{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T11:41:14Z","timestamp":1781350874651,"version":"3.54.1"},"reference-count":64,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,6,28]],"date-time":"2020-06-28T00:00:00Z","timestamp":1593302400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Static substitution-boxes in fixed structured block ciphers may make the system vulnerable to cryptanalysis. However, key-dependent dynamic substitution-boxes (S-boxes) assume to improve the security and robustness of the whole cryptosystem. This paper proposes to present the construction of key-dependent dynamic S-boxes having high nonlinearity. The proposed scheme involves the evolution of initially generated S-box for improved nonlinearity based on the fractional-order time-delayed Hopfield neural network. The cryptographic performance of the evolved S-box is assessed by using standard security parameters, including nonlinearity, strict avalanche criterion, bits independence criterion, differential uniformity, linear approximation probability, etc. The proposed scheme is able to evolve an S-box having mean nonlinearity of 111.25, strict avalanche criteria value of 0.5007, and differential uniformity of 10. The performance assessments demonstrate that the proposed scheme and S-box have excellent features, and are thus capable of offering high nonlinearity in the cryptosystem. The comparison analysis further confirms the improved security features of anticipated scheme and S-box, as compared to many existing chaos-based and other S-boxes.<\/jats:p>","DOI":"10.3390\/e22070717","type":"journal-article","created":{"date-parts":[[2020,6,29]],"date-time":"2020-06-29T03:40:07Z","timestamp":1593402007000},"page":"717","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["Evolving Dynamic S-Boxes Using Fractional-Order Hopfield Neural Network Based Scheme"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4915-9325","authenticated-orcid":false,"given":"Musheer","family":"Ahmad","sequence":"first","affiliation":[{"name":"Department of Computer Engineering, Jamia Millia Islamia, New Delhi 110025, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eesa","family":"Al-Solami","sequence":"additional","affiliation":[{"name":"Department of Information Security, University of Jeddah, Jeddah 21493, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,28]]},"reference":[{"key":"ref_1","unstructured":"Forouzan, B.A. (2007). Cryptography & Network Security, McGraw-Hill, Inc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1917","DOI":"10.1007\/s11760-014-0683-y","article-title":"Efficient confusion\u2013diffusion chaotic image cryptosystem using enhanced standard map","volume":"9","author":"Faragallah","year":"2014","journal-title":"Signal Image Video Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"32585","DOI":"10.1007\/s11042-019-07866-w","article-title":"A new construction of confusion component of block ciphers","volume":"78","author":"Ali","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/s11277-018-5698-1","article-title":"A method of generating 8 \u00d7 8 substitution boxes based on elliptic curves","volume":"101","author":"Hayat","year":"2018","journal-title":"Wirel. Pers. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2018\/9389065","article-title":"Sine-Cosine Optimization-Based Bijective Substitution-Boxes Construction Using Enhanced Dynamics of Chaotic Map","volume":"2018","author":"Alzaidi","year":"2018","journal-title":"Complexity"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ahmad, M., Chugh, H., Goel, A., and Singla, P. (2013). A chaos based method for efficient cryptographic S-box design. International Symposium on Security in Computing and Communication, Springer.","DOI":"10.1007\/978-3-642-40576-1_13"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"173273","DOI":"10.1109\/ACCESS.2019.2956385","article-title":"Construction of Cryptographic S-Boxes Based on Mobius Transformation and Chaotic Tent-Sine System","volume":"7","author":"Jamal","year":"2019","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1016\/j.pisc.2016.06.001","article-title":"Efficient cryptographic substitution box design using travelling salesman problem and chaos","volume":"8","author":"Ahmad","year":"2016","journal-title":"Perspect. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1007\/s11071-016-3046-0","article-title":"Efficient cryptosystem approaches: S-boxes and permutation\u2013substitution-based encryption","volume":"87","author":"Belazi","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_10","first-page":"181","article-title":"S-box design method based on improved one-dimensional discrete chaotic map","volume":"2","year":"2018","journal-title":"J. Inf. Telecommun."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ahmad, M., Haleem, H., and Khan, P.M. (2014, January 20\u201321). A new chaotic substitution box design for block ciphers. Proceedings of the 2014 International Conference on Signal Processing and Integrated Networks (SPIN), Noida, India.","DOI":"10.1109\/SPIN.2014.6776958"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s40998-019-00230-6","article-title":"An Analysis and Generation Toolbox for Chaotic Substitution Boxes: A Case Study Based on Chaotic Labyrinth Rene Thomas System","volume":"44","author":"Ozkaynak","year":"2019","journal-title":"Iran. J. Sci. Technol. Trans. Electr. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ahmad, M., and Malik, M. (2016, January 3\u20135). Design of chaotic neural network based method for cryptographic substitution box. Proceedings of the International Conference on Electrical, Electronics, and Optimization Techniques (ICEEOT), Chennai, India.","DOI":"10.1109\/ICEEOT.2016.7754809"},{"key":"ref_14","first-page":"1","article-title":"A Novel Technique to Improve Nonlinearity of Substitution Box without Disturbing Its Mathematical Properties","volume":"111","author":"Razaq","year":"2019","journal-title":"Wirel. Pers. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1121","DOI":"10.1142\/S0218127499000778","article-title":"Application of discrete chaotic dynamical systems in cryptography\u2014Dcc method","volume":"9","author":"Kotulski","year":"1999","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/j.physleta.2006.03.069","article-title":"Cryptography based on delayed chaotic neural networks","volume":"356","author":"Yu","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_17","first-page":"413","article-title":"Hyperchaotic encryption based on multi-scroll piecewise linear systems","volume":"270","author":"Campos","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhu, H., Zhou, S., and Zhang, W. (2008, January 18\u201321). Chaos and synchronization of time-delayed fractional neuron network system. Proceedings of the 2008 the 9th International Conference for Young Computer Scientists, Hunan, China.","DOI":"10.1109\/ICYCS.2008.107"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.neucom.2012.11.034","article-title":"Dynamic analysis of a class of fractional-order neural networks with delay","volume":"111","author":"Chen","year":"2013","journal-title":"Neurocomputing"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3265","DOI":"10.3390\/e15083355","article-title":"Synchronization of a Class of Fractional-Order Chaotic Neural Networks","volume":"15","author":"Chen","year":"2013","journal-title":"Entropy"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1721","DOI":"10.1007\/s11071-015-1911-x","article-title":"A novel image encryption scheme based on an improper fractional-order chaotic system","volume":"80","author":"Zhao","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1016\/j.chaos.2009.02.015","article-title":"Synchronization of time-delayed systems with chaotic modulation and cryptography","volume":"42","author":"Banerjee","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3733","DOI":"10.1016\/j.physleta.2010.07.019","article-title":"A method for designing strong S-Boxes based on chaotic Lorenz system","volume":"374","author":"Ozkaynak","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/s11071-012-0675-9","article-title":"An efficient method for the construction of block cipher with multi-chaotic systems","volume":"71","author":"Khan","year":"2012","journal-title":"Nonlinear Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1007\/s11760-013-0577-4","article-title":"An efficient construction of substitution box with fractional chaotic system","volume":"9","author":"Khan","year":"2013","journal-title":"Signal Image Video Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1867","DOI":"10.1007\/s11071-015-2283-y","article-title":"Designing S-boxes based on 3-D four-wing autonomous chaotic system","volume":"82","author":"Liu","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_27","first-page":"232","article-title":"S-box: Six-dimensional compound hyperchaotic map and artificial bee colony algorithm","volume":"27","author":"Tian","year":"2016","journal-title":"J. Syst. Eng. Electron."},{"key":"ref_28","first-page":"1081","article-title":"A novel approach for strong S-Box generation algorithm design based on chaotic scaled Zhongtang system","volume":"87","author":"Zengin","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/s13319-017-0119-x","article-title":"Designing S-Box Based on 4D-4Wing Hyperchaotic System","volume":"8","author":"Islam","year":"2017","journal-title":"3D Res."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Al Solami, E., Ahmad, M., Volos, C., Doja, M.N., and Beg, M.M.S. (2018). A New Hyperchaotic System-Based Design for Efficient Bijective Substitution-Boxes. Entropy, 20.","DOI":"10.3390\/e20070525"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Wang, X., Akgul, A., Cavusoglu, U., Pham, V.-T., Hoang, D.V., and Nguyen, X.Q. (2018). A Chaotic System with Infinite Equilibria and Its S-Box Constructing Application. Appl. Sci., 8.","DOI":"10.3390\/app8112132"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Liu, L., Zhang, Y., and Wang, X. (2018). A Novel Method for Constructing the S-Box Based on Spatiotemporal Chaotic Dynamics. Appl. Sci., 8.","DOI":"10.3390\/app8122650"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/s11071-013-0987-4","article-title":"Designing chaotic S-boxes based on time-delay chaotic system","volume":"74","author":"Yavuz","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_34","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":"2016","journal-title":"Signal Image Video Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.optlastec.2019.03.005","article-title":"A novel image steganography technique based on quantum substitution boxes","volume":"116","year":"2019","journal-title":"Opt. Laser Technol."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Petr\u00e1\u0161, I. (2011). Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-18101-6"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2018\/3421725","article-title":"An Injective S-Box Design Scheme over an Ordered Isomorphic Elliptic Curve and Its Characterization","volume":"2018","author":"Azam","year":"2018","journal-title":"Secur. Commun. Netw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1007\/s11277-018-5787-1","article-title":"ABC Optimization Based Construction of Strong Substitution-Boxes","volume":"101","author":"Ahmad","year":"2018","journal-title":"Wirel. Pers. Commun."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Zahid, A.H., Arshad, M.J., and Ahmad, M. (2019). A Novel Construction of Efficient Substitution-Boxes Using Cubic Fractional Transformation. Entropy, 21.","DOI":"10.3390\/e21030245"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"39781","DOI":"10.1109\/ACCESS.2020.2975880","article-title":"Comparison of Pre and Post-Action of a Finite Abelian Group over Certain Nonlinear Schemes","volume":"8","author":"Yousaf","year":"2020","journal-title":"IEEE Access"},{"key":"ref_41","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","year":"2020","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1007\/s00521-015-1887-y","article-title":"Construction of S-box based on chaotic Boolean functions and its application in image encryption","volume":"27","author":"Khan","year":"2015","journal-title":"Neural Comput. Appl."},{"key":"ref_43","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_44","doi-asserted-by":"crossref","unstructured":"Cusick, T.W., and Stanica, P. (2009). Cryptographic Boolean Functions and Applications, Elsevier.","DOI":"10.1016\/B978-0-12-374890-4.00009-4"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"55405","DOI":"10.1109\/ACCESS.2018.2871557","article-title":"A New 1D Chaotic Map and $\\beta$-Hill Climbing for Generating Substitution-Boxes","volume":"6","author":"Alzaidi","year":"2018","journal-title":"IEEE Access"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/BF00203967","article-title":"The structured design of cryptographically good s-boxes","volume":"3","author":"Adams","year":"1990","journal-title":"J. Cryptol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/BF00630563","article-title":"Differential Cryptanalysis of DES-like Cryptosystems. Advances in Cryptology-CRYPT0\u2019 90 Lecture Notes in Computer Science 2\u201321","volume":"4","author":"Biham","year":"1991","journal-title":"J. Cryptol."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"75473","DOI":"10.1109\/ACCESS.2020.2989676","article-title":"A Novel Method for Generation of Strong Substitution-Boxes Based on Coset Graphs and Symmetric Groups","volume":"8","author":"Razaq","year":"2020","journal-title":"IEEE Access"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Matsui, M. (1993). Linear Cryptanalysis Method for DES Cipher. Advances in Cryptology\u2014EUROCRYPT \u201993 Lecture Notes in Computer Science 386\u2013397, Springer.","DOI":"10.1007\/3-540-48285-7_33"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2115","DOI":"10.1007\/s11071-018-4478-5","article-title":"Chaotic S-box: Six-dimensional fractional Lorenz\u2013Duffing chaotic system and O-shaped path scrambling","volume":"94","author":"Ye","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3317","DOI":"10.1007\/s00521-017-3287-y","article-title":"Construction of robust substitution boxes based on chaotic systems","volume":"31","author":"Ozkaynak","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_52","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_53","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":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1007\/s00521-016-2511-5","article-title":"A novel construction of substitution box for image encryption applications with Gingerbreadman chaotic map and S8 permutation","volume":"29","author":"Khan","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2017\/5101934","article-title":"A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map","volume":"2017","author":"Razaq","year":"2017","journal-title":"Secur. Commun. Netw."},{"key":"ref_56","first-page":"123","article-title":"Substitution box generation using Chaos: An image encryption application","volume":"332","author":"Carapia","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2033","DOI":"10.1007\/s11277-016-3436-0","article-title":"A Watermarking Technique with Chaotic Fractional S-Box Transformation","volume":"90","author":"Jamal","year":"2016","journal-title":"Wirel. Pers. Commun."},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Lambi\u0107, D. (2020). A new discrete-space chaotic map based on the multiplication of integer numbers and its application in S-box design. Nonlinear Dyn., 1\u201313.","DOI":"10.1007\/s11071-020-05503-y"},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Lu, Q., Zhu, C., and Wang, G. (2019). A Novel S-Box Design Algorithm Based on a New Compound Chaotic System. Entropy, 21.","DOI":"10.3390\/e21101004"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1378","DOI":"10.1631\/FITEE.1800434","article-title":"Efficient construction of a substitution box based on a Mordell elliptic curve over a finite field","volume":"20","author":"Azam","year":"2019","journal-title":"Front. Inf. Technol. Electron. Eng."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Khan, F.A., Ahmed, J., Khan, J.S., Ahmad, J., Khan, M.A., and Hwang, S.O. (2017, January 27\u201329). A new technique for designing 8 \u00d7 8 substitution box for image encryption applications. Proceedings of the 2017 9th Computer Science and Electronic Engineering (CEEC), Colchester, UK.","DOI":"10.1109\/CEEC.2017.8101591"},{"key":"ref_62","first-page":"1","article-title":"Generation of Dynamical S-Boxes for Block Ciphers via Extended Logistic Map","volume":"2020","author":"Campos","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Tran, M.T., Bui, D.K., and Duong, A.D. (2008, January 13\u201317). Gray S-Box for Advanced Encryption Standard. Proceedings of the 2008 International Conference on Computational Intelligence and Security, Suzhou, China.","DOI":"10.1109\/CIS.2008.205"},{"key":"ref_64","unstructured":"Daemen, J., and Rijmen, V. (1998, January 20\u201322). The Rijndael block cipher: AES proposal. Proceedings of the First Candidate Conference (AeS1), Ventura, CA, USA."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/717\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:44:13Z","timestamp":1760175853000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/7\/717"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,28]]},"references-count":64,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["e22070717"],"URL":"https:\/\/doi.org\/10.3390\/e22070717","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,28]]}}}