{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T21:15:45Z","timestamp":1784150145163,"version":"3.55.0"},"reference-count":51,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,14]],"date-time":"2023-05-14T00:00:00Z","timestamp":1684022400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council in Taiwan","award":["NSTC-109-2628-E-027-004\u2013MY3"],"award-info":[{"award-number":["NSTC-109-2628-E-027-004\u2013MY3"]}]},{"name":"National Science and Technology Council in Taiwan","award":["NSTC-111-2218-E-027-003"],"award-info":[{"award-number":["NSTC-111-2218-E-027-003"]}]},{"name":"National Science and Technology Council in Taiwan","award":["NSTC-111-2622-8-027-009"],"award-info":[{"award-number":["NSTC-111-2622-8-027-009"]}]},{"name":"National Science and Technology Council in Taiwan","award":["1112303249"],"award-info":[{"award-number":["1112303249"]}]},{"name":"Ministry of Education of Taiwan","award":["NSTC-109-2628-E-027-004\u2013MY3"],"award-info":[{"award-number":["NSTC-109-2628-E-027-004\u2013MY3"]}]},{"name":"Ministry of Education of Taiwan","award":["NSTC-111-2218-E-027-003"],"award-info":[{"award-number":["NSTC-111-2218-E-027-003"]}]},{"name":"Ministry of Education of Taiwan","award":["NSTC-111-2622-8-027-009"],"award-info":[{"award-number":["NSTC-111-2622-8-027-009"]}]},{"name":"Ministry of Education of Taiwan","award":["1112303249"],"award-info":[{"award-number":["1112303249"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>With the explosion of the generation, transmission and sharing of image data over the Internet and other unsecured networks, the need for and significance of the development of novel image encryption algorithms are unprecedented. In this research work, we propose a novel framework for image encryption that is based on two hyperchaotic maps utilized in conjunction with the single neuron model (SNM). The framework entails three successive stages, where in every stage a substitution box (S-box) is applied, then XORing with an encryption key is carried out. The S-boxes and the encryption keys are generated from the numerical solutions of the hyperchaotic maps and the SNM. The performance of the proposed framework is gauged through a number of metrics, reflecting superior performance and complete asymmetry between the plain images and their encrypted versions. The main advantages of this work are (1) vast key space and (2) high encryption efficiency. The superior key space of 22551 is the result of employing the two hyperchaotic maps, while the improved efficiency, resulting in an average encryption rate of 8.54 Mbps, is the result of using the SNM as well as the employment of optimized parallel processing techniques. In addition, the proposed encryption framework is shown to output encrypted images that pass the NIST SP 800 suite. Average achieved values for the metrics include MSE of 9626, PSNR of 8.3 dB, MAE of 80.99, entropy of 7.999, NPCR of 99.6% and UACI of 31.49%.<\/jats:p>","DOI":"10.3390\/sym15051081","type":"journal-article","created":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T08:28:16Z","timestamp":1684139296000},"page":"1081","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":82,"title":["Hyperchaotic Maps and the Single Neuron Model: A Novel Framework for Chaos-Based Image Encryption"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6159-4971","authenticated-orcid":false,"given":"Wassim","family":"Alexan","sequence":"first","affiliation":[{"name":"Communications Department, Faculty of Information Engineering and Technology, German University in Cairo, Cairo 11835, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7717-9393","authenticated-orcid":false,"given":"Yen-Lin","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Information Engineering, National Taipei University of Technology, Taipei 106344, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5865-1533","authenticated-orcid":false,"given":"Lip Yee","family":"Por","sequence":"additional","affiliation":[{"name":"Department of Computer System and Technology, Faculty of Computer Science and Information Technology, Universiti Malaya, Kuala Lumpur 50603, Malaysia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3690-9585","authenticated-orcid":false,"given":"Mohamed","family":"Gabr","sequence":"additional","affiliation":[{"name":"Computer Science Department, Faculty of Media Engineering and Technology, German University in Cairo, Cairo 11835, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1007\/s11042-021-11384-z","article-title":"A color image encryption technique using block scrambling and chaos","volume":"81","author":"Hosny","year":"2022","journal-title":"Multimed. Tools Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"11541","DOI":"10.1109\/ACCESS.2023.3242311","article-title":"Color Image Encryption through Chaos and KAA Map","volume":"11","author":"Alexan","year":"2023","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Alexan, W., Mamdouh, E., ElBeltagy, M., Ashraf, A., Moustafa, M., and Al-Qurashi, H. (2022, January 26\u201328). Social Engineering and Technical Security Fusion. Proceedings of the 2022 International Telecommunications Conference (ITC-Egypt), Alexandria, Egypt.","DOI":"10.1109\/ITC-Egypt55520.2022.9855761"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"117861","DOI":"10.1016\/j.eswa.2022.117861","article-title":"Color image encryption and authentication using dynamic DNA encoding and hyper chaotic system","volume":"206","author":"Jasra","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"17245","DOI":"10.1007\/s11042-022-12604-w","article-title":"An image block encryption algorithm based on hyperchaotic system and DNA encoding","volume":"81","author":"Fang","year":"2022","journal-title":"Multimed. Tools Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Alexan, W., ElBeltagy, M., and Aboshousha, A. (2022). Rgb image encryption through cellular automata, s-box and the lorenz system. Symmetry, 14.","DOI":"10.3390\/sym14030443"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Gabr, M., Younis, H., Ibrahim, M., Alajmy, S., Khalid, I., Azab, E., Elias, R., and Alexan, W. (2022). Application of DNA Coding, the Lorenz Differential Equations and a Variation of the Logistic Map in a Multi-Stage Cryptosystem. Symmetry, 14.","DOI":"10.3390\/sym14122559"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Gabr, M., Alexan, W., Moussa, K., Maged, B., and Mezar, A. (2022, January 26\u201328). Multi-Stage RGB Image Encryption. Proceedings of the 2022 International Telecommunications Conference (ITC-Egypt), Alexandria, Egypt.","DOI":"10.1109\/ITC-Egypt55520.2022.9855767"},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"8","DOI":"10.5120\/cae2017652600","article-title":"Key management techniques in wireless sensor networks","volume":"7","author":"Elqusy","year":"2017","journal-title":"Commun. Appl. Electron. (CAE)"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"167779","DOI":"10.1016\/j.ijleo.2021.167779","article-title":"Proposing a novel Dynamic AES for image encryption using a chaotic map key management approach","volume":"246","author":"Shariatzadeh","year":"2021","journal-title":"Optik"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Alexan, W., Alexan, N., and Gabr, M. (2023). Multiple-Layer Image Encryption Utilizing Fractional-Order Chen Hyperchaotic Map and Cryptographically Secure PRNGs. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7040287"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Alexan, W., ElBeltagy, M., and Aboshousha, A. (2021, January 21\u201322). Image Encryption Through Lucas Sequence, S-Box and Chaos Theory. Proceedings of the 2021 8th NAFOSTED Conference on Information and Computer Science (NICS), Hanoi, Vietnam.","DOI":"10.1109\/NICS54270.2021.9701533"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"100458","DOI":"10.1016\/j.padiff.2022.100458","article-title":"Rich complex dynamics in new fractional-order hyperchaotic systems using a modified Caputo operator based on the extended Gamma function","volume":"6","author":"Matouk","year":"2022","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Younas, I., and Khan, M. (2018). A new efficient digital image encryption based on inverse left almost semi group and Lorenz chaotic system. Entropy, 20.","DOI":"10.3390\/e20120913"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Alexan, W., ElBeltagy, M., and Aboshousha, A. (2021, January 19\u201322). Lightweight image encryption: Cellular automata and the lorenz system. Proceedings of the 2021 International Conference on Microelectronics (ICM), New Cairo City, Egypt.","DOI":"10.1109\/ICM52667.2021.9664961"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"25497","DOI":"10.1007\/s11042-022-12595-8","article-title":"Image encryption based on a combination of multiple chaotic maps","volume":"81","author":"Elkandoz","year":"2022","journal-title":"Multimed. Tools Appl."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"ElBeltagy, M., Alexan, W., Elkhamry, A., Moustafa, M., and Hussein, H.H. (2022, January 26\u201329). Image Encryption through R\u00f6ssler System, PRNG S-Box and Recam\u00e1n\u2019s Sequence. Proceedings of the 2022 IEEE 12th Annual Computing and Communication Workshop and Conference (CCWC), Las Vegas, NV, USA.","DOI":"10.1109\/CCWC54503.2022.9720905"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"118253","DOI":"10.1109\/ACCESS.2021.3106028","article-title":"On the Image Encryption Algorithm Based on the Chaotic System, DNA Encoding and Castle","volume":"9","author":"Iqbal","year":"2021","journal-title":"IEEE Access"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.ijleo.2018.01.064","article-title":"A color image encryption technique using exclusive-OR with DNA complementary rules based on chaos theory and SHA-2","volume":"159","author":"Liao","year":"2018","journal-title":"Optik"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"21803","DOI":"10.1007\/s11042-017-5590-0","article-title":"A new color image encryption scheme based on logistic map over the finite field ZN","volume":"77","author":"Yang","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_22","first-page":"18342","article-title":"Multi-channel chaotic encryption algorithm for color image based on DNA coding","volume":"79","author":"Wang","year":"2020","journal-title":"Multimed. Tools Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"106040","DOI":"10.1016\/j.optlaseng.2020.106040","article-title":"A new color image encryption scheme based on 2DNLCML system and genetic operations","volume":"128","author":"Zhang","year":"2020","journal-title":"Opt. Lasers Eng."},{"key":"ref_24","first-page":"102428","article-title":"Colour image encryption algorithm combining, Arnold map, DNA sequence operation and a Mandelbrot set","volume":"50","author":"Jithin","year":"2020","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"172275","DOI":"10.1109\/ACCESS.2020.3024994","article-title":"A Color Image Encryption Algorithm Based on One Time Key, Chaos Theory and Concept of Rotor Machine","volume":"8","author":"Rehman","year":"2020","journal-title":"IEEE Access"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8911","DOI":"10.1007\/s11042-017-4786-7","article-title":"A novel image encryption scheme based on logistic map and dynatomic modular curve","volume":"77","author":"Li","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"12452","DOI":"10.1109\/ACCESS.2020.2965740","article-title":"Color image encryption algorithm based on dynamic chaos and matrix convolution","volume":"8","author":"Hu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6102824","DOI":"10.1155\/2020\/6102824","article-title":"An image encryption algorithm based on hyperchaotic system and variable-step Josephus problem","volume":"2020","author":"Zhang","year":"2020","journal-title":"Int. J. Opt."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.optlastec.2019.01.039","article-title":"An image compression and encryption algorithm based on chaotic system and compressive sensing","volume":"115","author":"Gong","year":"2019","journal-title":"Opt. Laser Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"118845","DOI":"10.1016\/j.eswa.2022.118845","article-title":"A novel pixel-split image encryption scheme based on 2D Salomon map","volume":"213","author":"Lai","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"119076","DOI":"10.1016\/j.eswa.2022.119076","article-title":"2D hyperchaotic system based on Schaffer function for image encryption","volume":"213","author":"Erkan","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.matcom.2022.07.029","article-title":"A parallel image encryption algorithm using intra bitplane scrambling","volume":"204","author":"Song","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1599","DOI":"10.1016\/S0960-0779(04)00379-0","article-title":"Coexisting chaotic attractors in a single neuron model with adapting feedback synapse","volume":"23","author":"Li","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2016","DOI":"10.1016\/j.chaos.2007.09.095","article-title":"A four-wing attractor and its analysis","volume":"40","author":"Qi","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"905","DOI":"10.3934\/math.2023044","article-title":"A new 4D hyperchaotic system and its control","volume":"8","author":"Cui","year":"2023","journal-title":"AIMS Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1850057","DOI":"10.1142\/S0218127418500578","article-title":"A new 7D hyperchaotic system with five positive Lyapunov exponents coined","volume":"28","author":"Yang","year":"2018","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Bassham, L.E.I., Rukhin, A.L., Soto, J., Nechvatal, J.R., Smid, M.E., Barker, E.B., Leigh, S.D., Levenson, M.D., Vangel, M., and Banks, D.L. (2010). A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, Revised 800-22.","DOI":"10.6028\/NIST.SP.800-22r1a"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"26203","DOI":"10.1007\/s11042-019-07818-4","article-title":"A novel chaotic image encryption technique based on multiple discrete dynamical maps","volume":"78","author":"Khan","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1007\/s00521-014-1800-0","article-title":"An efficient chaotic image encryption scheme","volume":"26","author":"Khan","year":"2015","journal-title":"Neural Comput. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Liu, H., Zhao, B., and Huang, L. (2019). Quantum image encryption scheme using Arnold transform and S-box scrambling. Entropy, 21.","DOI":"10.3390\/e21040343"},{"key":"ref_41","first-page":"31","article-title":"NPCR and UACI randomness tests for image encryption","volume":"1","author":"Wu","year":"2011","journal-title":"Cyber J. Multidiscip. J. Sci. Technol. J. Sel. Areas Telecommun. (JSAT)"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"30993","DOI":"10.1007\/s11042-018-6145-8","article-title":"A novel chaotic image cryptosystem based on DNA sequence operations and single neuron model","volume":"77","author":"Slimane","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"37873","DOI":"10.1007\/s11042-022-13095-5","article-title":"A novel colour image encryption scheme using dynamic DNA coding, chaotic maps and SHA-2","volume":"81","author":"Paul","year":"2022","journal-title":"Multimed. Tools Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"12349","DOI":"10.1007\/s11042-017-4885-5","article-title":"A robust and lossless DNA encryption scheme for color images","volume":"77","author":"Wu","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2129","DOI":"10.1142\/S0218127406015970","article-title":"Some basic cryptographic requirements for chaos-based cryptosystems","volume":"16","author":"Alvarez","year":"2006","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/10408340500526766","article-title":"The correlation coefficient: An overview","volume":"36","author":"Asuero","year":"2006","journal-title":"Crit. Rev. Anal. Chem."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"35682","DOI":"10.1109\/ACCESS.2020.2973679","article-title":"Generation of Highly Nonlinear and Dynamic AES Substitution-Boxes (S-Boxes) Using Chaos-Based Rotational Matrices","volume":"8","author":"Ali","year":"2020","journal-title":"IEEE Access"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1007\/s12652-021-03675-y","article-title":"Novel encryption for color images using fractional-order hyperchaotic system","volume":"13","author":"Hosny","year":"2022","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_49","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_50","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_51","doi-asserted-by":"crossref","unstructured":"Siddiqui, N., Yousaf, F., Murtaza, F., Ehatisham-ul-Haq, M., Ashraf, M.U., Alghamdi, A.M., and Alfakeeh, A.S. (2020). A highly nonlinear substitution-box (S-box) design using action of modular group on a projective line over a finite field. PLoS ONE, 15.","DOI":"10.1371\/journal.pone.0241890"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1081\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:34:39Z","timestamp":1760124879000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1081"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,14]]},"references-count":51,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15051081"],"URL":"https:\/\/doi.org\/10.3390\/sym15051081","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,14]]}}}