The resistance state of a memristor can be influenced by external stimuli, and these variations can be converted into a pseudorandom sequence through appropriate circuitry and control mechanisms. By leveraging this property, a reliable and complex pseudorandom number generator suitable for encryption can be designed. To enhance the chaotic complexity of memristor-based discrete systems, this paper introduces a three-dimensional hyperchaotic map based on a memristor (3D-HMBM), which integrates a sine-function nonlinearity with a discrete memristor model. Analyzing its dynamical properties via Lyapunov exponents, the 3D-HMBM exhibits evolution from periodicity to chaos and hyperchaos. The complexity of its iterated sequences is verified through metrics such as Spectral Entropy and C0 complexity. Furthermore, the 3D-HMBM displays a unique phenomenon of infinite coexisting attractors. As initial values vary, the system generates attractors at different positions, suggesting that—in theory—an infinite number of attractors exist. Finally, the simulation results are validated via digital-circuit implementation. Building on this foundation, we propose a multi-image encryption algorithm based on the 3D-HMBM, offering a more secure solution for encrypting large volumes of data. Through statistical testing and cryptographic analysis, we confirm the significant potential of the keystream generated by the 3D-HMBM for cryptographic applications.
REFERENCES
1.
W.
Feng
, J.
Yang
, X.
Zhao
, Z.
Qin
, J.
Zhang
, Z.
Zhu
, H.
Wen
, and K.
Qian
, “A novel multi-channel image encryption algorithm leveraging pixel reorganization and hyperchaotic maps
,” Mathematics
12
(24
), 3917
(2024
). 2.
F.
Yu
, Y.
Lin
, W.
Yao
, S.
Cai
, H.
Lin
, and Y.
Li
, “Multiscroll Hopfield neural network with extreme multistability and its application in video encryption for IIoT
,” Neural Netw.
182
, 106904
(2025
). 3.
O.
Kocak
, U.
Erkan
, and I.
Babaoglu
, “Design and practical implementation of a novel hyperchaotic system generator based on Apéry’s constant
,” Integration
103
, 102399
(2025
). 4.
S.
Gao
, J.
Liu
, H. H.-C.
Iu
, U.
Erkan
, S.
Zhou
, R.
Wu
, and X.
Tang
, “Development of a video encryption algorithm for critical areas using 2D extended Schaffer function map and neural networks
,” Appl. Math. Model.
134
, 520
–537
(2024
). 5.
S.
Gao
, H. H.-C.
Iu
, J.
Mou
, U.
Erkan
, J.
Liu
, R.
Wu
, and X.
Tang
, “Temporal action segmentation for video encryption
,” Chaos, Solitons Fractals
183
, 114958
(2024
). 6.
M.
Muñoz-Guillermo
, “Multiscale two-dimensional permutation entropy to analyze encrypted images
,” Chaos
33
(1
), 013112
(2023
).7.
A.
Toktas
, U.
Erkan
, D.
Ustun
, and Q.
Lai
, “Multiobjective design of 2D hyperchaotic system using leader Pareto grey wolf optimizer
,” IEEE Trans. Syst., Man, Cybern.: Syst.
54
(9
), 5237
–5247
(2024
). 8.
B.
Liu
, W.
Song
, M.
Zheng
, C.
Fu
, J.
Chen
, and X.
Wang
, “Semantically enhanced selective image encryption scheme with parallel computing
,” Expert Syst. Appl.
279
, 127404
(2025
). 9.
S.
Gao
, H. H.-C.
Iu
, U.
Erkan
, C.
Şimşek
, J.
Mou
, A.
Toktas
, R.
Wu
, and X.
Tang
, “Design, dynamical analysis, and hardware implementation of a novel memcapacitive hyperchaotic logistic map
,” IEEE Internet Things J.
11
(18
), 30368
–30375
(2024
). 10.
Y.
Yuan
, F.
Yu
, B.
Tan
, Y.
Huang
, W.
Yao
, S.
Cai
, and H.
Lin
, “A class of
-D Hamiltonian conservative chaotic systems with three-terminal memristor: Modeling, dynamical analysis, and FPGA implementation
,” Chaos
35
(1
), 013121
(2025
). 11.
F.
Yang
, X.
Song
, and J.
Ma
, “A memristive map neuron under noisy electric field
,” Chin. J. Phys.
91
, 287
–298
(2024
). 12.
A. L. M.
Biamou
, V. K.
Tamba
, G. C. G.
Kuate
, F. K.
Tagne
, A. C. N.
Takougang
, and H. B.
Fotsin
, “Initial states-induced complex behaviors in a memristive coupled Hopfield neural network model and its application in biomedical image encryption
,” Phys. Scr.
99
(1
), 015215
(2024
). 13.
U.
Kumari
and R.
Yadav
, “Design a chaotic circuit using tunable and power efficient memristor emulator circuit for image encryption
,” Phys. Scr.
100
(1
), 015282
(2025
). 14.
Z.
Yu
, K.
Zhu
, Y.
Wang
, and F.
Yang
, “Dynamics of a neuron with a hybrid memristive ion channel
,” Chaos, Solitons Fractals
194
, 116233
(2025
). 15.
F.
Yu
, D.
Su
, S.
He
, Y.
Wu
, S.
Zhang
, and H.
Yin
, “Resonant tunneling diode cellular neural network with memristor coupling and its application in police forensic digital image protection
,” Chin. Phys. B
34
(5
), 050502
(2025
).16.
B.
Sakthi Kumar
and R.
Revathi
, “A comprehensive review on image encryption techniques using memristor based chaotic system for multimedia application
,” IETE J. Res.
70
(11
), 8160
–8183
(2024
). 17.
F.
Yang
, J.
Ma
, and F.
Wu
, “Review on memristor application in neural circuit and network
,” Chaos, Solitons Fractals
187
, 115361
(2024
). 18.
F.
Yu
, S.
Zhang
, D.
Su
, Y.
Wu
, Y. M.
Gracia
, and H.
Yin
, “Dynamic analysis and implementation of FPGA for a new 4D fractional-order memristive Hopfield neural network
,” Fractal Fract.
9
(2
), 115
(2025
). 19.
D.
Ding
, L.
Jiang
, Y.
Hu
, Z.
Yang
, Q.
Li
, Z.
Zhang
, and Q.
Wu
, “Hidden coexisting firings in fractional-order hyperchaotic memristor-coupled HR neural network with two heterogeneous neurons and its applications
,” Chaos
31
(8
), 083107
(2021
).20.
S.
Gao
, H. H.-C.
Iu
, U.
Erkan
, C.
Simsek
, A.
Toktas
, Y.
Cao
, R.
Wu
, J.
Mou
, Q.
Li
, and C.
Wang
, “A 3D memristive cubic map with dual discrete memristors: Design, implementation, and application in image encryption
,” IEEE Trans. Circuits Syst. Video Technol.
1
(2025
).21.
A. A.
Al-Barakati
, F.
Mesdoui
, S.
Bekiros
, S.
Kaçar
, and H.
Jahanshahi
, “A variable-order fractional memristor neural network: Secure image encryption and synchronization via a smooth and robust control approach
,” Chaos, Solitons Fractals
186
, 115135
(2024
). 22.
A.
Akther
, Y.
Ushakov
, A.
Balanov
, and S.
Savel’ev
, “Deterministic modeling of the diffusive memristor
,” Chaos
31
(7
), 073111
(2021
). 23.
H.
Lin
, C.
Wang
, Y.
Sun
, and T.
Wang
, “Generating n-scroll chaotic attractors from a memristor-based magnetized Hopfield neural network
,” IEEE Trans. Circuits Syst. II: Express Briefs
70
(1
), 311
–315
(2022
).24.
X.
Leng
, L.
Zhang
, C.
Zhang
, and B.
Du
, “Modeling and complexity analysis of a fractional-order memristor conservative chaotic system
,” Phys. Scr.
98
(7
), 075206
(2023
). 25.
E. A.
Hagras
and M.
Saber
, “Low power and high-speed FPGA implementation for 4D memristor chaotic system for image encryption
,” Multimedia Tools Appl.
79
, 23203
–23222
(2020
). 26.
C.
Wang
, D.
Tang
, H.
Lin
, F.
Yu
, and Y.
Sun
, “High-dimensional memristive neural network and its application in commercial data encryption communication
,” Expert Syst. Appl.
242
, 122513
(2024
). 27.
X.
An
, S.
Liu
, L.
Xiong
, J.
Zhang
, and X.
Li
, “Mixed gray-color images encryption algorithm based on a memristor chaotic system and 2D compression sensing
,” Expert Syst. Appl.
243
, 122899
(2024
). 28.
Z.
Lin
and H.
Liu
, “An improved key expansion algorithm based on a 2D discrete hyper chaotic map with memristor
,” Eur. Phys. J. Spec. Top.
233
(6
), 1373
–1380
(2024
). 29.
Q.
Lai
, L.
Yang
, and G.
Chen
, “Design and performance analysis of discrete memristive hyperchaotic systems with stuffed cube attractors and ultraboosting behaviors
,” IEEE Trans. Ind. Electron.
71
(7
), 7819
–7828
(2024
). 30.
S.
He
, D.
Vignesh
, L.
Rondoni
, and S.
Banerjee
, “Chaos and multi-layer attractors in asymmetric neural networks coupled with discrete fractional memristor
,” Neural Netw.
167
, 572
–587
(2023
). 31.
Q.
Lai
, Y.
Liu
, and L.
Yang
, “Image encryption using memristive hyperchaos
,” Appl. Intell.
53
(19
), 22863
–22881
(2023
). 32.
M.
Wang
, M.
An
, S.
He
, X.
Zhang
, H.
Ho-Ching Iu
, and Z.
Li
, “Two-dimensional memristive hyperchaotic maps with different coupling frames and its hardware implementation
,” Chaos
33
(7
), 073129
(2023
). 33.
H.
Bao
, Z.
Hua
, H.
Li
, M.
Chen
, and B.
Bao
, “Discrete memristor hyperchaotic maps
,” IEEE Trans. Circuits Syst. I: Reg. Papers
68
(11
), 4534
–4544
(2021
).34.
Q.
Xu
, L.
Huang
, N.
Wang
, H.
Bao
, H.
Wu
, and M.
Chen
, “Initial-offset-boosted coexisting hyperchaos in a 2D memristive Chialvo neuron map and its application in image encryption
,” Nonlinear Dyn.
111
(21
), 20447
–20463
(2023
). 35.
M. S.
Feali
, “Chaotic dynamics of discrete memristor-coupled Sinh map
,” Chaos, Solitons Fractals
196
, 116480
(2025
). 36.
Q.
Lai
, L.
Yang
, and Y.
Liu
, “Design and realization of discrete memristive hyperchaotic map with application in image encryption
,” Chaos, Solitons Fractals
165
, 112781
(2022
). 37.
L.
Fu
, X.
Wu
, S.
He
, H.
Wang
, and K.
Sun
, “A memristive Hénon map based on the state variable difference and its analog circuit implementation
,” IEEE Trans. Ind. Electron.
71
(8
), 9668
–9676
(2024
). 38.
C.
Şimşek
, U.
Erkan
, A.
Toktas
, Q.
Lai
, and S.
Gao
, “Hexadecimal permutation and 2D cumulative diffusion image encryption using hyperchaotic sinusoidal exponential memristive system
,” Nonlinear Dyn.
113
, 17177
–17208
(2025
).39.
U.
Erkan
, A.
Toktas
, S.
Memis
, F.
Toktas
, Q.
Lai
, H.
Wen
, and S.
Gao
, “OSMRD-IE: Octal-based shuffling and multi-layer rotational diffusing image encryption using 2D hybrid Michalewicz-Ackley map
,” IEEE Internet Things J.
11
(21
), 35113
–35123
(2024
). 40.
S.
Gao
, R.
Wu
, X.
Wang
, J.
Liu
, Q.
Li
, and X.
Tang
, “EFR-CSTP: Encryption for face recognition based on the chaos and semi-tensor product theory
,” Inf. Sci.
621
, 766
–781
(2023
). 41.
Z.
Huang
, F.
Zhang
, L.
Kou
, Q.
Yuan
, and Y.
Liu
, “A real-time image encryption algorithm for a distributed energy system based on the 13-D complex chaotic sequence
,” IEEE MultiMedia
30
(4
), 71
–81
(2023
). 42.
C.
Ye
, X.
Tong
, M.
Zhang
, and Z.
Wang
, “A data hiding scheme in encrypted images using non-degenerate chaotic system and hybrid coding
,” Nonlinear Dyn.
113
, 18771
–18793
(2025
). 43.
Q.
Lai
and Y.
Chen
, “Effect of electromagnetic radiation on double-loop neural networks and its application to image encryption
,” Chaos, Solitons Fractals
194
, 116208
(2025
). 44.
K.
Shahna
, “Novel chaos based cryptosystem using four-dimensional hyper chaotic map with efficient permutation and substitution techniques
,” Chaos, Solitons Fractals
170
, 113383
(2023
). 45.
Z.
Man
, J.
Li
, X.
Di
, R.
Zhang
, X.
Li
, and X.
Sun
, “Research on cloud data encryption algorithm based on bidirectional activation neural network
,” Inf. Sci.
622
, 629
–651
(2023
). 46.
Z.
Zhang
, Y.
Cao
, N.
Zhou
, X.
Xu
, and J.
Mou
, “Novel discrete initial-boosted Tabu learning neuron: Dynamical analysis, DSP implementation, and batch medical image encryption
,” Appl. Intell.
55
(1
), 61
(2025
). 47.
W.
Song
, C.
Fu
, Y.
Zheng
, J.
Chen
, and Y.
Zhang
, “Neural network-driven parallel accelerated selective image encryption with semantic understanding
,” Eur. Phys. J. Spec. Top.
(2025
).48.
S.
Gao
, Z.
Zhang
, H. H.-C.
Iu
, S.
Ding
, J.
Mou
, U.
Erkan
, A.
Toktas
, Q.
Li
, C.
Wang
, and Y.
Cao
, “A parallel color image encryption algorithm based on a 2D logistic-Rulkov neuron map
,” IEEE Internet Things J.
12
(11
), 18115
–18124
(2025
). 49.
G.
Murtaza
and U.
Hayat
, “Efficient image encryption algorithm based on ECC and dynamic S-box
,” J. Inf. Secur. Appl.
90
, 104004
(2025
). 50.
K. N.
Singh
, N.
Baranwal
, O. P.
Singh
, and A. K.
Singh
, “Deepenc: Deep learning-based RoI selection for encryption of medical images through key generation with multimodal information fusion
,” IEEE Trans. Consum. Electron.
70
(3
), 6149
–6156
(2024
). © 2025 Author(s). Published under an exclusive license by AIP Publishing.
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