman and Hall/CRC. doi: 10.1201/9781003298335.
[22] Sheikhpour, S., Mahani, A., & Bagheri, N. (2021). Reliable advanced encryption standard hardware implementation: 32- bit and 64-bit data-paths. Microprocessors and Microsystems, 81, article number 103740. doi: 10.1016/j.micpro.2020.103740.
[23] Shvydkyi, V., Shcherba, A., Kharin, O., Lavdanskyi, A., & Faure, E. (2021). Basics theory of inseparable factorial data coding. Kharkiv: Novyi Kurs.
[24] Turchyn, V. (2014). Probability theory and mathematical statistics. Dnipro: IMA-Press.
[25] Valentine, D.T., & Hahn, B.H. (2023). Essential MATLAB for engineers and scientists. London: Academic Press.