{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T22:51:48Z","timestamp":1771023108445,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,21]],"date-time":"2020-09-21T00:00:00Z","timestamp":1600646400000},"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>Generating Boolean Functions (BFs) with high nonlinearity is a complex task that is usually addresses through algebraic constructions. Metaheuristics have also been applied extensively to this task. However, metaheuristics have not been able to attain so good results as the algebraic techniques. This paper proposes a novel diversity-aware metaheuristic that is able to excel. This proposal includes the design of a novel cost function that combines several information from the Walsh Hadamard Transform (WHT) and a replacement strategy that promotes a gradual change from exploration to exploitation as well as the formation of clusters of solutions with the aim of allowing intensification steps at each iteration. The combination of a high entropy in the population and a lower entropy inside clusters allows a proper balance between exploration and exploitation. This is the first memetic algorithm that is able to generate 10-variable BFs of similar quality than algebraic methods. Experimental results and comparisons provide evidence of the high performance of the proposed optimization mechanism for the generation of high quality BFs.<\/jats:p>","DOI":"10.3390\/e22091052","type":"journal-article","created":{"date-parts":[[2020,9,21]],"date-time":"2020-09-21T08:18:01Z","timestamp":1600676281000},"page":"1052","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Metaheuristics in the Optimization of Cryptographic Boolean Functions"],"prefix":"10.3390","volume":"22","author":[{"given":"Isaac","family":"L\u00f3pez-L\u00f3pez","sequence":"first","affiliation":[{"name":"Centro de Investigaci\u00f3n en Matem\u00e1ticas A.C. (CIMAT). \u00c1rea de Computaci\u00f3n, Jalisco S\/N, Col. Valenciana, Guanajuato 36023, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7793-896X","authenticated-orcid":false,"given":"Guillermo","family":"Sosa-G\u00f3mez","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias Econ\u00f3micas y Empresariales, Universidad Panamericana, \u00c1lvaro del Portillo 49, Zapopan, Jalisco 45010, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5431-5927","authenticated-orcid":false,"given":"Carlos","family":"Segura","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n en Matem\u00e1ticas A.C. (CIMAT). \u00c1rea de Computaci\u00f3n, Jalisco S\/N, Col. Valenciana, Guanajuato 36023, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8781-7993","authenticated-orcid":false,"given":"Diego","family":"Oliva","sequence":"additional","affiliation":[{"name":"IN3-Computer Science Department, Universitat Oberta de Catalunya, 08860 Castelldefels, Spain"},{"name":"Depto. de Ciencias Computacionales, Universidad de Guadalajara, CUCEI, Av. Revoluci\u00f3n 1500, Guadalajara C.P. 44430, Jalisco, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0681-3833","authenticated-orcid":false,"given":"Omar","family":"Rojas","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias Econ\u00f3micas y Empresariales, Universidad Panamericana, \u00c1lvaro del Portillo 49, Zapopan, Jalisco 45010, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1109\/18.6037","article-title":"A spectral characterization of correlation-immune combining functions","volume":"34","author":"Xiao","year":"1988","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","unstructured":"Katz, J., and Lindell, Y. (2007). Introduction to Modern Cryptography (Chapman 8 Hall\/Crc Cryptography and Network Security Series), Chapman & Hall\/CRC."},{"key":"ref_3","unstructured":"Millan, W., Clark, A., and Dawson, E. (1997). Smart hill climbing finds better boolean functions. Workshop on Selected Areas in Cryptology 1997, Workshop Record, Citeseer."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Picek, S., Jakobovic, D., Miller, J.F., Marchiori, E., and Batina, L. (2015). Evolutionary methods for the construction of cryptographic boolean functions. European Conference on Genetic Programming, Springer.","DOI":"10.1007\/978-3-319-16501-1_16"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Picek, S., Ege, B., Batina, L., Jakobovic, D., Chmielewski, \u0141., and Golub, M. (2014). On using genetic algorithms for intrinsic side-channel resistance: The case of aes s-box. Proceedings of the First Workshop on Cryptography and Security in Computing Systems, Association for Computing Machinery.","DOI":"10.1145\/2556315.2556319"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1162\/EVCO_a_00190","article-title":"Evolutionary algorithms for boolean functions in diverse domains of cryptography","volume":"24","author":"Picek","year":"2016","journal-title":"Evol. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Sarkar, P., and Maitra, S. (2000). Nonlinearity bounds and constructions of resilient Boolean functions. Annual International Cryptology Conference, Springer.","DOI":"10.1007\/3-540-44598-6_32"},{"key":"ref_8","unstructured":"Doganaksoy, A., D\u00fcndar, B.G., G\u00f6loglu, F., Sayg\u0131, Z., Sulak, F., and Uguz, M. (2005, January 18\u201320). Constructions of highly nonlinear balanced boolean functions. Proceedings of the Ulusal Kriptoloji Sempozyumu, Ankara, Turkey."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1109\/TEVC.2016.2641477","article-title":"Improving diversity in evolutionary algorithms: New best solutions for frequency assignment","volume":"21","author":"Segura","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Romero Ruiz, E., and Segura, C. (2018). Memetic algorithm with hungarian matching based crossover and diversity preservation. Comput. Sist., 22.","DOI":"10.13053\/cys-22-2-2951"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"151","DOI":"10.3390\/e17010151","article-title":"A clustering method based on the maximum entropy principle","volume":"17","year":"2015","journal-title":"Entropy"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"88","DOI":"10.47277\/10.47277\/IJCNCS\/1(3)3","article-title":"State of the art in Boolean functions cryptographic assessment","volume":"1","author":"Elhosary","year":"2013","journal-title":"Int. J. Comput. Netw. Commun. Secur."},{"key":"ref_13","unstructured":"MacWilliams, F.J., and Sloane, N.J.A. (1977). The Theory of Error-Correcting Codes, Elsevier."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Millan, W., Clark, A., and Dawson, E. (1997). An effective genetic algorithm for finding highly nonlinear Boolean functions. International Conference on Information and Communications Security, Springer.","DOI":"10.1007\/BFb0028471"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Millan, W., Clark, A., and Dawson, E. (1998). Heuristic design of cryptographically strong balanced Boolean functions. International Conference on the Theory and Applications of Cryptographic Techniques, Springer.","DOI":"10.1007\/BFb0054148"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Clark, J.A., and Jacob, J.L. (2000). Two-stage optimisation in the design of Boolean functions. Australasian Conference on Information Security and Privacy, Springer.","DOI":"10.1007\/10718964_20"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2173","DOI":"10.1109\/CEC.2003.1299941","article-title":"Almost Boolean functions: The design of Boolean functions by spectral inversion","volume":"Volume 3","author":"Clark","year":"2003","journal-title":"Proceedings of the 2003 Congress on Evolutionary Computation, 2003. CEC \u201903"},{"key":"ref_18","first-page":"231","article-title":"Simpler methods for generating better Boolean functions with good cryptographic properties","volume":"29","author":"Burnett","year":"2004","journal-title":"Australas. J. Comb."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Izbenko, Y., Kovtun, V., and Kuznetsov, A. (2009, January 27\u201329). The design of boolean functions by modified hill climbing method. Proceedings of the 2009 Sixth International Conference on Information Technology: New Generations, Las Vegas, NV, USA.","DOI":"10.1109\/ITNG.2009.102"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0097-3165(73)90031-9","article-title":"A family of difference sets in non-cyclic groups","volume":"15","author":"McFarland","year":"1973","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Picek, S., McKay, R.I., Santana, R., and Gedeon, T.D. (2015, January 8\u201311). Fighting the symmetries: The structure of cryptographic boolean function spaces. Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, Coimbra, Portugal.","DOI":"10.1145\/2739480.2754739"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1016\/j.asoc.2015.10.066","article-title":"Cryptographic Boolean functions: One output, many design criteria","volume":"40","author":"Picek","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Picek, S., Santana, R., and Jakobovic, D. (2016, January 24\u201329). Maximal nonlinearity in balanced boolean functions with even number of inputs, revisited. Proceedings of the 2016 IEEE Congress on Evolutionary Computation (CEC), Vancouver, BC, Canada.","DOI":"10.1109\/CEC.2016.7744197"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s13198-012-0089-6","article-title":"An evolutionary approach to construct cryptographically strong Boolean functions","volume":"3","author":"Goyal","year":"2012","journal-title":"Int. J. Syst. Assur. Eng. Manag."},{"key":"ref_25","first-page":"35:1","article-title":"Exploration and Exploitation in Evolutionary Algorithms: A Survey","volume":"45","author":"Liu","year":"2013","journal-title":"ACM Comput. Surv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3233","DOI":"10.1109\/TCYB.2015.2501726","article-title":"A novel diversity-based replacement strategy for evolutionary algorithms","volume":"46","author":"Segura","year":"2016","journal-title":"IEEE Trans. Cybern."},{"key":"ref_27","unstructured":"Clark, J.A., Jacob, J.L., and Stepney, S. (2004, January 19\u201323). Searching for cost functions. Proceedings of the IEEE Congress on Evolutionary Computation, Portland, OR, USA."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Talbi, E.G. (2009). Metaheuristics: From Design to Implementation, John Wiley & Sons.","DOI":"10.1002\/9780470496916"},{"key":"ref_29","first-page":"1989","article-title":"On evolution, search, optimization, genetic algorithms and martial arts: Towards memetic algorithms","volume":"826","author":"Moscato","year":"1989","journal-title":"Caltech Concurr. Comput. Program, C3p Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1109\/TEVC.2011.2132725","article-title":"A multi-facet survey on memetic computation","volume":"15","author":"Chen","year":"2011","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_31","unstructured":"Eiben, A.E., and Smith, J.E. (2008). Introduction to Evolutionary Computing (Natural Computing Series), Springer."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s10623-011-9587-9","article-title":"Construction of balanced Boolean functions with high nonlinearity and good autocorrelation properties","volume":"67","author":"Tang","year":"2013","journal-title":"Des. Codes Cryptogr."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Asghar, A., Iqbal, M.M., Ahmed, W., Ali, M., Parvez, H., and Rashid, M. (2017, January 15\u201316). Logic algebra for exploiting shared SRAM-table based FPGAs for large LUT inputs. Proceedings of the 2017 First International Conference on Latest Trends in Electrical Engineering and Computing Technologies (INTELLECT), Karachi, Pakistan.","DOI":"10.1109\/INTELLECT.2017.8277632"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Asghar, A., Iqbal, M.M., Ahmed, W., Ali, M., Parvez, H., and Rashid, M. (2017). Exploring Shared SRAM Tables in FPGAs for Larger LUTs and Higher Degree of Sharing. Int. J. Reconfig. Comput., 2017.","DOI":"10.1155\/2017\/7021056"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/1052\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:11:58Z","timestamp":1760177518000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/1052"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,21]]},"references-count":34,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["e22091052"],"URL":"https:\/\/doi.org\/10.3390\/e22091052","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,21]]}}}