{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T16:25:21Z","timestamp":1790785521276,"version":"4.1.0"},"reference-count":53,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:00:00Z","timestamp":1783900800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"S&amp;T Program of Chengde","award":["202404B067"],"award-info":[{"award-number":["202404B067"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>With the widespread adoption of digital images in network transmission and storage, the demand for image privacy protection keeps rising. We propose a robust scheme combining a four-dimensional variable-order fractional hyperchaotic system (4D-VOFHS) and a chess-game play-inspired dynamic mechanism. Firstly, we construct 4D-VOFHS, to overcome inherent limitations of constant-order systems: unlike constant-order systems that are vulnerable to deep-learning-based parameter identification attacks, this system introduces time-varying orders and high-dimensional coupling to enrich nonlinear dynamics. Secondly, inspired by the dynamic strategic interactions within chess gameplay, we design a synchronous encryption framework with a tightly coupled permutation\u2013diffusion mechanism. This design not only significantly enhances the nonlinear complexity, confusion and diffusion performance of the algorithm, but also enables parallel synchronous processing to improve computational throughput. Finally, we propose a block-based collaborative scrambling strategy with multi-chess-piece rules, wherein traversal rules and scrambling operations are not predefined; instead, they are dynamically updated according to the real-time state evolution of the 4D-VOFHS. Through comprehensive correlation analysis and differential attack tests, the presented encryption framework achieves outstanding performance metrics: an average NPCR of 99.6%, a UACI of 33.4%, and an average information entropy of 7.9993. Overall, these results verify the strong cryptographic robustness and practical applicability of the scheme, highlighting its great potential for deployment in real-world color image encryption systems.<\/jats:p>","DOI":"10.3390\/e28070795","type":"journal-article","created":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T02:34:44Z","timestamp":1783996484000},"page":"795","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Color Image Encryption Using a 4D Variable-Order Fractional Hyperchaotic System and Chess-Gameplay-Inspired Dynamic Mechanism"],"prefix":"10.3390","volume":"28","author":[{"given":"Xiaomeng","family":"Cui","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Hebei Minzu Normal University, Chengde 067000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7686-2841","authenticated-orcid":false,"given":"Xiaoqiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiaqi","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Hebei Minzu Normal University, Chengde 067000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"110402","DOI":"10.1016\/j.compeleceng.2025.110402","article-title":"A novel chaotic-based image encryption scheme using multi-layer confusion and conditional diffusion","volume":"124","author":"Ali","year":"2025","journal-title":"Comput. 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