{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T19:39:05Z","timestamp":1770061145392,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T00:00:00Z","timestamp":1631836800000},"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>Chaos is considered as a natural candidate for encryption systems owing to its sensitivity to initial values and unpredictability of its orbit. However, some encryption schemes based on low-dimensional chaotic systems exhibit various security defects due to their relatively simple dynamic characteristics. In order to enhance the dynamic behaviors of chaotic maps, a novel 3D infinite collapse map (3D-ICM) is proposed, and the performance of the chaotic system is analyzed from three aspects: a phase diagram, the Lyapunov exponent, and Sample Entropy. The results show that the chaotic system has complex chaotic behavior and high complexity. Furthermore, an image encryption scheme based on 3D-ICM is presented, whose security analysis indicates that the proposed image encryption scheme can resist violent attacks, correlation analysis, and differential attacks, so it has a higher security level.<\/jats:p>","DOI":"10.3390\/e23091221","type":"journal-article","created":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T21:23:29Z","timestamp":1631913809000},"page":"1221","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Three-Dimensional Infinite Collapse Map with Image Encryption"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9237-7996","authenticated-orcid":false,"given":"Wenhao","family":"Yan","sequence":"first","affiliation":[{"name":"Electronic Engineering College, Heilongjiang University, Harbin 150080, China"}]},{"given":"Zijing","family":"Jiang","sequence":"additional","affiliation":[{"name":"Electronic Engineering College, Heilongjiang University, Harbin 150080, China"}]},{"given":"Xin","family":"Huang","sequence":"additional","affiliation":[{"name":"Electronic Engineering College, Heilongjiang University, Harbin 150080, China"}]},{"given":"Qun","family":"Ding","sequence":"additional","affiliation":[{"name":"Electronic Engineering College, Heilongjiang University, Harbin 150080, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,17]]},"reference":[{"key":"ref_1","unstructured":"Schonberger, V.M. (2013). Big Data: A Revolution That Will Transform How We Live, Work, and Think, Houghton Mifflin Harcourt."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.sigpro.2019.02.016","article-title":"A new hybrid digital chaotic system with applications in image encryption","volume":"160","author":"Alawida","year":"2019","journal-title":"Signal Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"22778","DOI":"10.1109\/ACCESS.2018.2828319","article-title":"ACryp-Proc: Flexible Asymmetric Crypto Processor for Point Multiplication","volume":"6","author":"Imran","year":"2018","journal-title":"IEEE Access"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"8660","DOI":"10.1109\/ACCESS.2018.2890116","article-title":"Image encryption using Josephus problem and filtering diffusion","volume":"7","author":"Hua","year":"2019","journal-title":"IEEE Access"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1109\/TMM.2016.2605499","article-title":"A novel data hiding algorithm for high dynamical range images","volume":"19","author":"Lin","year":"2017","journal-title":"IEEE Trans. Multimed."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1244","DOI":"10.1109\/TIP.2015.2395724","article-title":"On local prediction based reversible watermarking","volume":"24","author":"Dragoi","year":"2015","journal-title":"IEEE Trans. Image Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3039","DOI":"10.1016\/j.sigpro.2013.04.021","article-title":"Image encryption using a new parametric switching chaotic system","volume":"93","author":"Zhou","year":"2013","journal-title":"Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.sigpro.2017.03.011","article-title":"A new color image encryption using combination of the 1D chaotic map","volume":"138","author":"Pak","year":"2017","journal-title":"Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"140876","DOI":"10.1109\/ACCESS.2020.3012912","article-title":"Chaos-Based Confusion and Diffusion of Image Pixels Using Dynamic Substitution","volume":"8","author":"Qayyum","year":"2020","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2322","DOI":"10.1109\/TCSI.2018.2888688","article-title":"Dynamic analysis of digital chaotic maps via state-mapping networks","volume":"66","author":"Li","year":"2019","journal-title":"IEEE Trans. Circuit. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.chaos.2004.02.008","article-title":"Parameter estimation only from the symbolic sequences generated by chaos system","volume":"22","author":"Wu","year":"2004","journal-title":"Chaos Soliton. Fract."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Masood, F., Driss, M., Boulila, W., Ahmad, J., Rehman, S.U., Jan, S.U., Qayyum, A., and Buchanan, W.J. (2021). A Lightweight Chaos-Based Medical Image Encryption Scheme Using Random Shuffling and XOR Operations. Wirel. Pers. Commun.","DOI":"10.1007\/s11277-021-08584-z"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"033118","DOI":"10.1063\/1.2242052","article-title":"Encryption and decryption of images with chaotic map lattices","volume":"16","author":"Pisarchik","year":"2006","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"159732","DOI":"10.1109\/ACCESS.2020.3020917","article-title":"DNA and Plaintext Dependent Chaotic Visual Selective Image Encryption","volume":"8","author":"Khan","year":"2020","journal-title":"IEEE Access"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s11071-019-04809-w","article-title":"Enhancing unimodal digital chaotic maps through hybridization","volume":"96","author":"Alawida","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Fan, C.L., and Ding, Q. (2018). A Novel Image Encryption Scheme Based on Self-Synchronous Chaotic Stream Cipher and Wavelet Transform. Entropy, 20.","DOI":"10.3390\/e20060445"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhang, R., Yu, L., Jiang, D., Ding, W., Song, J., He, K., and Ding, Q. (2021). A Novel Plaintext-Related Color Image Encryption Scheme Based on Cellular Neural Network and Chen\u2019s Chaotic System. Symmetry, 13.","DOI":"10.3390\/sym13030393"},{"key":"ref_18","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. Bifurcat. Chaos"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.optlaseng.2016.07.010","article-title":"Chaos-based partial image encryption scheme based on linear fractional and lifting wavelet transforms","volume":"88","author":"Belazi","year":"2017","journal-title":"Opt. Lasers Eng."},{"key":"ref_20","first-page":"337","article-title":"Efficient cryptosystem approaches: S-boxes and permutation-substitution-based encryption","volume":"87","author":"Belazi","year":"2017","journal-title":"Wirel. Personal Commun."},{"key":"ref_21","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_22","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/S0375-9601(02)00522-4","article-title":"Parameters identification and synchronization of chaotic systems based upon adaptive control","volume":"299","author":"Chen","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/j.sigpro.2017.08.020","article-title":"A novel bit-level image encryption algorithm based on 2D-LICM hyperchaotic map","volume":"143","author":"Cao","year":"2018","journal-title":"Signal Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.optlaseng.2016.03.019","article-title":"A fast image encryption algorithm based on chaotic map","volume":"84","author":"Liu","year":"2016","journal-title":"Opt. Lasers Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3837209","DOI":"10.1155\/2020\/3837209","article-title":"A color image encryption scheme based on a novel 3d chaotic mapping","volume":"2020","author":"Liu","year":"2020","journal-title":"Complexity"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1950206","DOI":"10.1142\/S0218127419502067","article-title":"A New 3D Discrete Hyperchaotic System and Its Application in Secure Transmission","volume":"29","author":"Li","year":"2019","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"107457","DOI":"10.1016\/j.sigpro.2020.107457","article-title":"Designing a 2D infinite collapse map for image encryption","volume":"17","author":"Cao","year":"2020","journal-title":"Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"P. Nat. Acad. Sci. USA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"112977","DOI":"10.1109\/ACCESS.2019.2935304","article-title":"A new matrix projective synchronization and its application in secure communication","volume":"7","author":"Yan","year":"2019","journal-title":"IEEE Access"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"900","DOI":"10.1109\/81.933333","article-title":"Chaotic characteristics of a one-dimensional iterative map with infinite collapses","volume":"48","author":"He","year":"2001","journal-title":"IEEE Trans. Circuit. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol. Heart. C"},{"key":"ref_32","first-page":"314","article-title":"Circular inter-intra pixels bit-level permutation and chaos-based image encryption","volume":"355","author":"Dragoi","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.ins.2017.02.036","article-title":"Design of image cipher using block-based scrambling and image filtering","volume":"396","author":"Hua","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3039","DOI":"10.1016\/j.sigpro.2013.10.034","article-title":"A new 1D chaotic map for image encryption","volume":"97","author":"Zhou","year":"2014","journal-title":"Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.optlaseng.2015.09.007","article-title":"A novel bit-level image encryption algorithm based on chaotic maps","volume":"78","author":"Xu","year":"2016","journal-title":"Opt. Lasers Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1550093","DOI":"10.1142\/S012918311550093X","article-title":"Two-dimensional DFA scaling analysis applied to encrypted images","volume":"26","author":"Rosu","year":"2015","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Reyes-L\u00f3pez, L.E., Murgu\u00eda, J.S., Gonz\u00e1lez-Aguilar, H., Ram\u00edrez-Torres, M.T., Mej\u00eda-Carlos, M., and Armijo-Correa, J.O. (2021). Scaling Analysis of an Image Encryption Scheme Based on Chaotic Dynamical Systems. Entropy, 23.","DOI":"10.3390\/e23060672"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"106165","DOI":"10.1016\/j.optlastec.2020.106165","article-title":"An improved visually meaningful encrypted image scheme","volume":"127","year":"2020","journal-title":"Opt. Laser Technol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1221\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:01:25Z","timestamp":1760166085000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1221"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,17]]},"references-count":38,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["e23091221"],"URL":"https:\/\/doi.org\/10.3390\/e23091221","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9,17]]}}}