{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T00:14:02Z","timestamp":1782951242925,"version":"3.54.5"},"reference-count":47,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T00:00:00Z","timestamp":1778198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Planning Project of Gansu","award":["23JRRA894, 24JRRA243"],"award-info":[{"award-number":["23JRRA894, 24JRRA243"]}]},{"name":"Humanities and Social Sciences Research Planning Fund of the Ministry of Education of China","award":["24YJAZH242"],"award-info":[{"award-number":["24YJAZH242"]}]},{"name":"Open Foundation of Key Laboratory of Computing Power Network and Information Security","award":["SKLCN-2023-02"],"award-info":[{"award-number":["SKLCN-2023-02"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Most existing low-dimensional chaotic maps suffer from a limited dynamical complexity and dynamic degradation, which restrict their effectiveness in image encryption. To address this issue, a novel three-dimensional chaotic map (3D-TCM) was constructed to improve dynamical complexity and stability, and its superiority was verified through a dynamical analysis. Based on these advantages, a plaintext-related image encryption scheme was designed by combining bit-level permutation and S-box-based diffusion. The experimental results show that the proposed scheme achieved high information entropy, a low pixel correlation, and desirable NPCR and UACI values, and successfully passed NIST SP800-22 statistical tests, demonstrating a strong resistance to differential attacks and overall robustness.<\/jats:p>","DOI":"10.3390\/e28050535","type":"journal-article","created":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T20:11:13Z","timestamp":1778271073000},"page":"535","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["3D-TCM-Driven Bit-Level Image Encryption via S-Box Feedback Algorithm"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-9403-0696","authenticated-orcid":false,"given":"Jie","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Electronics and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenjie","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Electronics and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mingxu","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Information Engineering, Dalian Ocean University, Dalian 116023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4159-3132","authenticated-orcid":false,"given":"Yiting","family":"Lin","sequence":"additional","affiliation":[{"name":"University of Electronic Science and Technology of China Zhongshan Institute, Zhongshan 528406, China"},{"name":"University of Electronic Science and Technology of China, Chengdu 611731, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"121452","DOI":"10.1016\/j.eswa.2023.121452","article-title":"PSO-based image encryption scheme using modular integrated logistic exponential map","volume":"237","author":"Kocak","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5237","DOI":"10.1109\/TSMC.2024.3401412","article-title":"Multiobjective design of 2D hyperchaotic system using Leader Pareto Grey Wolf optimizer","volume":"54","author":"Toktas","year":"2024","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1142\/S021812749800098X","article-title":"Symmetric ciphers based on two-dimensional chaotic maps","volume":"8","author":"Fridrich","year":"1998","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"9469","DOI":"10.1038\/s41598-026-38483-x","article-title":"A novel 1D powered Chebyshev quadratic map-based image encryption using dynamic permutation-diffusion","volume":"16","author":"Sarra","year":"2026","journal-title":"Sci. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"124891","DOI":"10.1016\/j.eswa.2024.124891","article-title":"Image privacy protection scheme based on high-quality reconstruction DCT compression and nonlinear dynamics","volume":"257","author":"Lin","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"115987","DOI":"10.1016\/j.chaos.2024.115987","article-title":"Degn-Harrison map: Dynamical and network behaviours with applications in image encryption","volume":"192","author":"Vismaya","year":"2025","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"40268","DOI":"10.1109\/JIOT.2024.3450547","article-title":"A novel Fibonacci-Sequence-based chaotification model for enhancing chaos in 1-D maps","volume":"11","author":"Nazish","year":"2025","journal-title":"IEEE Internet Things J."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lin, Y., Liao, Y., Zeng, W., Wei, Y., Chen, D., Yuan, X., Li, Y., Erkan, U., Toktas, A., and Zhang, C. (2026). 3D non-degenerate hyperchaos: Design, analysis, and application in image encryption. IEEE Trans. Consum. Electron.","DOI":"10.1109\/TCE.2026.3672135"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"55182","DOI":"10.1109\/JIOT.2025.3624032","article-title":"Using 3D-LMM-based encryption to secure digital images with 3D S-box and Fibonacci Q-Matrix","volume":"12","author":"Liao","year":"2025","journal-title":"IEEE Internet Things J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1140\/epjp\/s13360-026-07421-1","article-title":"SRA: A sine-based reconstruction approach to improve 2D chaotic map performance","volume":"141","author":"Ding","year":"2026","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"073105","DOI":"10.1063\/5.0270220","article-title":"A three-dimensional memristor-based hyperchaotic map for pseudorandom number generation and multi-image encryption","volume":"35","author":"Gao","year":"2025","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Lin, Y., Wei, Y., Chen, D., Li, Y., Erkan, U., Toktas, A., Gao, S., and Zhang, Y. (2026). Cryptanalysis and Improvement of a Video Cryptosystem via Chaos and S-Box. ACM Trans. Multimed. Comput. Commun. Appl.","DOI":"10.1145\/3808699"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/s11227-025-08196-5","article-title":"Design of a three-dimensional Logistic map and its application to seafood image encryption","volume":"82","author":"Ding","year":"2026","journal-title":"J. Supercomput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4464","DOI":"10.1109\/TCE.2025.3534859","article-title":"DDCM: Cracking anonymized facial images using denoising diffusion cryptanalytic model","volume":"71","author":"Jiang","year":"2025","journal-title":"IEEE Trans. Consum. Electron."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"19590","DOI":"10.1109\/JIOT.2026.3663610","article-title":"A fast multi-image encryption scheme using hyperchaotic map and parallel algorithm for vehicle detection applications","volume":"13","author":"Lai","year":"2026","journal-title":"IEEE Internet Things J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/s00224-026-10274-x","article-title":"Algebraically enhanced 3D chaotic map with hash-based initialization for secure image encryption","volume":"70","author":"Kareem","year":"2026","journal-title":"Theory Comput. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Aibai, A., Nuermaimaiti, M., Tuersun, Y., and Ghopur, D. (2026). A novel 2D hyperchaotic map for secure financial data encryption. Entropy, 28.","DOI":"10.3390\/e28030262"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s11071-025-12116-w","article-title":"A multi-image encryption scheme for gray-color images based on 2D chaotic map and dual S-boxes","volume":"114","author":"Duan","year":"2026","journal-title":"Nonlinear Dyn."},{"key":"ref_19","first-page":"104389","article-title":"A 3D chaotic map-based novel intra and inter-level bit plane image content encryption","volume":"98","author":"Deepankumar","year":"2026","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"105941","DOI":"10.1016\/j.dsp.2026.105941","article-title":"Bit-level quad-block shuffling and sequential summing dispersing image encryption based on hyperchaotic 2D Euler Pi crossed Sine map","volume":"173","author":"Kocak","year":"2026","journal-title":"Digit. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"124238","DOI":"10.1016\/j.eswa.2024.124238","article-title":"Cryptanalysis and construction of keyed strong S-Box based on random affine transformation matrix and 2D hyper chaotic map","volume":"252","author":"Liu","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"21603","DOI":"10.1038\/s41598-025-04968-4","article-title":"A color image encryption scheme utilizing a logistic-sine chaotic map and cellular automata","volume":"15","author":"Sun","year":"2025","journal-title":"Sci. Rep."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"31691","DOI":"10.1007\/s11071-025-11690-3","article-title":"A chaotic multi-image encryption scheme based on block space jump scrambling and dynamic sliding queue diffusion","volume":"113","author":"Gao","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"14543","DOI":"10.1109\/TIE.2025.3581192","article-title":"Exploiting Compressed Sensing and DNA Coding for multi-image encryption scheme with application to industrial internet of things","volume":"72","author":"Wang","year":"2025","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"107790","DOI":"10.1016\/j.jfranklin.2025.107790","article-title":"A novel image encryption scheme based on a 3D enhanced chaotic map and DNA computing model","volume":"362","author":"Wang","year":"2025","journal-title":"J. Frankl. Inst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"124304","DOI":"10.1063\/5.0229908","article-title":"Designing a novel image encryption scheme based on an improved 2D logistic map","volume":"136","author":"Liu","year":"2024","journal-title":"J. Appl. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4689","DOI":"10.1109\/TCSVT.2023.3246520","article-title":"Visualized multiple image selection encryption based on Log chaos system and multilayer cellular automata saliency detection","volume":"33","author":"Su","year":"2023","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_28","first-page":"128340","article-title":"A robust bit-level image encryption based on Bessel map","volume":"462","author":"Toktas","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.ins.2022.05.032","article-title":"Image encryption algorithm based on 2D-CLSS hyperchaotic map using simultaneous permutation and diffusion","volume":"605","author":"Teng","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1109\/TLA.2024.10472959","article-title":"An audio encryption scheme based on empirical mode decomposition and 2D Cosine Logistic map","volume":"22","author":"Maity","year":"2024","journal-title":"IEEE Lat. Am. Trans."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"129574","DOI":"10.1016\/j.eswa.2025.129574","article-title":"A robust medical image encryption technique using inverse cosine chaotic map","volume":"298","author":"Jackson","year":"2026","journal-title":"Expert Syst. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"125328","DOI":"10.1016\/j.eswa.2024.125328","article-title":"A new 2D cross hyperchaotic Sine-modulation-Logistic map and its application in bit-level image encryption","volume":"261","author":"Wang","year":"2025","journal-title":"Expert Syst. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"\u00d6zpolat, E., \u00c7elik, V., and G\u00fclten, A. (2025). Hyperchaotic system-based PRNG and S-box design for a novel secure image encryption. Entropy, 27.","DOI":"10.3390\/e27030299"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Qiu, D., Zhang, T., Liu, J., Liu, S., and He, P. (2025). An innovative image encryption algorithm based on the DNAS_box and hyperchaos. Entropy, 27.","DOI":"10.3390\/e27030239"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Sun, M., Yuan, J., Li, X., Liu, D., and Wei, X. (2024). LVGG-IE: A novel lightweight VGG-based image encryption scheme. Entropy, 26.","DOI":"10.3390\/e26121013"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2150146","DOI":"10.1142\/S0218127421501467","article-title":"Constructing Keyed Strong S-Box Using an Enhanced Quadratic Map","volume":"31","author":"Si","year":"2021","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_37","first-page":"125153","article-title":"Cryptanalysis and constructing S-Box based on chaotic map and backtracking","volume":"376","author":"Liu","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_38","first-page":"619","article-title":"Highly dispersive substitution box (S-Box) design using chaos","volume":"42","author":"Faheem","year":"2020","journal-title":"Electron. Telecommun. Res. Inst. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1930","DOI":"10.1038\/s41598-020-58636-w","article-title":"Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications","volume":"10","author":"Amin","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/j.matcom.2025.06.009","article-title":"Color image encryption scheme based on a novel 2D-CLCM chaotic system and RNA encoding","volume":"239","author":"Liu","year":"2026","journal-title":"Math. Comput. Simul."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"25411","DOI":"10.1038\/s41598-025-10485-1","article-title":"A novel 2D MTMHM based key generation for enhanced security in medical image communication","volume":"15","author":"Devi","year":"2025","journal-title":"Sci. Rep."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"128237","DOI":"10.1016\/j.eswa.2025.128237","article-title":"A reliable color image encryption scheme based on a novel dual-wing hyperchaotic map","volume":"289","author":"Obaid","year":"2025","journal-title":"Expert Syst. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"82049","DOI":"10.1007\/s11042-024-18741-8","article-title":"Constructing a non-degeneracy 3D hyperchaotic map and application in image encryption","volume":"83","author":"Lin","year":"2024","journal-title":"Multimed. Tools Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1007\/s10489-025-06414-4","article-title":"New 2D hyperchaotic Cubic-Tent map and improved 3D Hilbert diffusion for image encryption","volume":"55","author":"Xu","year":"2025","journal-title":"Appl. Intell."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2511","DOI":"10.1109\/TCSI.2023.3250713","article-title":"Globally coupled private image encryption algorithm based on infinite interval spatiotemporal chaotic system","volume":"70","author":"Liu","year":"2023","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"23179","DOI":"10.1007\/s11042-022-14133-y","article-title":"A novel image encryption scheme based on 2D SILM and improved permutation-confusion-diffusion operations","volume":"82","author":"Liu","year":"2023","journal-title":"Multimed. Tools Appl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"075203","DOI":"10.1088\/1402-4896\/ae3fde","article-title":"Spatiotemporal chaos in a recursively permuted lattice with dual maps on odd and even sites and its application in image encryption","volume":"101","author":"Su","year":"2026","journal-title":"Phys. Scr."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/535\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T04:38:22Z","timestamp":1779251902000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/5\/535"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,8]]},"references-count":47,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["e28050535"],"URL":"https:\/\/doi.org\/10.3390\/e28050535","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,8]]}}}