{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T07:37:56Z","timestamp":1768289876225,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,7,31]],"date-time":"2023-07-31T00:00:00Z","timestamp":1690761600000},"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>It is well known that, compared to low-dimension chaotic systems, three-dimensional chaotic systems have a wider parameter range, more complicated behaviour, and better unpredictability. This fact motivated us to introduce a novel image encryption method that employs a three-dimensional chaotic system. We proposed a novel three-dimensional conservative system that can exhibit chaotic behaviour involving hyperbolic functions. The dynamical behaviours of the proposed system are discovered by calculating Lyapunov exponents and bifurcation diagrams. Thereafter, we designed an image encryption method based on the proposed system and a 4\u00d74 self-invertible matrix. A modified Diffie\u2013Hellman key exchange protocol was utilised to generate the self-invertible key matrix Km employed in the diffusion stage. Our approach has three main stages. In the first stage, the proposed three-dimensional system utilises the original image to create three sequences, two of which are chosen for confusion and diffusion processes. The next stage involves confusing the image\u2019s pixels by changing the positions of pixels using these sequences. In the third stage, the confused image is split into sub-blocks of size 4\u00d74, and each block is encrypted by multiplying it with Km. Simulation findings demonstrated that the proposed image scheme has a high level of security and is resistant to statistical analysis, noise, and other attacks.<\/jats:p>","DOI":"10.3390\/sym15081511","type":"journal-article","created":{"date-parts":[[2023,7,31]],"date-time":"2023-07-31T09:22:38Z","timestamp":1690795358000},"page":"1511","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["A Novel Conservative Chaotic System Involved in Hyperbolic Functions and Its Application to Design an Efficient Colour Image Encryption Scheme"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4833-781X","authenticated-orcid":false,"given":"Ammar Ali","family":"Neamah","sequence":"first","affiliation":[{"name":"Faculty of Computer Science and Mathematics, University of Kufa, Najaf 54001, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3513-9216","authenticated-orcid":false,"given":"Ali A.","family":"Shukur","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science and Mathematics, University of Kufa, Najaf 54001, Iraq"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,31]]},"reference":[{"key":"ref_1","first-page":"739","article-title":"Selective image encryption using singular value decomposition and arnold transform","volume":"15","author":"Naik","year":"2018","journal-title":"Int. Arab J. Inf. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1950383","DOI":"10.1142\/S0217979219503831","article-title":"A novel 4D chaotic system with nonhyperbolic hyperbolic shape equilibrium points: Analysis, circuit implementation and color image encryption","volume":"33","author":"Cai","year":"2019","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Liu, H., Zhao, B., and Huang, L.Q. (2019). Quantum image encryption scheme using arnold transform and s-box scrambling. Entropy, 21.","DOI":"10.3390\/e21040343"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1007\/s10773-014-2245-z","article-title":"Quantum image encryption algorithm based on image correlation decomposition","volume":"54","author":"Hua","year":"2015","journal-title":"Int. J. Theor. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2050022","DOI":"10.1142\/S0219749920500227","article-title":"Quantum image encryption based on Lorenz hyper-chaotic system","volume":"18","author":"Zhou","year":"2020","journal-title":"Int. J. Quantum Inf."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"21044","DOI":"10.1038\/s41598-020-78127-2","article-title":"An image encryption scheme based on public key cryptosystem and quantum logistic system","volume":"10","author":"Ye","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2191","DOI":"10.1007\/s11042-017-4370-1","article-title":"A fast and efficient approach to color image encryption based on compressive sensing and fractional Fourier transform","volume":"77","author":"Zhang","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"58751","DOI":"10.1109\/ACCESS.2019.2914722","article-title":"Color image compression-encryption algorithm based on fractional-order memristor chaotic circuit","volume":"7","author":"Yang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.optlaseng.2016.10.019","article-title":"Color image encryption based on hybrid hyper-chaotic system and cellular automata","volume":"90","author":"Niyat","year":"2017","journal-title":"Opt. Lasers Eng."},{"key":"ref_11","first-page":"117","article-title":"A secure image encryption scheme based on cellular automata and chaotic skew tent map","volume":"45","author":"Mondal","year":"2019","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"115829","DOI":"10.1016\/j.image.2020.115829","article-title":"A novel method for lossless image compression and encryption based on LWT, SPIHT and cellular automata","volume":"84","author":"Zhang","year":"2020","journal-title":"Signal Process. Image Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6954","DOI":"10.3390\/e17106954","article-title":"A novel image encryption algorithm based on DNA encoding and spatiotemporal chaos","volume":"17","author":"Song","year":"2015","journal-title":"Entropy"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.sigpro.2018.06.008","article-title":"Image encryption using 2D Henon-Sine map and DNA approach","volume":"153","author":"Wu","year":"2018","journal-title":"Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wan, Y., Gu, S., and Du, B. (2020). A new image encryption algorithm based on composite chaos and hyperchaos combined with DNA coding. Entropy, 22.","DOI":"10.3390\/e22020171"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Lone, P.N., Singh, D., Stoffova, V., Mishra, D.C., Mir, U.H., and Kumar, N. (2022). Cryptanalysis and Improved Image Encryption Scheme Using Elliptic Curve and Affine Hill Cipher. Mathematics, 10.","DOI":"10.3390\/math10203878"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Hayat, U., Ullah, I., Azam, N.A., and Azhar, S. (2022). A Novel Image Encryption Scheme Based on Elliptic Curves over Finite Rings. Entropy, 24.","DOI":"10.3390\/e24050571"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5483","DOI":"10.1109\/ACCESS.2022.3230096","article-title":"An integrated image encryption scheme based on elliptic curve","volume":"11","author":"Khalid","year":"2022","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Borah, M., and Roy, B.K. (2018, January 1\u20132). A novel multi-wing fractional-order chaotic system, its synchronisation control and application in secure communication. Proceedings of the 2018 2nd International Conference on Power, Energy and Environment: Towards Smart Technology (ICEPE), Shillong, India.","DOI":"10.1109\/EPETSG.2018.8658986"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"106202","DOI":"10.1016\/j.optlaseng.2020.106202","article-title":"Image encryption based on chaotic sub-block scrambling and chaotic digit selection diffusion","volume":"134","author":"Xian","year":"2020","journal-title":"Opt. Lasers Eng."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Nestor, T., Belazi, A., Abd-El-Atty, B., Aslam, M.N., Volos, C., Dieu, N.J.D., and El-Latif, A.A. (2022). A new 4D hyperchaotic system with dynamics analysis, synchronization, and application to image encryption. Symmetry, 14.","DOI":"10.3390\/sym14020424"},{"key":"ref_22","first-page":"238","article-title":"An image encryption scheme based on a seven-dimensional hyperchaotic system and Pascal\u2019s matrix","volume":"35","author":"Neamah","year":"2023","journal-title":"J. King Saud Univ.-Comput. Inf. Sci."},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1038\/261459a0","article-title":"Simple mathematical models with very complicated dynamics","volume":"261","author":"May","year":"1976","journal-title":"Nature"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/0375-9601(76)90101-8","article-title":"An equation for continuous chaos","volume":"57","year":"1976","journal-title":"Phys. Lett. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1142\/S0218127499001024","article-title":"Yet another chaotic attractor","volume":"9","author":"Chen","year":"1999","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3747","DOI":"10.1140\/epjst\/e2018-00063-9","article-title":"Fractional-order systems with diverse dynamical behaviour and their switching-parameter hybrid-synchronisation","volume":"226","author":"Borah","year":"2017","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_29","first-page":"102829","article-title":"A novel hybrid permutation substitution base colored image encryption scheme for multimedia data","volume":"59","author":"Naseer","year":"2021","journal-title":"J. Inf. Secur."},{"key":"ref_30","first-page":"9926","article-title":"FPGA implementation of a chaos-based image encryption algorithm","volume":"34","author":"Maazouz","year":"2022","journal-title":"J. King Saud Univ. Comput. Inf. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2310","DOI":"10.1049\/iet-ipr.2019.1340","article-title":"Novel image encryption by combining dynamic DNA sequence encryption and the improved 2D logistic sine map","volume":"14","author":"Zheng","year":"2020","journal-title":"IET Image Process."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3749","DOI":"10.1007\/s11042-020-09854-x","article-title":"A 2D logistic map and Lorenz-Rossler chaotic system based RGB image encryption approach","volume":"80","author":"Kumar","year":"2021","journal-title":"Multimed. Tools Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"18317","DOI":"10.1007\/s11042-020-08742-8","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_34","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_35","doi-asserted-by":"crossref","first-page":"1830009","DOI":"10.1142\/S0218127418300094","article-title":"Chen system as a controlled weather model-physical principle, engineering design and real applications","volume":"28","author":"Sooraksa","year":"2018","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"726","DOI":"10.3390\/sym15030726","article-title":"Chaotification of 1D Maps by Multiple Remainder Operator Additions-Application to B-Spline Curve Encryption","volume":"15","author":"Moysis","year":"2023","journal-title":"Symmetry"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1038\/370615a0","article-title":"Controlling chaos in the brain","volume":"370","author":"Schiff","year":"1994","journal-title":"Nature"},{"key":"ref_38","first-page":"49","article-title":"Do We Need More Chaos Examples?","volume":"2","author":"Sprott","year":"2020","journal-title":"Chaos Theory Appl. Appl. Sci. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1750152","DOI":"10.1142\/S0218127417501528","article-title":"A new chaotic attractor around a pre-located ring","volume":"27","author":"Wang","year":"2017","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Diffie, W., and Hellman, M.E. (1976, January 7\u201310). Multiuser cryptographic techniques. Proceedings of the National Computer Conference and Exposition, New York, NY, USA.","DOI":"10.1145\/1499799.1499815"},{"key":"ref_41","first-page":"14","article-title":"Novel methods of generating self-invertible matrix for hill cipher algorithm","volume":"1","author":"Acharya","year":"2007","journal-title":"Int. J. Secur."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1511\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:23:03Z","timestamp":1760127783000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1511"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,31]]},"references-count":41,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["sym15081511"],"URL":"https:\/\/doi.org\/10.3390\/sym15081511","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,31]]}}}