{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T15:51:18Z","timestamp":1769961078700,"version":"3.49.0"},"reference-count":45,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,18]],"date-time":"2020-11-18T00:00:00Z","timestamp":1605657600000},"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 property of nonlinearity has high importance for the design of strong substitution boxes. Therefore, the development of new techniques to produce substitution boxes with high values of nonlinearity is essential. Many research papers have shown that optimization algorithms are an efficient technique to obtain good solutions. However, there is no reference in the public literature showing that a heuristic method obtains optimal nonlinearity unless seeded with optimal initial solutions. Moreover, the majority of papers with the best nonlinearity reported for pseudo-random seeding of the algorithm(s) often achieve their results with the help of some cost function(s) over the Walsh\u2013Hadamard spectrum of the substitution. In the sense, we proposed to present, in this paper, a novel external parameter independent cost function for evolving bijective s-boxes of high nonlinearity, which is highly correlated to this property. Several heuristic approaches including GaT (genetic and tree), LSA (local search algorithm), and the Hill Climbing algorithm have been investigated to assess the performance of evolved s-boxes. A performance comparison has been done to show the advantages of our new cost function, with respect to cost functions for s-boxes like Clark\u2019s and Picek\u2019s cost functions.<\/jats:p>","DOI":"10.3390\/sym12111896","type":"journal-article","created":{"date-parts":[[2020,11,19]],"date-time":"2020-11-19T06:23:52Z","timestamp":1605767032000},"page":"1896","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["An External Parameter Independent Novel Cost Function for Evolving Bijective Substitution-Boxes"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0537-9430","authenticated-orcid":false,"given":"Alejandro","family":"Freyre-Echevarr\u00eda","sequence":"first","affiliation":[{"name":"Institute of Cryptography, University of Havana, Havana 10400, Cuba"}]},{"given":"Ahmad","family":"Alanezi","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Faculty of Engineering and Technology, Liverpool John Moores University (LJMU), Liverpool L3 3AF, UK"}]},{"given":"Ismel","family":"Mart\u00ednez-D\u00edaz","sequence":"additional","affiliation":[{"name":"Institute of Cryptography, University of Havana, Havana 10400, Cuba"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4915-9325","authenticated-orcid":false,"given":"Musheer","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, Jamia Millia Islamia, New Delhi 110025, India"}]},{"given":"Ahmed A.","family":"Abd El-Latif","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt"}]},{"given":"Hoshang","family":"Kolivand","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Faculty of Engineering and Technology, Liverpool John Moores University (LJMU), Liverpool L3 3AF, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1898-4082","authenticated-orcid":false,"given":"Abdul","family":"Razaq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54770, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1147\/rd.383.0243","article-title":"The Data Encryption Standard (DES) and its strength against attacks","volume":"38","author":"Coppersmith","year":"1994","journal-title":"IBM J. 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