A multi-image encryption algorithm based on hybrid chaotic map and computer-generated holography

Yingfang Zhu; Erxi Zhu; Yingfang Zhu; Erxi Zhu

doi:10.3934/math.2025947

AIMS Mathematics

2025, Volume 10, Issue 9: 21209-21239. doi: 10.3934/math.2025947

Previous Article Next Article

Research article

A multi-image encryption algorithm based on hybrid chaotic map and computer-generated holography

Yingfang Zhu ¹,
Erxi Zhu ^{2,3
,
,}

1.
College of Internet of Things Engineering, Jiangsu Vocational College of Information Technology, No.1 qianou Road, Huishan District, Jiangsu Wuxi, 214153, China
2.
College of Information Engineering, Changzhou Vocational Institute of Industry Technology, No. 28, Mingxin Middle Road, Wujin District, Changzhou City, Jiangsu Province, 213164, China
3.
Key Laboratory of Intelligent Connected Vehicle Driverless Driving and Network Security Technology, No. 28, Mingxin Middle Road, Wujin District, Changzhou City, Jiangsu Province, 213164, China

Received: 09 July 2025 Revised: 28 August 2025 Accepted: 01 September 2025 Published: 15 September 2025
MSC : 68T45, 94A08

In response to the critical challenges of securing multi-image transmissions in cloud and 5G/6G networks, this paper proposes an innovative encryption algorithm that synergistically combines hybrid chaotic maps with computer-generated holography (CGH). The authors introduce three groundbreaking contributions: (1) A novel integrated chaotic system (NICS) fusing logistic, sinus, and tent mappings through modular arithmetic to achieve full-range chaos with a $ 10^{84} $ key space and 40% higher Lyapunov exponents; (2) An enhanced Gerchberg Saxton algorithm incorporating adaptive feedback to accelerate convergence by 35% while enabling parallel encryption of eight $ 512\times512 $ images; (3) A dynamic secure hash algorithm 256 (SHA-256) bits based key binding mechanism that resists chosen plaintext attacks. Extensive experiments validate the exceptional performance metrics: Information entropy approaching the theoretical maximum (7.992±0.005), near-zero adjacent pixel correlation (< 0.004), and robust resistance to noise (20%) and cropping attacks (60% recovery at 60% loss). The algorithm's practical superiority is demonstrated through 2.3× faster processing speeds compared with conventional methods, along with successful deployments in medical imaging and military communication systems, establishing a new benchmark for secure multi-image transmission in next-generation networks.
- multi-image encryption,
- chaotic map,
- computer-generated holography,
- security analysis,
- dynamical analysis,
- phase retrieval
Citation: Yingfang Zhu, Erxi Zhu. A multi-image encryption algorithm based on hybrid chaotic map and computer-generated holography[J]. AIMS Mathematics, 2025, 10(9): 21209-21239. doi: 10.3934/math.2025947

Related Papers:

Abstract

In response to the critical challenges of securing multi-image transmissions in cloud and 5G/6G networks, this paper proposes an innovative encryption algorithm that synergistically combines hybrid chaotic maps with computer-generated holography (CGH). The authors introduce three groundbreaking contributions: (1) A novel integrated chaotic system (NICS) fusing logistic, sinus, and tent mappings through modular arithmetic to achieve full-range chaos with a $ 10^{84} $ key space and 40% higher Lyapunov exponents; (2) An enhanced Gerchberg Saxton algorithm incorporating adaptive feedback to accelerate convergence by 35% while enabling parallel encryption of eight $ 512\times512 $ images; (3) A dynamic secure hash algorithm 256 (SHA-256) bits based key binding mechanism that resists chosen plaintext attacks. Extensive experiments validate the exceptional performance metrics: Information entropy approaching the theoretical maximum (7.992±0.005), near-zero adjacent pixel correlation (< 0.004), and robust resistance to noise (20%) and cropping attacks (60% recovery at 60% loss). The algorithm's practical superiority is demonstrated through 2.3× faster processing speeds compared with conventional methods, along with successful deployments in medical imaging and military communication systems, establishing a new benchmark for secure multi-image transmission in next-generation networks.

References

[1]	K. V. Oskolok, O. V. Monogarova, A. V. Garmay, V. S. Milkina, One- and two-component digital image HDR-colorimetry: Technical challenges and analytical capabilities, Talanta, 288 (2025), 127736. https://doi.org/10.1016/j.talanta.2025.127736 doi: 10.1016/j.talanta.2025.127736
[2]	M. Nadeem, O. A. Arqub, Soliton solutions and stability analysis to the Hirota-bilinear-like model using the unified Riccati equation expansion approach, Modern Phys. Lett. B, 39 (2025), 2550138. https://doi.org/10.1142/S0217984925501386 doi: 10.1142/S0217984925501386
[3]	F. Yu, S. He, W. Yao, S. Cai, Q. Xu, Bursting firings in memristive hopfield neural network with image encryption and hardware implementation, In: IEEE transactions on computer-aided design of integrated circuits and systems, IEEE, 2025, 1. https://doi.org/10.1109/TCAD.2025.3567878
[4]	S. M. U. Din, T. Shah, F. Alblehai, S. Nooh, S. S. Jamal, Publisher correction: A combinatory approach of non-chain ring and henon map for image encryption application, Sci. Rep., 15 (2025), 7882. https://doi.org/10.1038/s41598-025-92489-5 doi: 10.1038/s41598-025-92489-5
[5]	H. Lin, S. Yu, W. Feng, Y. Chen, J. Zhang, Z. Qin, et al., Exploiting dynamic vector-level operations and a 2D-enhanced logistic modular map for efficient chaotic image encryption, Entropy, 25 (2023), 1147. https://doi.org/10.3390/e25081147 doi: 10.3390/e25081147
[6]	J. Zeng, X. Chen, L. Wei, D. Li, Bifurcation analysis of a fractional-order eco-epidemiological system with two delays, Nonlinear Dyn., 112 (2024), 22505–22527. https://doi.org/10.1007/s11071-024-10184-y doi: 10.1007/s11071-024-10184-y
[7]	Y. Yuan, F. Yu, B. Tan, Y. Huang, W. Yao, S. Cai, et al., A class of n-D Hamiltonian conservative chaotic systems with three-terminal memristor: Modeling, dynamical analysis, and FPGA implementation, Chaos, 35 (2025), 013121. https://doi.org/10.1063/5.0238893 doi: 10.1063/5.0238893
[8]	W. Feng, J. Zhang, Y. Chen, Z. Qin, Y. Zhang, M. Ahmad, et al., Exploiting robust quadratic polynomial hyperchaotic map and pixel fusion strategy for efficient image encryption, Expert Syst. Appl., 246 (2024), 123190. https://doi.org/10.1016/j.eswa.2024.123190 doi: 10.1016/j.eswa.2024.123190
[9]	B. Gleb, J. Simon, Choquet integral optimisation with constraints and the buoyancy property for fuzzy measures, Inform. Sci., 578 (2021), 22–36. https://doi.org/10.1016/j.ins.2021.07.032 doi: 10.1016/j.ins.2021.07.032
[10]	Y. Cao, Z. Li, S. He, Complex hidden dynamics in a memristive map with delta connection and its application in image encryption, Nonlinear Dyn., 112 (2024), 7597–7613. https://doi.org/10.1007/s11071-024-09344-x doi: 10.1007/s11071-024-09344-x
[11]	A. Aviles, Testing gravity with the full-shape galaxy power spectrum: First constraints on scale-dependent modified gravity, Phys. Rev. D, 111 (2025), L021301. https://doi.org/10.1103/PhysRevD.111.L021301 doi: 10.1103/PhysRevD.111.L021301
[12]	P. Refregier, B. Javidi, Optical image encryption based on input plane and Fourier plane random encoding, Opt. Lett., 20 (1995), 767–769. https://doi.org/10.1364/OL.20.000767 doi: 10.1364/OL.20.000767
[13]	Q. Wang, A. Ge, X. Chen, J. Wu, S. Liu, D. Zhu, Text information security protection method based on computer-generated holograms, Appl. Optics, 63 (2024), 4165–4174. https://doi.org/10.1364/AO.523616 doi: 10.1364/AO.523616
[14]	Y. Su, Z. Wang, Y. Wang, R. Xue, B. Wang, W. Zhong, et al., Multiple-image encryption based on authenticable phase and phase retrieval under structured light illumination, Opt. Commun., 564 (2024), 130603. https://doi.org/10.1016/j.optcom.2024.130603 doi: 10.1016/j.optcom.2024.130603
[15]	K. P. Zhai, X. Y. Zhang, S. Zhu, Y. Liu, H. S. Wen, N. H. Zhu, Secure optical communication system based on polarization regulation of the data fragmentation multipath transmission technology, Opt. Lett., 49 (2024), 3226–3229. https://doi.org/10.1364/OL.524408 doi: 10.1364/OL.524408
[16]	W. Feng, Z, Qin, J. Zhang, M. Ahmad, Cryptanalysis and improvement of the image encryption scheme based on Feistel network and dynamic DNA encoding, IEEE Access, 9 (2021), 145459–145470. https://doi.org/10.1109/ACCESS.2021.3123571 doi: 10.1109/ACCESS.2021.3123571
[17]	W. Feng, Y. He, H. Li, C. Li, Cryptanalysis and improvement of the image encryption scheme based on 2D logistic-adjusted-sine map, IEEE Access, 7 (2019), 12584–12597. https://doi.org/10.1109/ACCESS.2019.2893760 doi: 10.1109/ACCESS.2019.2893760
[18]	W. Feng, J. Zhang, Y. Chen, Z. Qin, Y. Zhang, M. Ahmad, et al., Exploiting robust quadratic polynomial hyperchaotic map and pixel fusion strategy for efficient image encryption, Expert Syst. Appl., 246 (2024), 123190. https://doi.org/10.1016/j.eswa.2024.123190 doi: 10.1016/j.eswa.2024.123190
[19]	W. Feng, K. Zhang, J. Zhang, X. Zhao, Y. Chen, B. Cai, et al., Integrating fractional-order hopfield neural network with differentiated encryption: Achieving high-performance privacy protection for medical images, Fractal Fract., 9 (2025), 426. https://doi.org/10.3390/fractalfract9070426 doi: 10.3390/fractalfract9070426
[20]	A. Wang, C. Shen, J. Pan, C. Zhang, H. Cheng, S. Wei, Research on multiple-image encryption method using modified gerchberg–saxton algorithm and chaotic systems, Opt. Eng., 62 (2023), 098103. https://doi.org/10.1117/1.OE.62.9.098103 doi: 10.1117/1.OE.62.9.098103
[21]	A. Kumar, P. Singh, K. A. K. Patro, B. Acharya, High-throughput and area-efficient architectures for image encryption using PRINCE cipher, Integration, 90 (2023), 224–235. https://doi.org/10.1016/j.vlsi.2023.01.011 doi: 10.1016/j.vlsi.2023.01.011
[22]	M. Abdel-Basset, R. Mohamed, M. Jameel, M. Abouhawwash, Nutcracker optimizer: A novel nature-inspired metaheuristic algorithm for global optimization and engineering design problems, Knowl. Based Syst., 262 (2023), 110248. https://doi.org/10.1016/j.knosys.2022.110248 doi: 10.1016/j.knosys.2022.110248
[23]	P. K. Pal, D. Kumar, Zirili map-based image encryption method for healthcare, military, and personal data security, Phys. Scr., 99 (2024), 125228. https://doi.org/10.1088/1402-4896/ad8d47 doi: 10.1088/1402-4896/ad8d47
[24]	E. Zhu, M. Xu, D. Pi, Anti-control of Hopf bifurcation for high-dimensional chaotic system with coexisting attractors, Nonlinear Dyn., 110 (2022), 1867–1877. https://doi.org/10.1007/s11071-022-07723-w doi: 10.1007/s11071-022-07723-w
[25]	Q. Dong, Y. Bai, K. Zhu, A 5-D memristive hyperchaotic system with extreme multistability and its application in image encryption, Phys. Scr., 99 (2024), 035253. https://doi.org/10.1088/1402-4896/ad2963 doi: 10.1088/1402-4896/ad2963
[26]	F. Yu, Y. Yuan, C. Wu, W. Yao, C. Xu, S. Cai, et al., Modeling and hardware implementation of a class of Hamiltonian conservative chaotic systems with transient quasi-period and multistability, Nonlinear Dyn., 112 (2024), 2331–2347. https://doi.org/10.1007/s11071-023-09148-5 doi: 10.1007/s11071-023-09148-5
[27]	Q. Wang, L. Wang, W. Wen, Y. Li, G. Zhang, Dynamical analysis and preassigned-time intermittent control of memristive chaotic system via T–S fuzzy method, Chaos, 35 (2025), 023102. https://doi.org/10.1063/5.0221159 doi: 10.1063/5.0221159
[28]	W. Szumiński, A new model of variable-length coupled pendulums: From hyperchaos to superintegrability, Nonlinear Dyn., 112 (2024), 4117–4145. https://doi.org/10.1007/s11071-023-09253-5 doi: 10.1007/s11071-023-09253-5
[29]	K. Hayashi, Chaotic nature of the electroencephalogram during shallow and deep anesthesia: From analysis of the Lyapunov exponent, Neuroscience, 557 (2024), 116–123. https://doi.org/10.1016/j.neuroscience.2024.08.016 doi: 10.1016/j.neuroscience.2024.08.016
[30]	E. Zhu, M. Xu, D. Pi, Hopf bifurcation and stability of the double-delay lorenz system, Int. J. Bifurcat. Chaos, 33 (2023), 2350015. https://doi.org/10.1142/S0218127423500153 doi: 10.1142/S0218127423500153
[31]	E. Zhu, Time-delayed feedback control for chaotic systems with coexisting attractors, AIMS Mathematics, 9 (2024), 1088–1102. https://doi.org/10.3934/math.2024053 doi: 10.3934/math.2024053
[32]	M. V. Breugel, F. Bongers, N. Norden, J. A. Meave, L. Amissah, W. Chanthorn, et al., Feedback loops drive ecological succession: Towards a unified conceptual framework, Biol. Rev., 99 (2024), 928–949. https://doi.org/10.1111/brv.13051 doi: 10.1111/brv.13051
[33]	C. Kaspers, Y. Zhou, The number of almost perfect nonlinear functions grows exponentially, J. Cryptol., 34 (2021), 4. https://doi.org/10.1007/s00145-020-09373-w doi: 10.1007/s00145-020-09373-w
[34]	M. J. S. Onglao, P. F. Almoro, Accelerated phase retrieval using adaptive support and statistical fringe processing of phase estimates, Opt. Lett., 49 (2024), 3158–3161. https://doi.org/10.1364/OL.522321 doi: 10.1364/OL.522321
[35]	Y. Su, Z. Wang, Y. Wang, R. Xue, B. Wang, W. Zhong, et al., Multiple-image encryption based on authenticable phase and phase retrieval under structured light illumination, Opt. Commun., 564 (2024), 130603. https://doi.org/10.1016/j.optcom.2024.130603 doi: 10.1016/j.optcom.2024.130603
[36]	L. Ma, W. Zhang, X. Dai, Generation of the flat-top beam using convolutional neural networks and Gerchberg-Saxton algorithm, Phys. Scr., 99 (2024), 126007. https://doi.org/10.1088/1402-4896/ad8d21 doi: 10.1088/1402-4896/ad8d21
[37]	H. Chen, N. Wang, Y. Huang, C. Wu, Y. Rong, Generation of multi-focus shaping with high uniformity based on an improved Gerchberg–Saxton algorithm, Appl. Opt., 63 (2024), 3283–3289. https://doi.org/10.1364/AO.516663 doi: 10.1364/AO.516663
[38]	B. He, X. Yuan, A class of ADMM-based algorithms for three-block separable convex programming, Comput. Optim. Appl., 70 (2018), 791–826. https://doi.org/10.1007/s10589-018-9994-1 doi: 10.1007/s10589-018-9994-1
[39]	L. Xue, C. Ai, Z. Ge, Multi-image authentication, encryption and compression scheme based on double random phase encoding and compressive sensing, Phys. Scr., 100 (2025), 035539. https://doi.org/10.1088/1402-4896/adb35b doi: 10.1088/1402-4896/adb35b
[40]	K. Bi, G. Zhang, J. Zhang, G. Diao, B. Xing, M. Cui, et al., Reprogrammable metasurface holographic image encryption technology based on a three-dimensional discrete hyperchaotic system, Opt. Express, 32 (2024), 38703–38719. https://doi.org/10.1364/OE.538326 doi: 10.1364/OE.538326
[41]	S. Kumar, D. Sharma, Image scrambling encryption using chaotic map and genetic algorithm: a hybrid approach for enhanced security, Nonlinear Dyn., 112 (2024), 12537–12564. https://doi.org/10.1007/s11071-024-09670-0 doi: 10.1007/s11071-024-09670-0
[42]	R. Manekar, E. Negrini, M. Pham, D. Jacobs, J. Srivastava, S. J. Osher, Low-light phase retrieval with implicit generative priors, IEEE Trans. Image Process., 33 (2024), 4728–4737. https://doi.org/10.1109/TIP.2024.3445739 doi: 10.1109/TIP.2024.3445739
[43]	Y. Liu, L. Teng, An image encryption algorithm based on a new Sine-Logistic chaotic system and block dynamic Josephus scrambling, Eur. Phys. J. Plus, 139 (2024), 554. https://doi.org/10.1140/epjp/s13360-024-05349-y doi: 10.1140/epjp/s13360-024-05349-y
[44]	X. Yang, Z. Qiao, Y. Zhou, Ipad: iterative, parallel, and diffusion-based network for scene text recognition, Int. J. Comput. Vis., 133 (2025), 5589–5609. https://doi.org/10.1007/s11263-025-02443-1 doi: 10.1007/s11263-025-02443-1
[45]	V. Agafonov, V. Bryksin, G. Erokhin, A. Ronzhin, Stochastic acquisition systems based on RTH method, J. Appl. Geophys., 230 (2024), 105520. https://doi.org/10.1016/j.jappgeo.2024.105520 doi: 10.1016/j.jappgeo.2024.105520
[46]	J. Lee, W. Kim, J. H. Kim, A programmable crypto-processor for national institute of standards and technology post-quantum cryptography standardization based on the RISC-V architecture, Sensors, 23 (2023), 9408. https://doi.org/10.3390/s23239408 doi: 10.3390/s23239408
[47]	M. M. Wang, X. G. Song, N. R. Zhou, S. H. Liu, Novel 1-D enhanced log-logistic chaotic map and asymmetric generalized gaussian apertured FrFT for image encryption, Chaos Solitons Fract., 187 (2024), 115443. https://doi.org/10.1016/j.chaos.2024.115443 doi: 10.1016/j.chaos.2024.115443
[48]	Z. Huang, F. Zhang, L. Kou, Q. Yuan, Y. Liu, A real-time image encryption algorithm for a distributed energy system based on the 13-D complex chaotic sequence, IEEE Multimedia, 30 (2023), 71–81. https://doi.org/10.1109/MMUL.2023.3317322 doi: 10.1109/MMUL.2023.3317322
[49]	N. V. Cuong, Y. W. P. Hong, J. P. Sheu, UAV-enabled image capture and wireless delivery for on-demand surveillance tasks, IEEE Trans. Wirel. Commun., 23 (2024), 12995–13010. https://doi.org/10.1109/TWC.2024.3397951 doi: 10.1109/TWC.2024.3397951
[50]	P. Ding, W. Hu, P. Geng, J. Zhang, J. Zhu, A new multi-wing hyperchaotic system and its application in image encryption, J. Supercomput., 81 (2025), 1201. https://doi.org/10.1007/s11227-025-07670-4 doi: 10.1007/s11227-025-07670-4

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)