{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T12:38:32Z","timestamp":1761395912765,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,19]],"date-time":"2020-12-19T00:00:00Z","timestamp":1608336000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The substitution box (S-box) is the only nonlinear components in the symmetric block cipher. Its performance directly determines the security strength of the block cipher. With the dynamic characteristics degradation and the local periodic phenomenon of digital chaos, and the security problems caused by them becoming more and more prominent, how to efficiently generate an S-box with security guarantee based on chaos has gradually attracted the attention of cryptographers. In this paper, a chaotic S-box construction method is proposed based on a memorable simulated annealing algorithm (MSAA). The chaotic S-box set is produced by using the nonlinearity and randomness of the dynamic iteration of digital cascaded chaotic mapping. The composite objective function is constructed based on the analysis of the performance indexes of S-box. The MSAA is used to efficiently optimize the S-box set. The matrix segmentation and scrambling operations are carried out on the optimized S-box. The cryptographic performance of chaotic S-box is tested and analyzed, and compared with the mainstream chaotic S-box of the same kind. The results show that the S-box constructed in this paper can not only stably and efficiently generate chaotic S-box with better performance, but also make an effective exploration of the construction of chaotic S-boxes based on intelligent algorithms.<\/jats:p>","DOI":"10.3390\/sym12122115","type":"journal-article","created":{"date-parts":[[2020,12,21]],"date-time":"2020-12-21T04:12:01Z","timestamp":1608523921000},"page":"2115","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm"],"prefix":"10.3390","volume":"12","author":[{"given":"Juan","family":"Wang","sequence":"first","affiliation":[{"name":"College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China"},{"name":"College of Electronic and Information Engineering, Heilongjiang University of Science and Technology, Harbin 150027, China"}]},{"given":"Yangqing","family":"Zhu","sequence":"additional","affiliation":[{"name":"College of Electronic and Information Engineering, Heilongjiang University of Science and Technology, Harbin 150027, China"}]},{"given":"Chao","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Electronic and Information Engineering, Heilongjiang University of Science and Technology, Harbin 150027, China"}]},{"given":"Zhiming","family":"Qi","sequence":"additional","affiliation":[{"name":"State Grid Liaoning Maintenance Company, Shenyang 110006, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1007\/s11071-013-1011-8","article-title":"Literature survey on nonlinear components and chaotic nonlinear components of block ciphers","volume":"74","author":"Hussain","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lu, J.Q., and Seo, H. (2017, January 5\u20138). An investigation of an S-box mechanism in modern block cipher design. Proceedings of the TENCON 2017\u20142017 IEEE Region 10 Conference, Penang, Malaysia.","DOI":"10.1109\/TENCON.2017.8227852"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Matheis, K., Steinwandt, R., and Corona, A.S. (2019). Algebraic Properties of the Block Cipher DESL. Symmetry Basel, 11.","DOI":"10.3390\/sym11111411"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Mohamed, K., Pauzi, M.N.M., Ali, F.H.H.M., and Ariffin, S. (2014, January 2\u20134). Study of S-box properties in block cipher. Proceedings of the 2014 International Conference on Computer, Communications, and Control Technology (I4CT), Langkawi, Malaysia.","DOI":"10.1109\/I4CT.2014.6914206"},{"key":"ref_5","first-page":"1","article-title":"A Key Selected S-Box Mechanism and Its Investigation in Modern Block Cipher Design","volume":"2020","author":"Lu","year":"2020","journal-title":"Secur. Commun. Netw."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2017\/6969312","article-title":"Chaotic S-Box: Intertwining Logistic Map and Bacterial Foraging Optimization","volume":"2017","author":"Tian","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1016\/S1004-4132(07)60108-X","article-title":"One AES S-box to increase complexity and its cryptanalysis","volume":"18","author":"Liu","year":"2007","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_8","first-page":"2291","article-title":"An improved AES S-Box and its performance analysis","volume":"7","author":"Cui","year":"2011","journal-title":"Int. J. Innov. Comput. Inf. Control."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Hussain, I., Anees, A., Al-Maadeed, T.A., and Mustafa, M.T. (2019). Construction of S-Box Based on Chaotic Map and Algebraic Structures. Symmetry Basel, 11.","DOI":"10.3390\/sym11030351"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1007\/s11760-016-1007-1","article-title":"A new S-box construction method based on the fractional-order chaotic Chen system","volume":"11","author":"Ozkaynak","year":"2017","journal-title":"Signal Image Video Process."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lu, Q., Zhu, C.X., and Wang, G.J. (2019). A Novel S-Box Design Algorithm Based on a New Compound Chaotic System. Entropy, 21.","DOI":"10.3390\/e21101004"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1867","DOI":"10.1007\/s11071-015-2283-y","article-title":"Designing S-boxes based on 3D four-wing autonomous chaotic system","volume":"82","author":"Liu","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2407","DOI":"10.1007\/s11071-016-3199-x","article-title":"A novel method of S-box design based on discrete chaotic map","volume":"87","author":"Lambic","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1438","DOI":"10.1016\/j.ijleo.2016.11.152","article-title":"A simple yet efficient S-box method based on chaotic sine map","volume":"130","author":"Belazi","year":"2017","journal-title":"Optik"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Guesmi, R., Ben Farah, M.A., Kachouri, A., and Samet, M. (2014, January 10\u201313). A Novel Design of Chaos Based S-Boxes Using Genetic Algorithm Techniques. Proceedings of the 2014 IEEE\/ACS 11th International Conference on Computer Systems and Applications (AICCSA), Doha, Qatar.","DOI":"10.1109\/AICCSA.2014.7073265"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"7201","DOI":"10.1007\/s00521-018-3557-3","article-title":"A novel efficient substitution-box design based on firefly algorithm and discrete chaotic map","volume":"31","author":"Ahmed","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1016\/j.ins.2020.03.025","article-title":"A genetic algorithm for constructing bijective substitution boxes with high nonlinearity","volume":"523","author":"Wang","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/S0898-1221(96)00221-0","article-title":"Simulated annealing simulated","volume":"33","author":"Fabian","year":"1997","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.engappai.2009.10.001","article-title":"Exhaustive and heuristic search approaches for learning a software defect prediction model","volume":"23","author":"Pendharkar","year":"2010","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_20","first-page":"1","article-title":"Multiobjective Simulated Annealing: Principles and Algorithm Variants","volume":"2019","author":"Amine","year":"2019","journal-title":"Adv. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"124072","DOI":"10.1016\/j.physa.2019.124072","article-title":"On the effect of chaotic system in performance characteristics of chaos based s-box designs","volume":"550","author":"Ozkaynak","year":"2020","journal-title":"Phys. a-Stat. Mech. Its Appl."},{"key":"ref_22","first-page":"97","article-title":"The Design of S-box Based on Cascaded Integer Chaos Applied to Wireless Sensor Network","volume":"9","author":"Wang","year":"2016","journal-title":"Int. J. Future Gener. Commun. Netw."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1630001","DOI":"10.1142\/S0218213016300015","article-title":"Simulated Annealing, Its Variants and Engineering Applications","volume":"25","author":"Siddique","year":"2016","journal-title":"Int. J. Artif. Intell. Tools"},{"key":"ref_24","first-page":"1","article-title":"Multiple-Try Simulated Annealing Algorithm for Global Optimization","volume":"2018","author":"Shao","year":"2018","journal-title":"Math. Probl. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"673","DOI":"10.17159\/2411-9717\/2016\/v116n7a9","article-title":"An improved meta-heuristic approach to extraction sequencing and block routing","volume":"116","author":"Sari","year":"2016","journal-title":"J. South. Afr. Inst. Min. Metall."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1016\/j.procs.2015.07.394","article-title":"A Novel Ant Colony Optimization Based Scheme for Substitution Box Design","volume":"57","author":"Ahmad","year":"2015","journal-title":"Procedia Comput. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1950158","DOI":"10.1142\/S021812741950158X","article-title":"Construction Method and Performance Analysis of Chaotic S-Box Based on Fireworks Algorithm","volume":"29","author":"Wang","year":"2019","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7970","DOI":"10.1109\/TIT.2014.2360880","article-title":"Highly Nonlinear Balanced S-Boxes With Good Differential Properties","volume":"60","author":"Zhang","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Webster, A.F., and Tavares, S.E. (1986). On the Design of S-Boxes. Proceedings of Advances in Cryptology\u2014CRYPTO \u201985 Proceedings, Springer.","DOI":"10.1007\/3-540-39799-X_41"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1109\/81.904880","article-title":"Chaos and cryptography: Block encryption ciphers based on chaotic maps","volume":"48","author":"Jakimoski","year":"2001","journal-title":"IEEE Trans. Circuits Syst. Fundam. Theory Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2115\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:47:24Z","timestamp":1760179644000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2115"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,19]]},"references-count":30,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["sym12122115"],"URL":"https:\/\/doi.org\/10.3390\/sym12122115","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,12,19]]}}}