Color image encryption based on a square-term enhanced 4D chaotic system

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The Original Article was published on

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Abstract

Chaotic systems confer distinct advantages for image encryption due to their remarkable responsiveness to initial parameters, inherent unpredictability, and characteristics of pseudo-randomness. However, many existing chaos-based image encryption schemes still suffer from limitations such as narrow chaotic parameter intervals, insufficient trajectory uniformity, and inadequate diffusion depth, which may weaken the randomness of generated sequences and reduce encryption robustness. To address these issues, a square-term enhanced four-dimensional chaotic system was constructed, and a color image encryption scheme integrating hierarchical scrambling and symmetric bidirectional diffusion was developed. Dynamic analyses, including phase portraits, bifurcation diagrams, maximum Lyapunov exponents, time-series distribution, and complexity evaluation, indicate that the proposed system exhibits a broad chaotic interval, strong initial-value sensitivity, and improved trajectory uniformity. It can well-adapt to the image encryption requirements of different categories. Simulation experiments conducted using a variety of test images indicate that the encryption scheme offers an adequately large key space, with pixel correlation measured at less than 0.03, information entropy exceeding 7.9974, and robust resistance to differential attacks. The computed mean values for number of pixels change rate (NPCR) and unified average changing intensity (UACI) were found to be 99.6073% and 33.4626%, respectively, demonstrating a strong alignment with the optimal benchmarks. The algorithm achieves competitive performance in terms of efficiency and security, can effectively resist common attacks, and is suitable for the secure transmission and storage of sensitive images.

Keywords

four-dimensional chaotic system / dynamic analysis / two-level scrambling / multidirectional random diffusion / image encryption

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References

[1] A. Wajid, Z. P. Zhang, H. Muhammad, A novel chaotic-based image encryption scheme using multi-layer confusion and conditional diffusion, Comput. Electr. Eng., 124 (2025), 110402. https://doi.org/10.1016/j.compeleceng.2025.110402

[2] M. Yao, H. W. Deng, Z. Chen, P. Zhang, Efficient and secure image compression and encryption based on compressive sensing and four-dimensional hyperchaotic system, Inf. Sci., 719 (2025), 122430. https://doi.org/10.1016/j.ins.2025.122430

[3] Z. L. Man, J. Q. Li, X. Q. Di, Y. H. Sheng, Z. F. Liu, Double image encryption algorithm based on neural network and chaos, Chaos, Solitons Fractals, 152 (2021), 111318. https://doi.org/10.1016/j.chaos.2021.111318

[4] B. Abd-El-Atty, A dual medical image encryption algorithm based on dynamic DNA operations, Comput. Electr. Eng., 130 (2026), 110899. https://doi.org/10.1016/j.compeleceng.2025.110899

[5] C. F. Zhao, Z. B. Zhai, B. Zeng, Delayed chaotic image encryption algorithm using cross-layer and DNA coding techniques, Comput. Stand. Interfaces, 93 (2025), 103974. https://doi.org/10.1016/j.csi.2025.103974

[6] M. Alawida, A novel DNA tree-based chaotic image encryption algorithm, J. Inf. Secur. Appl., 83 (2024), 103791. https://doi.org/10.1016/j.jisa.2024.103791

[7] S. M. Wang, Q. Q. Peng, B. X. Du, Chaotic color image encryption based on 4D chaotic maps and DNA sequence, Opt. Laser Technol., 148 (2022), 107753. https://doi.org/10.1016/j.optlastec.2021.107753

[8] S. H. Yan, L. Li, W. L. Zhao, B. X. Gu, Design of a new four-dimensional chaotic system and its application to color image encryption, Nonlinear Dyn., 111 (2023), 17519-17545. https://doi.org/10.1007/s11071-023-08726-x

[9] B. Yogi, A. K. Khan, EHIES-ECCCA: An efficient hybrid image encryption scheme using ECC and cellular automata with secure shared key generation, Signal Process. Image Commun., 144 (2026), 117527. https://doi.org/10.1016/j.image.2026.117527

[10] C. F. Zhao, X. C. He, Y. F. Li, B. Zeng, Chaotic encryption algorithm for satellite images based on novel RNA coding, Adv. Space Res., 75 (2025), 8334-8356. https://doi.org/10.1016/j.asr.2025.03.057

[11] T. He, F. Yu, Y. Lin, S. Q. He, W. Yao, S. Cai, et al., Multi-scroll hopfield neural network excited by memristive self-synapses and its application in image encryption, Chin. Phys. B, 34 (2025), 120506. https://doi.org/10.1088/1674-1056/adfeff

[12] Z. Huang, Z. Li, Q. Wang, W. J. Tan, X. M. Wu, A novel hyperchaotic multistable heterogeneous neural network with hidden attractors and its application in image encryption, Chin. J. Phys., 96 (2025), 851-874. https://doi.org/10.1016/j.cjph.2025.05.037

[13] F. Yu, X. Q. Wang, R. Y. Guo, Z. J. Ying, Y. He, Q. Zou, Discrete neuron models and memristive neural network mapping: A comprehensive review, Chin. Phys. B, 34 (2025), 120501. https://doi.org/10.1088/1674-1056/ae0a3b

[14] L. Q. Huang, W. J. Li, X. M. Xiong, R. Yu, Q. X. Wang, S. T. Cai, Designing a double-way spread permutation framework utilizing chaos and S-box for symmetric image encryption, Opt. Commun., 517 (2022), 128365. https://doi.org/10.1016/j.optcom.2022.128365

[15] M. U. Safdar, T. Shah, A. Ali, Multiple-image encryption algorithm based on S- boxes and DNA Sequences, Signal Process. Image Commun., 138 (2025), 117353. https://doi.org/10.1016/j.image.2025.117353

[16] L. Shu, Y. Zhang, Z. W. Yu, L. T. Yang, M. Hauswirth, N. X. Xiong, Context-aware cross-layer optimized video streaming in wireless multimedia sensor networks, J. Supercomput., 54 (2010), 94-121. https://doi.org/10.1007/s11227-009-0321-6

[17] F. A. Khan, J. Ahmed, S. A. Alsuhibany, A new multi chaos-based compression sensing image encryption, Comput. Mater. Continua, 76 (2023), 437-453. https://doi.org/10.32604/cmc.2023.032236

[18] X. L. An, S. Y. Liu, L. Xiong, J. G. Zhang, X. Y. Li, Mixed gray-color images encryption algorithm based on a memristor chaotic system and 2D compression sensing, Expert Syst. Appl., 243 (2024), 122899. https://doi.org/10.1016/j.eswa.2023.122899

[19] Y. L. Chen, T. X. Lu, G. R. Chen, Thumbnail-preserving encryption based on binary-decimal hybrid and a generalized coupled logistic map, Int. J. Bifurcation Chaos, 34 (2024), 2450193. https://dx.doi.org/10.1142/S0218127424501931

[20] S. Gao, R. Wu, H. Ho-Ching Iu, U. Erkan, Y. Cao, Q. Li, et al., Chaos- based video encryption techniques: A review, Comput. Sci. Rev., 58 (2025), 100816. https://doi.org/10.1016/j.cosrev.2025.100816

[21] M. A. Ben Farah, R. Guesmi, A. Kachouri, M. Samet, A novel chaos based optical image encryption using fractional Fourier transform and DNA sequence operation, Opt. Laser Technol., 121 (2020), 105777. https://doi.org/10.1016/j.optlastec.2019.105777

[22] Rohit, S. K. Tripathi, B. Gupta, S. Lamba, A companion matrix-based efficient image encryption method, Signal Process., 228 (2025), 109753. https://doi.org/10.1016/j.sigpro.2024.109753

[23] Z. Wang, T. Li, N. X. Xiong, Y. Pan, A novel dynamic network data replication scheme based on historical access record and proactive deletion, J. Supercomput., 62 (2012), 227-250. https://doi.org/10.1007/s11227-011-0708-z

[24] L. D. Singh, R. Thingbaijam, R. Patgiri, K. M. Singh, Cryptanalysis of cross-coupled chaotic maps multi-image encryption scheme, J. Inf. Secur. Appl., 80 (2024), 103694. https://doi.org/10.1016/j.jisa.2023.103694

[25] J. W. Yu, W. Xie, Z. Y. Zhong, H. Wang, Image encryption algorithm based on hyperchaotic system and a new DNA sequence operation, Chaos, Solitons Fractals, 162 (2022), 112456. https://doi.org/10.1016/j.chaos.2022.112456

[26] Y. Wang, K. W. Wong, X. F. Liao, T. Xiang, G. R. Chen, A chaos-based image encryption algorithm with variable control parameters, Chaos, Solitons Fractals, 41 (2009), 1773-1783. https://doi.org/10.1016/j.chaos.2008.07.031

[27] R. Parvaz, Y. K. Yengejeh, Y. Behroo, A new 4D chaos system with an encryption algorithm for color and grayscale images, Int. J. Bifurcation Chaos, 32 (2022), 2250214. https://doi.org/10.1142/S0218127422502145

[28] X. Z. Yang, H. Tian, F. Wang, J. P. Ni, J. Q. Zhang, J. J. Jia, Bounded-chaos Duffing detector for weak occlusion signals in laser light screen system, Opt. Lasers Eng., 203 (2026), 109824. https://doi.org/10.1016/j.optlaseng.2026.109824

[29] M. Vogl, Chaos measure dynamics in a multifactor model for financial market predictions, Commun. Nonlinear Sci. Numer. Simul., 130 (2024), 107760. https://doi.org/10.1016/j.cnsns.2023.107760

[30] L. H. Gong, H. X. Luo, R. Q. Wu, N. R. Zhou, New 4D chaotic system with hidden attractors and self-excited attractors and its application in image encryption based on RNG, Physica A: Stat. Mech. Appl., 591 (2022), 126793. https://doi.org/10.1016/j.physa.2021.126793

[31] Z. Wan, N. X. Xiong, N. Ghani, A. Vasilakos, L. Zhou, Adaptive unequal protection for wireless video transmission over IEEE 802.11e networks, Multimedia Tools Appl., 72 (2014), 541-571. https://doi.org/10.1007/s11042-013-1378-z

[32] W. Y. Liang, L. M. Zhang, Z. B. Yang, T. T. Yu, J. J. Li, X. L. Li, Image encryption algorithm based on hyperchaotic system and dynamic DNA encoding, Phys. Scr., 98 (2023), 115215. https://doi.org/10.1088/1402-4896/acfc71

[33] G. R. Chen, Y. B. Mao, C. K. Chui, A symmetric image encryption scheme based on 3D chaotic cat maps, Chaos, Solitons Fractals, 21 (2004), 749-761. https://doi.org/10.1016/j.chaos.2003.12.022

[34] S. A. Mehdi, Image encryption algorithm based on a novel 4D chaotic system, Int. J. Inf. Secur. Privacy, 15 (2021), 118-131. https://doi.org/10.4018/IJISP.2021100107

[35] L. M. Wu, Z. J. Tian, W. Chen, Color image encryption scheme based on 5D fractional-order complex chaotic system and eight-base DNA cubes, Expert Syst. Appl., 299 (2026), 129950. https://doi.org/10.1016/j.eswa.2025.129950

[36] A. Panwar, G. Biban, R. Chugh, A. Tassaddiq, R. Alharbi, An efficient image encryption model based on 6D hyperchaotic system and symmetric matrix for color and gray images, Heliyon, 10 (2024), e31618. https://doi.org/10.1016/j.heliyon.2024.e31618

[37] X. J. Tong, L. M. Cheng, Y. H. Wang, Color image encryption algorithm based on a novel five-dimensional chaotic system, Phys. Scr., 100 (2025), 095202. https://doi.org/10.1088/1402-4896/adfca7

[38] M. X. Gong, X. L. Chai, Y. Lu, Y. S. Zhang, Exploiting four-dimensional chaotic systems with dissipation and optimized logical operations for secure image compression and encryption, IEEE Trans. Circuits Syst. Video Technol., 34 (2024), 7628-7642. https://doi.org/10.1109/TCSVT.2024.3375868

[39] W. Song, C. Fu, Y. Zheng, Y. F. Zhang, J. X. Chen, P. P. Wang, Batch image encryption using cross image permutation and diffusion, J. Inf. Secur. Appl., 80 (2024), 103686. https://doi.org/10.1016/j.jisa.2023.103686

[40] B. Y. Liu, W. Song, M. Y. Zheng, C. Fu, J. X. Chen, X. W. Wang, Semantically enhanced selective image encryption scheme with parallel computing, Expert Syst. Appl., 279 (2025), 127404. https://doi.org/10.1016/j.eswa.2025.127404

[41] F. Yu, X. Q. Wang, Y. Xiao, Y. He, W. Yao, S. Cai, et al., Extreme multistability in discrete memristive neuron maps and implications for dual-field applications, Integration, 109 (2026), 102688. https://doi.org/10.1016/j.vlsi.2026.102688

[42] Q. Wang, H. W. Sang, P. Wang, X. Yu, Z. Y. Yang, A novel 4D chaotic system coupling with dual-memristors and application in image encryption, Sci. Rep., 14 (2024), 29615. https://doi.org/10.1038/s41598-024-80445-8

[43] A. A. Elsadany, S. Hussein, A. Al-Khedhairi, A. Elsonbaty, On dynamics of 4-D blinking chaotic system and voice encryption application, Alexandria Eng. J., 70 (2023), 701-718. https://doi.org/10.1016/j.aej.2023.03.024