{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T16:50:26Z","timestamp":1777913426470,"version":"3.51.4"},"reference-count":46,"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\/"}],"funder":[{"name":"Research Foundation of Education Bureau of Hunan Province of China","award":["20A219"],"award-info":[{"award-number":["20A219"]}]},{"name":"Research Foundation of Education Bureau of Hunan Province of China","award":["2021KSA01008"],"award-info":[{"award-number":["2021KSA01008"]}]},{"name":"Research Foundation of Education Bureau of Hunan Province of China","award":["GJXHXXH21-YB-27"],"award-info":[{"award-number":["GJXHXXH21-YB-27"]}]},{"name":"Research Foundation of Education Bureau of Hunan Province of China","award":["2020ZD-S-40"],"award-info":[{"award-number":["2020ZD-S-40"]}]},{"name":"Science and Technology Development Center Project of Chinese Ministry of Education","award":["20A219"],"award-info":[{"award-number":["20A219"]}]},{"name":"Science and Technology Development Center Project of Chinese Ministry of Education","award":["2021KSA01008"],"award-info":[{"award-number":["2021KSA01008"]}]},{"name":"Science and Technology Development Center Project of Chinese Ministry of Education","award":["GJXHXXH21-YB-27"],"award-info":[{"award-number":["GJXHXXH21-YB-27"]}]},{"name":"Science and Technology Development Center Project of Chinese Ministry of Education","award":["2020ZD-S-40"],"award-info":[{"award-number":["2020ZD-S-40"]}]},{"name":"Project of the Sichuan Higher Education Society of China","award":["20A219"],"award-info":[{"award-number":["20A219"]}]},{"name":"Project of the Sichuan Higher Education Society of China","award":["2021KSA01008"],"award-info":[{"award-number":["2021KSA01008"]}]},{"name":"Project of the Sichuan Higher Education Society of China","award":["GJXHXXH21-YB-27"],"award-info":[{"award-number":["GJXHXXH21-YB-27"]}]},{"name":"Project of the Sichuan Higher Education Society of China","award":["2020ZD-S-40"],"award-info":[{"award-number":["2020ZD-S-40"]}]},{"name":"Guiding Science and Technology Plan Project of Panzhihua City","award":["20A219"],"award-info":[{"award-number":["20A219"]}]},{"name":"Guiding Science and Technology Plan Project of Panzhihua City","award":["2021KSA01008"],"award-info":[{"award-number":["2021KSA01008"]}]},{"name":"Guiding Science and Technology Plan Project of Panzhihua City","award":["GJXHXXH21-YB-27"],"award-info":[{"award-number":["GJXHXXH21-YB-27"]}]},{"name":"Guiding Science and Technology Plan Project of Panzhihua City","award":["2020ZD-S-40"],"award-info":[{"award-number":["2020ZD-S-40"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Over the past few years, chaotic image encryption has gained extensive attention. Nevertheless, the current studies on chaotic image encryption still possess certain constraints. To break these constraints, we initially created a two-dimensional enhanced logistic modular map (2D-ELMM) and subsequently devised a chaotic image encryption scheme based on vector-level operations and 2D-ELMM (CIES-DVEM). In contrast to some recent schemes, CIES-DVEM features remarkable advantages in several aspects. Firstly, 2D-ELMM is not only simpler in structure, but its chaotic performance is also significantly better than that of some newly reported chaotic maps. Secondly, the key stream generation process of CIES-DVEM is more practical, and there is no need to replace the secret key or recreate the chaotic sequence when handling different images. Thirdly, the encryption process of CIES-DVEM is dynamic and closely related to plaintext images, enabling it to withstand various attacks more effectively. Finally, CIES-DVEM incorporates lots of vector-level operations, resulting in a highly efficient encryption process. Numerous experiments and analyses indicate that CIES-DVEM not only boasts highly significant advantages in terms of encryption efficiency, but it also surpasses many recent encryption schemes in practicality and security.<\/jats:p>","DOI":"10.3390\/e25081147","type":"journal-article","created":{"date-parts":[[2023,7,31]],"date-time":"2023-07-31T09:22:38Z","timestamp":1690795358000},"page":"1147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":94,"title":["Exploiting Dynamic Vector-Level Operations and a 2D-Enhanced Logistic Modular Map for Efficient Chaotic Image Encryption"],"prefix":"10.3390","volume":"25","author":[{"given":"Hongmin","family":"Li","sequence":"first","affiliation":[{"name":"Key Laboratory of Hunan Province on Information Photonics and Freespace Optical Communications, Hunan Institute of Science and Technology, Yueyang 414006, China"},{"name":"College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China"}]},{"given":"Shuqi","family":"Yu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Hunan Province on Information Photonics and Freespace Optical Communications, Hunan Institute of Science and Technology, Yueyang 414006, China"},{"name":"College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3023-5225","authenticated-orcid":false,"given":"Wei","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}]},{"given":"Yao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9834-081X","authenticated-orcid":false,"given":"Jing","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1583-6164","authenticated-orcid":false,"given":"Zhentao","family":"Qin","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}]},{"given":"Zhengguo","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9073-5347","authenticated-orcid":false,"given":"Marcin","family":"Wozniak","sequence":"additional","affiliation":[{"name":"Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2022.01.042","article-title":"An image encryption algorithm based on new generalized fusion fractal structure","volume":"592","author":"Ahmad","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_2","first-page":"77","article-title":"Security analysis and enhancements of an image cryptosystem based on hyperchaotic system","volume":"33","author":"Ahmad","year":"2021","journal-title":"J. King Saud Univ.-Comput. Inf. Sci."},{"key":"ref_3","first-page":"1535","article-title":"An effective multiple-image encryption algorithm based on 3D cube and hyperchaotic map","volume":"34","author":"Gao","year":"2022","journal-title":"J. King Saud-Univ.-Comput. Inf. Sci."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"112017","DOI":"10.1016\/j.chaos.2022.112017","article-title":"Hidden coexisting hyperchaos of new memristive neuron model and its application in image encryption","volume":"158","author":"Lai","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3225","DOI":"10.1140\/epjs\/s11734-022-00638-y","article-title":"Bit-level image encryption algorithm based on random-time S-Box substitution","volume":"231","author":"Wang","year":"2022","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"1392","DOI":"10.3390\/electronics10121392","article-title":"Efficient chaos-based substitution-box and its application to image encryption","volume":"10","author":"Belazi","year":"2021","journal-title":"Electronics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2303","DOI":"10.1109\/ACCESS.2022.3233321","article-title":"Applicable Image Security Based on Computational Genetic Approach and Self-Adaptive Substitution","volume":"11","author":"Shaltout","year":"2022","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2434","DOI":"10.3390\/math10142434","article-title":"A Novel Chaos-Based Cryptography Algorithm and Its Performance Analysis","volume":"10","author":"Ramadoss","year":"2022","journal-title":"Mathematics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.sigpro.2019.06.010","article-title":"An overview of encryption algorithms in color images","volume":"164","author":"Ghadirli","year":"2019","journal-title":"Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1007\/s11071-021-06654-2","article-title":"An amplitude-controllable 3D hyperchaotic map with homogenous multistability","volume":"105","author":"Zhou","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"341","DOI":"10.3390\/e23030341","article-title":"A new algorithm for digital image encryption based on chaos theory","volume":"23","author":"Pourasad","year":"2021","journal-title":"Entropy"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4028","DOI":"10.1109\/TCSVT.2021.3108767","article-title":"Double parameters fractal sorting matrix and its application in image encryption","volume":"32","author":"Xian","year":"2021","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"37855","DOI":"10.1109\/ACCESS.2021.3063237","article-title":"A New Image Encryption Algorithm for Grey and Color Medical Images","volume":"9","author":"Kamal","year":"2021","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4505","DOI":"10.1007\/s11071-021-06472-6","article-title":"Color image encryption using orthogonal Latin squares and a new 2D chaotic system","volume":"104","author":"Hua","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_17","first-page":"102844","article-title":"A novel image encryption scheme based on non-adjacent parallelable permutation and dynamic DNA-level two-way diffusion","volume":"61","author":"Li","year":"2021","journal-title":"J. Inf. Secur. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"24993","DOI":"10.1007\/s11042-020-09111-1","article-title":"An image encryption scheme based on a hybrid model of DNA computing, chaotic systems and hash functions","volume":"79","author":"Zefreh","year":"2020","journal-title":"Multimed. Tools Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"718","DOI":"10.3389\/fphy.2022.963795","article-title":"A novel image encryption scheme based on memristive chaotic system and combining bidirectional bit-level cyclic shift and dynamic DNA-level diffusion","volume":"10","author":"Qian","year":"2022","journal-title":"Front. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2751","DOI":"10.3390\/math10152751","article-title":"Image Encryption Algorithm Based on Plane-Level Image Filtering and Discrete Logarithmic Transform","volume":"10","author":"Feng","year":"2022","journal-title":"Mathematics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"108489","DOI":"10.1016\/j.sigpro.2022.108489","article-title":"A stable meaningful image encryption scheme using the newly-designed 2D discrete fractional-order chaotic map and Bayesian compressive sensing","volume":"195","author":"Zhu","year":"2022","journal-title":"Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"112133","DOI":"10.1016\/j.chaos.2022.112133","article-title":"Design and DSP implementation of a fractional-order detuned laser hyperchaotic circuit with applications in image encryption","volume":"159","author":"Li","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1937","DOI":"10.1109\/TSP.2020.2979596","article-title":"Two-dimensional modular chaotification system for improving chaos complexity","volume":"68","author":"Hua","year":"2020","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"45","DOI":"10.51537\/chaos.1069002","article-title":"Lyapunov exponent enhancement in chaotic maps with uniform distribution modulo one transformation","volume":"4","author":"Ablay","year":"2022","journal-title":"Chaos Theory Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2916","DOI":"10.1109\/TIE.2022.3174288","article-title":"Buffeting chaotification model for enhancing chaos and its hardware implementation","volume":"70","author":"Zhang","year":"2022","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4402","DOI":"10.1109\/TSMC.2021.3096967","article-title":"Two-dimensional parametric polynomial chaotic system","volume":"52","author":"Hua","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"113108","DOI":"10.1063\/5.0090518","article-title":"New 4D and 3D models of chaotic systems developed from the dynamic behavior of nuclear reactors","volume":"32","author":"Ablay","year":"2022","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2705","DOI":"10.1007\/s11071-022-07335-4","article-title":"Remote sensing image compression and encryption based on block compressive sensing and 2D-LCCCM","volume":"108","author":"Nan","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"59313","DOI":"10.1109\/ACCESS.2021.3070350","article-title":"2D-SCMCI Hyperchaotic Map for Image Encryption Algorithm","volume":"9","author":"Sun","year":"2021","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1109\/MMUL.2021.3114589","article-title":"Cryptanalyzing two image encryption algorithms based on a first-order time-delay system","volume":"29","author":"Liu","year":"2022","journal-title":"IEEE Multimed."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"103424","DOI":"10.1016\/j.jvcir.2021.103424","article-title":"Security Measurement of a Medical Image Communication Scheme based on Chaos and DNA","volume":"83","author":"Chen","year":"2022","journal-title":"J. Vis. Commun. Image Represent."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"145459","DOI":"10.1109\/ACCESS.2021.3123571","article-title":"Cryptanalysis and Improvement of the Image Encryption Scheme Based on Feistel Network and Dynamic DNA Encoding","volume":"9","author":"Feng","year":"2021","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1016\/j.ijleo.2018.12.103","article-title":"Cryptanalysis of the integrated chaotic systems based image encryption algorithm","volume":"186","author":"Feng","year":"2019","journal-title":"Optik"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Zhang, C., Chen, J., and Chen, D. (2022). Cryptanalysis of an Image Encryption Algorithm Based on a 2D Hyperchaotic Map. Entropy, 24.","DOI":"10.3390\/e24111551"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"085211","DOI":"10.1088\/1402-4896\/acdeb1","article-title":"An image encryption scheme based on elementary and life-liked cellular automatons","volume":"98","author":"Lv","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3887","DOI":"10.1007\/s11071-022-08021-1","article-title":"A robustness-improved image encryption scheme utilizing Life-liked cellular automaton","volume":"111","author":"Lv","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.1007\/s11071-021-06993-0","article-title":"Parameter identification for discrete memristive chaotic map using adaptive differential evolution algorithm","volume":"107","author":"Peng","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"153539","DOI":"10.1016\/j.aeue.2020.153539","article-title":"A higher dimensional chaotic map with discrete memristor","volume":"129","author":"Peng","year":"2021","journal-title":"Aeu-Int. J. Electron. Commun."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"112519","DOI":"10.1016\/j.chaos.2022.112519","article-title":"n-Dimensional Chaotic Map with application in secure communication","volume":"163","author":"Cao","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2591","DOI":"10.1103\/PhysRevA.28.2591","article-title":"Estimation of the Kolmogorov entropy from a chaotic signal","volume":"28","author":"Grassberger","year":"1983","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"108041","DOI":"10.1016\/j.sigpro.2021.108041","article-title":"Combining improved genetic algorithm and matrix semi-tensor product (STP) in color image encryption","volume":"183","author":"Chai","year":"2021","journal-title":"Signal Process."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2671","DOI":"10.1007\/s11071-022-07328-3","article-title":"An image encryption scheme based on multi-objective optimization and block compressed sensing","volume":"108","author":"Chai","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_43","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_44","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s11071-018-4056-x","article-title":"Brief review on application of nonlinear dynamics in image encryption","volume":"92","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.ins.2018.12.048","article-title":"Cosine-transform-based chaotic system for image encryption","volume":"480","author":"Hua","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.sigpro.2018.03.010","article-title":"2D Logistic-Sine-coupling map for image encryption","volume":"149","author":"Hua","year":"2018","journal-title":"Signal Process."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1147\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:23:15Z","timestamp":1760127795000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1147"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,31]]},"references-count":46,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["e25081147"],"URL":"https:\/\/doi.org\/10.3390\/e25081147","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,31]]}}}