{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,8]],"date-time":"2025-11-08T18:01:26Z","timestamp":1762624886203,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2022,8,14]],"date-time":"2022-08-14T00:00:00Z","timestamp":1660435200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Cryptographic algorithms are used to ensure confidentiality, integrity and authenticity of data in information systems. One of the important areas of modern cryptography is that of symmetric key ciphers. They convert the input plaintext into ciphertext, representing it as a random sequence of characters. S-boxes are designed to complicate the input\u2013output relationship of the cipher. In other words, S-boxes introduce nonlinearity into the encryption process, complicating the use of different methods of cryptanalysis (linear, differential, statistical, correlation, etc.). In addition, S-boxes must be random. This property means that nonlinear substitution cannot be represented as simple algebraic constructions. Random S-boxes are designed to protect against algebraic methods of cryptanalysis. Thus, generation of random S-boxes is an important area of research directly related to the design of modern cryptographically strong symmetric ciphers. This problem has been solved in many related works, including some using the simulated annealing (SA) algorithm. Some works managed to generate 8-bit bijective S-boxes with a nonlinearity index of 104. However, this required enormous computational resources. This paper presents the results of our optimization of SA via various parameters. We were able to significantly reduce the computational complexity of substitution generation with SA. In addition, we also significantly increased the probability of generating the target S-boxes with a nonlinearity score of 104.<\/jats:p>","DOI":"10.3390\/s22166073","type":"journal-article","created":{"date-parts":[[2022,8,15]],"date-time":"2022-08-15T23:44:03Z","timestamp":1660607043000},"page":"6073","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Optimization of a Simulated Annealing Algorithm for S-Boxes Generating"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2331-6326","authenticated-orcid":false,"given":"Alexandr","family":"Kuznetsov","sequence":"first","affiliation":[{"name":"Department of Political Sciences, Communication and International Relations, University of Macerata, Via Crescimbeni, 30\/32, 62100 Macerata, Italy"},{"name":"Department of Information and Communication Systems Security, Faculty of Comupter Science, V. N. Karazin Kharkiv National University, 4 Svobody Sq., 61022 Kharkiv, Ukraine"},{"name":"Department of Information Systems and Technologies Security, JSC \u201cInstitute of Information Technologies\u201d, Bakulin St., 12, 61166 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4866-6122","authenticated-orcid":false,"given":"Lukasz","family":"Wieclaw","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Automatics, Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biala, Willowa 2, 43-300 Bielsko-Biala, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikolay","family":"Poluyanenko","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Systems Security, Faculty of Comupter Science, V. N. Karazin Kharkiv National University, 4 Svobody Sq., 61022 Kharkiv, Ukraine"},{"name":"Department of Information Systems and Technologies Security, JSC \u201cInstitute of Information Technologies\u201d, Bakulin St., 12, 61166 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lukasz","family":"Hamera","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Automatics, Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biala, Willowa 2, 43-300 Bielsko-Biala, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey","family":"Kandiy","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Systems Security, Faculty of Comupter Science, V. N. Karazin Kharkiv National University, 4 Svobody Sq., 61022 Kharkiv, Ukraine"},{"name":"Department of Information Systems and Technologies Security, JSC \u201cInstitute of Information Technologies\u201d, Bakulin St., 12, 61166 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yelyzaveta","family":"Lohachova","sequence":"additional","affiliation":[{"name":"Department of Information and Communication Systems Security, Faculty of Comupter Science, V. N. Karazin Kharkiv National University, 4 Svobody Sq., 61022 Kharkiv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Menezes, A.J., van Oorschot, P.C., Vanstone, S.A., van Oorschot, P.C., and Vanstone, S.A. (2018). Handbook of Applied Cryptography, CRC Press.","DOI":"10.1201\/9781439821916"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kuznetsov, A.A., Potii, O.V., Poluyanenko, N.A., Gorbenko, Y.I., and Kryvinska, N. (2022). Stream Ciphers in Modern Real-Time IT Systems: Analysis, Design and Comparative Studies, Springer International Publishing. Studies in Systems, Decision and Control.","DOI":"10.1007\/978-3-030-79770-6"},{"key":"ref_3","first-page":"113","article-title":"Lightweight Stream Ciphers for Green IT Engineering","volume":"Volume 171","author":"Kharchenko","year":"2019","journal-title":"Green IT Engineering: Social, Business and Industrial Applications"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1002\/j.1538-7305.1949.tb00928.x","article-title":"Communication Theory of Secrecy Systems","volume":"28","author":"Shannon","year":"1949","journal-title":"Bell Syst. Tech. J."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Rubinstein-Salzedo, S. (2018). Cryptography, Springer International Publishing.","DOI":"10.1007\/978-3-319-94818-8"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zheng, Y. (2002). Cryptanalysis of Block Ciphers with Overdefined Systems of Equations. Advances in Cryptology\u2014ASIACRYPT 2002, Springer.","DOI":"10.1007\/3-540-36178-2"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Galbraith, S.D. (2007). Algebraic Cryptanalysis of the Data Encryption Standard. Cryptography and Coding, Springer.","DOI":"10.1007\/978-3-540-77272-9"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Daemen, J., and Rijmen, V. (2020). Specification of Rijndael. The Design of Rijndael: The Advanced Encryption Standard (AES), Springer. Information Security and Cryptography.","DOI":"10.1007\/978-3-662-60769-5"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Bard, G.V. (2009). Algebraic Cryptanalysis, Springer US.","DOI":"10.1007\/978-0-387-88757-9"},{"key":"ref_10","unstructured":"Nover, H. (2005). Algebraic Cryptanalysis of Aes: An Overview, University of Wisconsin."},{"key":"ref_11","unstructured":"Clark, J.A., Jacob, J.L., and Stepney, S. (2004, January 19\u201323). The Design of S-Boxes by Simulated Annealing. Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753), Portland, OR, USA."},{"key":"ref_12","unstructured":"McLaughlin, J., and Clark, J.A. (2013). Using Evolutionary Computation to Create Vectorial Boolean Functions with Low Differential Uniformity and High Nonlinearity. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.asoc.2017.05.052","article-title":"Designing Bijective S-Boxes Using Algorithm Portfolios with Limited Time Budgets","volume":"59","author":"Souravlias","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Wang, J., Zhu, Y., Zhou, C., and Qi, Z. (2020). Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm. Symmetry, 12.","DOI":"10.3390\/sym12122115"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Delahaye, D., Chaimatanan, S., and Mongeau, M. (2019). Simulated Annealing: From Basics to Applications, Springer.","DOI":"10.1007\/978-3-319-91086-4_1"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Eremia, M., Liu, C.-C., and Edris, A.-A. (2016). Heuristic Optimization Techniques. Advanced Solutions in Power Systems: HVDC, FACTS, and Artificial Intelligence, IEEE.","DOI":"10.1002\/9781119175391"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Helleseth, T. (1994). On Almost Perfect Nonlinear Permutations. Advances in Cryptology\u2014EUROCRYPT \u201993, Springer.","DOI":"10.1007\/3-540-48285-7"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Helleseth, T. (1994). Differentially Uniform Mappings for Cryptography. Advances in Cryptology\u2014EUROCRYPT \u201993, Springer.","DOI":"10.1007\/3-540-48285-7"},{"key":"ref_19","unstructured":"Clark, A.J. (1998). Optimisation Heuristics for Cryptology. [Ph.D. Thesis, Queensland University of Technology]."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Kose, U., Kose, U., Guraksin, G.E., and Deperlioglu, O. (2018). Nature-Inspired Intelligent Techniques for Solving Biomedical Engineering Problems, IGI Global. Advances in Bioinformatics and Biomedical Engineering.","DOI":"10.4018\/978-1-5225-4769-3"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Information Resources Management Association (2017). Nature-Inspired Computing: Concepts, Methodologies, Tools, and Applications, IGI Global.","DOI":"10.4018\/978-1-5225-0788-8"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Korte, B., and Vygen, J. (2018). Combinatorial Optimization: Theory and Algorithms, Springer. [6th ed.].","DOI":"10.1007\/978-3-662-56039-6"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Oliva, D., Houssein, E.H., and Hinojosa, S. (2021). Metaheuristics in Machine Learning: Theory and Applications, Springer International Publishing. Studies in Computational Intelligence.","DOI":"10.1007\/978-3-030-70542-8"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"114616","DOI":"10.1016\/j.cma.2022.114616","article-title":"Starling Murmuration Optimizer: A Novel Bio-Inspired Algorithm for Global and Engineering Optimization","volume":"392","author":"Zamani","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2445","DOI":"10.1007\/s00170-020-05850-5","article-title":"Machine Learning and Optimization for Production Rescheduling in Industry 4.0","volume":"110","author":"Li","year":"2020","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"104314","DOI":"10.1016\/j.engappai.2021.104314","article-title":"QANA: Quantum-Based Avian Navigation Optimizer Algorithm","volume":"104","author":"Zamani","year":"2021","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"20764","DOI":"10.1109\/ACCESS.2022.3152062","article-title":"A Simulated Annealing Algorithm for the Vehicle Routing Problem With Parcel Lockers","volume":"10","author":"Yu","year":"2022","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.ijpe.2003.11.006","article-title":"Simulated Annealing in Lot Sizing Problems","volume":"88","author":"Tang","year":"2004","journal-title":"Int. J. Prod. Econ."},{"key":"ref_29","first-page":"23","article-title":"A New Method for Generating High Non-Linearity S-Boxes","volume":"19","author":"Tesar","year":"2010","journal-title":"Radioengineering"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1162\/EVCO_a_00191","article-title":"A New Cost Function for Evolution of S-Boxes","volume":"24","author":"Picek","year":"2016","journal-title":"Evol. Comput. Winter"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Freyre-Echevarr\u00eda, A., Alanezi, A., Mart\u00ednez-D\u00edaz, I., Ahmad, M., Abd El-Latif, A.A., Kolivand, H., and Razaq, A. (2020). An External Parameter Independent Novel Cost Function for Evolving Bijective Substitution-Boxes. Symmetry, 12.","DOI":"10.3390\/sym12111896"},{"key":"ref_32","unstructured":"Freyre Echevarr\u00eda, A., and Mart\u00ednez D\u00edaz, I. (2020). A New Cost Function to Improve Nonlinearity of Bijective S-Boxes, preprint."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Kuznetsov, A., Poluyanenko, N., Kandii, S., Zaichenko, Y., Prokopovich-Tkachenko, D., and Katkova, T. (2021, January 5\u20137). WHS Cost Function for Generating S-Boxes. Proceedings of the 2021 IEEE 8th International Conference on Problems of Infocommunications, Science and Technology (PIC S&T), Kharkiv, Ukraine.","DOI":"10.1109\/PICST54195.2021.9772133"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.chaos.2006.08.003","article-title":"A Novel Heuristic Method for Obtaining S-Boxes","volume":"36","author":"Chen","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Kuznetsov, A., Poluyanenko, N., Kandii, S., Zaichenko, Y., Prokopovich-Tkachenko, D., and Katkova, T. (2021, January 5\u20137). Optimizing the Local Search Algorithm for Generating S-Boxes. Proceedings of the 2021 IEEE 8th International Conference on Problems of Infocommunications, Science and Technology (PIC S&T), Kharkiv, Ukraine.","DOI":"10.1109\/PICST54195.2021.9772163"},{"key":"ref_36","unstructured":"McLaughlin, J. (2012). Applications of Search Techniques to Cryptanalysis and the Construction of Cipher Components. [Ph.D. Thesis, University of York]."},{"key":"ref_37","unstructured":"Pasalic, E., and Knudsen, L.R. (2016). Cryptographically Strong S-Boxes Generated by Modified Immune Algorithm. Cryptography and Information Security in the Balkans, Springer International Publishing."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/16\/6073\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:08:34Z","timestamp":1760141314000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/16\/6073"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,14]]},"references-count":37,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["s22166073"],"URL":"https:\/\/doi.org\/10.3390\/s22166073","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,8,14]]}}}