The influence of S-boxes on the arrival of the cipher to the state of random substitution

Iryna Lysytska, Kostiantyn Lysytskyi, Oleksii Nariezhnii, Tetiana Hrinenko

Abstract


The subject of this study is the process of developing block symmetric ciphers with improved cryptographic stability indicators for solving the problems of information protection in information and communication systems. The goal of this study is to develop a mathematical model of the optimal s-box design (non-linear substitution block) for modern block symmetric ciphers. Task: to show that the stability of block symmetric ciphers does not significantly depend on the indicators of the s-boxes included in these ciphers; to justify their use without reducing the stability of random substitutions as s-box constructions of modern block symmetric ciphers; and to prove the futility of searching for s-boxes with special properties. The methods used are: methods of probability theory; mathematical statistics; combinatorics and system analysis; methods of statistical tests; and methods of Boolean algebra. The following results were obtained. In modern ciphers, nonlinear substitution transformations are used as S-boxes (in most ciphers, byte substitutions are used). S-boxes perform the main function of encryption transformation - mixing of text bits. Dynamic indicators of the arrival of the cipher in the state of random substitution depend on their effectiveness. The presented results confirm that the indicators of cipher stability do not significantly depend on the indicators of the S-boxes included in the ciphers. It is not the individual properties of substitutions, but the results of their sequential interactions decisive for achieving stability indicators. It is shown that the fee for encryption is chosen to be three to four times greater than the depth of the avalanche effect - the minimum number of cycles of the cipher’s arrival to the state of random substitution) is hardly noticeable. Conclusions. The scientific novelty of the obtained results is as follows. This work represents a generalization of the transition to the use of random permutations as S-boxes is an increase of only one cycle in the number of cycles of the arrival of the cipher to the state of random permutations, and this is hardly noticeable against the background of the stability margin used in modern ciphers. The result of this generalization is a scientific statement that can be described as follows: random S-boxes can be used in all known modern ciphers without reducing stability. That is, the task of finding S-boxes with special properties loses any meaning, which means that the scientific direction related to the search for S-boxes with special properties becomes redundant.

Keywords


complete differential; randomness criteria; S-box; substitution; table of differential differences; table of linear approximations

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References


Dolgov, V. I., Lisitska, I. V., & Lisitskiy K. Yе. The new concept of block symmetric ciphers design. Telecommunications and Radio Engineering, 2017, vol. 76, no. 2, pp. 157184. DOI: 10.1615/TelecomRadEng.v76.i2.60.

Dolgov, V. I., & Lisitska, I. V., 2013. Blochnye simmetrichnye shifry. Metodologiia otsenki stoikosti k atakam differentsialnogo i lineinogo kriptoanaliza. [Block symmetric ciphers. Methodology for assessing the resistance to differential and linear cryptanalysis attacks]. Kharkiv, Fort Publ., 2013. 456 p.

Lisitskaya, I. V., Melnychuk, E. D., & Lisitskiy, K. E. Importance of S-Blocks in Modern Block Ciphers. International Journal of Computer Network and Information Security, 2012, vol .4, no. 10, pp. 1-12. DOI: 10.5815/ijcnis.2012.10.01.

Lambić, D., & Živković, M. Comparison of random S-box generation methods. Publications de L’institut Mathematique Nouvelle série, 2013, vol. 93, iss. 107, pp. 109-115. DOI: 10.2298/PIM1307109L.

Ruisanchez, C. P. A new algorithm to construct S-boxes with high diffusion. International Journal of Soft Computing, Mathematics and Control (IJSCMC), 2015, vol. 4, no. 3, pp. 41-50. DOI: 10.14810/ijscmc.2015.4303.

Kuznetsov, О. О., Gorbenko, Yu. І., Bilozertsev, І. М., Аndrushkevych, А. V., & Narizhnyi, О. P. Algebraic immunity of non-linear blocks of symmetric ciphers. Telecommunications and Radio Engineering, 2018, vol. 77, no. 4, pp. 309-325. DOI: 10.1615/TelecomRadEng.v77.i4.30.

DSTU 7624:2014. Informatsiyni tekhnolohiyi. Kryptohrafichnyy zakhyst informatsiyi. Alhorytm symetrychnoho blokovoho peretvorennya [DSTU 7624:2014, 2015. Information Technology. Cryptographic protection of information. Algorithm of symmetric block transformation]. Kyiv. Derzhspozhivstandard of Ukraine Publ., 2015. 238 p.

Rodinko, M. Yu, Oliynykov, R. V., & Hrinenko, T. O. Improvement of the method the optimal S-boxes generation. Applied Radio Electronics, 2015, vol. 14, no. 4, pp. 315-320. Available at: http://openarchive.nure.ua/handle/document/6469. (accessed 3.12.2022).

Lambić, D. S-box design method based on improved one-dimensional discrete chaotic map. Journal of Information and Telecommunication, 2018, vol. 2, iss. 2, pp. 181-191. DOI: 10.1080/24751839.2018.1434723.

Farhan, A. K., Ali, R. S., Yassein, H. R., Al-Saidi, N. M. G., & Abdul-Majeed, G. H. A new approach to generate multi S-boxes based on RNA computing. International Journal of Innovative Computing, Information and Control, 2020, vol. 16, no. 1, pp. 331-348. DOI: 10.24507/ijicic.16.01.331.

Sani, H. R., Behnia, S., & Akhshani, A. Creation of S-box based on a hierarchy of Julia sets: image encryption approach. Multidimensional Systems and Signal Processing, 2022, vol. 33, no. 1, pp. 39-62. DOI: 10.1007/s11045-021-00786-9.

Cassal-Quiroga, B. B., & Campos-Cantón, E. Generation of dynamical S-boxes for block ciphers via extended logistic map. Mathematical Problems in Engineering, 2020, vol. 2020, article no. 2702653, pp. 1-12. DOI: 10.1155/2020/2702653.

Kuznetsov, A., Wieclaw, L., Poluyanenko, N., Hamera, L., Kandiy, S., & Lohachova, Y. Optimization of a Simulated Annealing Algorithm for S-Boxes Generating. Sensors, 2022, vol. 22, iss. 16, article no. 6073. DOI: 10.3390/s22166073.

Marochok, S., & Zajac, P. Algorithm for Generating S-Boxes with Prescribed Differential Properties. Algorithms, 2023, vol. 16, iss. 3, article no. 157. DOI: 10.3390/a16030157.

Isa, H., Junid, S. A. A. S., Z’aba, M. R., Endut, R., Ammar, S. M., & Ali, N. Enhancement of Non-Permutation Binomial Power Functions to Construct Cryptographically Strong S-Boxes. Mathematics, 2023, vol. 11, iss. 2, article no. 446. DOI: 10.3390/math11020446.

Alsaif, H., Guesmi, R. Kalghoum, A., Alshammari, B. M., & Guesmi, T. A Novel Strong S-Box Design Using Quantum Crossover and Chaotic Boolean Functions for Symmetric Cryptosystems. Symmetry, 2023, vol. 15, iss. 4, article no. 833. DOI: 10.3390/sym15040833.

Lisickiy, K., Dolgov, V., Lisickaya, I., & Kuznetsova, K. Block Symmetric Cipher with Random S-boxes. International Journal of Computing, 2019, vol. 18, iss. 1, pp. 89-100. DOI: 10.47839/ijc.18.1.1278.

Evseev, S. P., Ostapov, S. E., & Korolev, R. V. Ispolzovanie mini-versii dlia otsenki stoikosti blochno-simmetrichnykh shifrov [Using mini-versions to assess the strength of block-symmetric ciphers]. Ukrainian Scientific Journal of Information Security, 2017, vol. 23, no. 2, pp. 100-108. DOI: 10.18372/2225-5036.23.11796.

Ruzhentsev, V. I. Proverka metoda dokazatelstva stoikosti blochnykh shifrov k atake nevypolnimykh differentsialov [Verification of a method for proving the resistance of block ciphers to the attack of impracticable differentials]. Applied Radio Electronics, 2016, vol. 15, no. 3, pp. 184-190. Available at: http://nbuv.gov.ua/UJRN/Prre_2016_15_3_10 (accessed 26.12.2022).




DOI: https://doi.org/10.32620/reks.2023.3.12

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