{"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":1782951242856,"version":"3.54.5"},"reference-count":56,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T00:00:00Z","timestamp":1773360000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LQ21A040007"],"award-info":[{"award-number":["LQ21A040007"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LQ23A050006"],"award-info":[{"award-number":["LQ23A050006"]}]},{"name":"Special Fund for Basic Scientific Research Operations of Zhejiang Provincial Universities","award":["QRK23013"],"award-info":[{"award-number":["QRK23013"]}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"crossref","award":["2023M741716"],"award-info":[{"award-number":["2023M741716"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In contemporary communication systems, digital images occupy an irreplaceable role; however, the privacy-related risks attendant to their prevalent application have grown increasingly salient. This paper presents an image encryption scheme integrating a novel two-dimensional Ackley-Sine chaotic map (2D-ASM) with dynamic DNA operations. First, a two-dimensional Ackley-Sine chaotic map, constructed based on the Ackley function and sine function, is designed and validated through a series of chaotic indicators. Results demonstrate that 2D-ASM exhibits superior chaotic properties compared to several existing state-of-the-art chaotic maps, with its maximum Lyapunov exponent (LE) exceeding 23, Permutation Entropy (PE) close to 1 in the full parameter range, and correlation dimension (CD) significantly higher than comparative chaotic systems. The proposed 2D-ASM-based image encryption scheme leverages the SHA-256 hash value of the plaintext image and four external keys to jointly generate the initial conditions and parameters of the 2D-ASM chaotic system, thereby ensuring a sufficiently large key space of 2256. Subsequently, chaotic sequences generated by 2D-ASM are employed to permute and diffuse the plaintext image, followed by dynamic DNA coding, operations, and decoding to obtain the encrypted image. Security analyses and comparisons with several existing representative algorithms confirm that the proposed encryption scheme achieves excellent encryption performance: the Number of Pixels Change Rate (NPCR) is above 99.6%, the Unified Average Changing Intensity (UACI) approaches 33.4%, and the information entropy of ciphertext images reaches 7.999 or higher. The scheme can effectively resist various potential attacks, including statistical and differential attacks, and outperforms representative algorithms in pixel correlation reduction and anti-interference performance.<\/jats:p>","DOI":"10.3390\/e28030322","type":"journal-article","created":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T09:26:38Z","timestamp":1773393998000},"page":"322","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Novel Image Encryption Scheme Based on Two-Dimensional Chaotic Map Constructed from Ackley Function and DNA Operations"],"prefix":"10.3390","volume":"28","author":[{"given":"Chao","family":"Jiang","sequence":"first","affiliation":[{"name":"College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiong","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China"},{"name":"School of Optoelectronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoqin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China"},{"name":"School of Information and Electronic Engineering (Sussex Artificial Intelligence Institute), Zhejiang Gongshang University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Schneier, B. (2015). Secrets and Lies: Digital Security in a Networked World, John Wiley & Sons.","DOI":"10.1002\/9781119183631"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s11831-018-9298-8","article-title":"A comprehensive review on image encryption techniques","volume":"27","author":"Kaur","year":"2020","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zhang, B., and Liu, L. (2023). Chaos-based image encryption: Review, application, and challenges. Mathematics, 11.","DOI":"10.3390\/math11112585"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1007\/s00371-022-02459-5","article-title":"A survey of image encryption algorithms based on chaotic system","volume":"39","author":"Fang","year":"2023","journal-title":"Vis. Comput."},{"key":"ref_5","unstructured":"Brock, W.A., Hsieh, D.A., and LeBaron, B.D. (1991). Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence, MIT Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/0161-118991863745","article-title":"On the derivation of a \u201cchaotic\u201d encryption algorithm","volume":"13","author":"Matthews","year":"1989","journal-title":"Cryptologia"},{"key":"ref_7","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":"171","author":"Cao","year":"2020","journal-title":"Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"107252","DOI":"10.1016\/j.optlastec.2021.107252","article-title":"Image encryption algorithm based on 2D hyperchaotic map","volume":"142","author":"Gao","year":"2021","journal-title":"Opt. Laser Technol."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"1859","DOI":"10.1007\/s11071-021-06663-1","article-title":"Color image encryption based on cross 2D hyperchaotic map using combined cycle shift scrambling and selecting diffusion","volume":"105","author":"Teng","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_11","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_12","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_13","doi-asserted-by":"crossref","first-page":"19487","DOI":"10.1007\/s11071-023-08545-0","article-title":"Novel 3-D hyperchaotic map with hidden attractor and its application in meaningful image encryption","volume":"111","author":"Hu","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"114440","DOI":"10.1016\/j.chaos.2023.114440","article-title":"Dynamics analysis, synchronization and FPGA implementation of multiscroll Hopfield neural networks with non-polynomial memristor","volume":"179","author":"Yu","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Feng, W., Tang, Z., Zhao, X., Qin, Z., Chen, Y., Cai, B., Zhu, Z., Wen, H., and Ye, C. (2025). State-Dependent Variable Fractional-Order Hyperchaotic Dynamics in a Coupled Quadratic Map: A Novel System for High-Performance Image Protection. Fractal Fract., 9.","DOI":"10.3390\/fractalfract9120792"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"12579","DOI":"10.1007\/s11071-024-09679-5","article-title":"A novel image encryption algorithm based on hyperchaotic system with cross-feedback structure and diffusive DNA coding operations","volume":"112","author":"Du","year":"2024","journal-title":"Nonlinear Dyn."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Zhang, H., Liu, X., Chen, K., Te, R., and Yan, F. (2025). Robust image encryption with 2D hyperchaotic map and dynamic DNA-zigzag encoding. Entropy, 27.","DOI":"10.3390\/e27060606"},{"key":"ref_18","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_19","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_20","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_21","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.ins.2016.01.017","article-title":"Image encryption using 2D Logistic-adjusted-Sine map","volume":"339","author":"Hua","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"033112","DOI":"10.1063\/1.2959102","article-title":"On the security of a new image encryption scheme based on chaotic map lattices","volume":"18","author":"Arroyo","year":"2008","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1007\/s11071-018-4440-6","article-title":"Applications of symbolic dynamics in counteracting the dynamical degradation of digital chaos","volume":"94","author":"Zheng","year":"2018","journal-title":"Nonlinear Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.ins.2015.01.028","article-title":"A general hybrid model for chaos robust synchronization and degradation reduction","volume":"305","author":"Deng","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_26","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. Circuits Syst. I Regul. Pap."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"28225","DOI":"10.1007\/s11042-018-6015-4","article-title":"Cryptanalyzing an image encryption scheme using reverse 2-dimensional chaotic map and dependent diffusion","volume":"77","author":"Farajallah","year":"2018","journal-title":"Multimed. Tools Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"18759","DOI":"10.1109\/ACCESS.2018.2817600","article-title":"Cryptanalyzing and improving a novel color image encryption algorithm using RT-enhanced chaotic tent maps","volume":"6","author":"Zhu","year":"2018","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1007\/s11071-017-3715-7","article-title":"Cryptanalyzing an image encryption algorithm with compound chaotic stream cipher based on perturbation","volume":"90","author":"Ge","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wen, H., Yu, S., and L\u00fc, J. (2019). Breaking an image encryption algorithm based on DNA encoding and spatiotemporal chaos. Entropy, 21.","DOI":"10.3390\/e21030246"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.optlaseng.2015.03.007","article-title":"A novel chaotic based image encryption using a hybrid model of deoxyribonucleic acid and cellular automata","volume":"71","author":"Enayatifar","year":"2015","journal-title":"Opt. Lasers Eng."},{"key":"ref_32","first-page":"177","article-title":"Parallel DNA arithmetic operation based on n-moduli set","volume":"212","author":"Zheng","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.optlaseng.2016.08.009","article-title":"A novel chaos-based image encryption algorithm using DNA sequence operations","volume":"88","author":"Chai","year":"2017","journal-title":"Opt. Lasers Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.optlaseng.2015.03.022","article-title":"A novel chaotic image encryption scheme using DNA sequence operations","volume":"73","author":"Wang","year":"2015","journal-title":"Opt. Lasers Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"116246","DOI":"10.1016\/j.image.2021.116246","article-title":"Image encryption based on compressed sensing and DNA encoding","volume":"95","author":"Wang","year":"2021","journal-title":"Signal Process. Image Commun."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106393","DOI":"10.1016\/j.optlaseng.2020.106393","article-title":"Chaotic image encryption algorithm based on hybrid multi-objective particle swarm optimization and DNA sequence","volume":"137","author":"Wang","year":"2021","journal-title":"Opt. Lasers Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1497","DOI":"10.1007\/s11042-019-08247-z","article-title":"Color image encryption based on hybrid chaotic system and DNA sequences","volume":"79","author":"Moattar","year":"2020","journal-title":"Multimed. Tools Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.sigpro.2018.09.029","article-title":"A color image cryptosystem based on dynamic DNA encryption and chaos","volume":"155","author":"Chai","year":"2019","journal-title":"Signal Process."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1457","DOI":"10.1016\/j.asoc.2012.01.016","article-title":"Image encryption using DNA complementary rule and chaotic maps","volume":"12","author":"Liu","year":"2012","journal-title":"Appl. Soft Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"109033","DOI":"10.1016\/j.optlastec.2022.109033","article-title":"A new one-dimensional chaotic map for image encryption scheme based on random DNA coding","volume":"160","author":"Liang","year":"2023","journal-title":"Opt. Laser Technol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.aeue.2013.08.007","article-title":"Improved algorithm for image encryption based on DNA encoding and multi-chaotic maps","volume":"68","author":"Zhang","year":"2014","journal-title":"AEU-Int. J. Electron. Commun."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.jss.2011.08.017","article-title":"A novel color image encryption algorithm based on DNA sequence operation and hyper-chaotic system","volume":"85","author":"Wei","year":"2012","journal-title":"J. Syst. Softw."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"107753","DOI":"10.1016\/j.optlastec.2021.107753","article-title":"Chaotic color image encryption based on 4D chaotic maps and DNA sequence","volume":"148","author":"Wang","year":"2022","journal-title":"Opt. Laser Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1007\/s11071-015-2392-7","article-title":"A novel chaos-based image encryption using DNA sequence operation and Secure Hash Algorithm SHA-2","volume":"83","author":"Guesmi","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_45","first-page":"32","article-title":"Color image encryption algorithm based on DNA code and alternating quantum random walk","volume":"70","author":"Wang","year":"2021","journal-title":"Acta Phys. Sin."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s11047-012-9334-9","article-title":"A novel text and image encryption method based on chaos theory and DNA computing","volume":"12","author":"Babaei","year":"2013","journal-title":"Nat. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.optlastec.2014.01.015","article-title":"Cryptanalyzing a RGB image encryption algorithm based on DNA encoding and chaos map","volume":"60","author":"Liu","year":"2014","journal-title":"Opt. Laser Technol."},{"key":"ref_48","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_49","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/S0076-6879(04)84011-4","article-title":"Sample entropy","volume":"Volume 384","author":"Richman","year":"2004","journal-title":"Methods in Enzymology"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"174102","DOI":"10.1103\/PhysRevLett.88.174102","article-title":"Permutation entropy: A natural complexity measure for time series","volume":"88","author":"Bandt","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2338","DOI":"10.1103\/PhysRevA.14.2338","article-title":"Kolmogorov entropy and numerical experiments","volume":"14","author":"Benettin","year":"1976","journal-title":"Phys. Rev. A"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Gottwald, G.A., and Melbourne, I. (2016). The 0\u20131 test for chaos: A review. Chaos Detection and Predictability, Springer.","DOI":"10.1007\/978-3-662-48410-4_7"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1865","DOI":"10.1142\/S021812740100305X","article-title":"Improved correlation dimension calculation","volume":"11","author":"Sprott","year":"2001","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_55","first-page":"10","article-title":"Analysis and Comparison of the NIST SP 800-53 and ISO\/IEC 27001: 2013","volume":"800","author":"Kurii","year":"2022","journal-title":"NIST Spec. Publ."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"e70036","DOI":"10.1002\/spy2.70036","article-title":"Design of a Fast Image Encryption Algorithm Based on a Novel 2D Chaotic Map and DNA Encoding","volume":"8","author":"Fu","year":"2025","journal-title":"Secur. Priv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/322\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T05:20:50Z","timestamp":1773552050000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/322"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,13]]},"references-count":56,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["e28030322"],"URL":"https:\/\/doi.org\/10.3390\/e28030322","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,13]]}}}