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Chin. Phys. B, 2020, Vol. 29(11): 110504    DOI: 10.1088/1674-1056/abbbfe
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Memristor-based hyper-chaotic circuit for image encryption

Jiao-Jiao Chen(陈娇娇)1,2, Deng-Wei Yan(闫登卫)1,2, Shu-Kai Duan(段书凯)1,2,3,4,5,6, Li-Dan Wang(王丽丹)1,2,3,4,5,†()
1 College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China
2 Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Southwest University, Chongqing 400715, China
3 Intelligent Transmission and Control Technology Joint Engineering Laboratory, Chongqing 400715, China
4 Brain-inspired Computing and Intelligent Control Chongqing Key Laboratory, Chongqing 400715, China
5 Chongqing Collaborative Innovation Center for Brain Science, Chongqing 400715, China
6 School of Artificial Intelligence, Southwest University, Chongqing 400715, China
Abstract  

The memristor is a kind of non-linear element with memory function, which can be applied to chaotic systems to increase signal randomness and complexity. In this paper, a new four-dimensional hyper-chaotic system is designed based on a flux controlled memristor model, which can generate complex chaotic attractors. The basic dynamic theory analysis and numerical simulations of the system, such as the stability of equilibrium points, the Lyapunov exponents and dimension, Poincare maps, the power spectrum, and the waveform graph prove that it has rich dynamic behaviors. Then, the circuit implementation of this system is established. The consistency of simulation program with integrated circuit emphasis (SPICE) simulation and numerical analysis proves the ability to generate chaos. Finally, a new image encryption scheme is designed by using the memristor-based hyper-chaotic system proposed in this paper. The scheme involves a total of two encryptions. By using different security analysis factors, the proposed algorithm is compared with other image encryption schemes, including correlation analysis, information entropy, etc. The results show that the proposed image encryption scheme has a large key space and presents a better encryption effect.

Keywords:  memristor      SPICE simulation      hyper-chaotic system      image encryption  
Received:  30 April 2020      Revised:  04 September 2020      Published:  03 November 2020
Fund: the National Key Research and Development Program of China (Grant No. 2018YFB1306600), the National Natural Science Foundation of China (Grant Nos. 62076207 and 62076208), and the Fundamental Science and Advanced Technology Research Foundation of Chongqing, China (Grant Nos. cstc2017jcyjBX0050).
Corresponding Authors:  Corresponding author. E-mail: ldwang@swu.edu.cn   

Cite this article: 

Jiao-Jiao Chen(陈娇娇), Deng-Wei Yan(闫登卫), Shu-Kai Duan(段书凯), Li-Dan Wang(王丽丹) Memristor-based hyper-chaotic circuit for image encryption 2020 Chin. Phys. B 29 110504

Fig. 1.  

Phase portraits of system (1) with a = 3, b = 1, c = 9, e = 0.2, m = 1 in (a) xy plane, (b) yz plane, (c) xz plane, and (d) xw plane.

x y z w
S1 −155885327979.4965 −519616.0680 5.1961 −300000.9767
S2 155885376219.5367 519616.1484 −5.1961 −300001.0232
S3 0 0 0 0
Table 1.  

Equilibrium points of formula (5) when x < C5.

x y z w
S4 −5.9999× 10−38 −2.0020× 10−38 −5.999× 10−38 0
S5 −0.00003216 −0.00001072 −0.00003216 0
S6 260428016091 −67162084306 −5.19615242 38776047452.94
S7 −26043446442 −67162915693 5.196152427 −38776527455.9
Table 2.  

Equilibrium points of formula (4) when C5xC6.

x y z w
S8 −3899109.2976 −2598.3098032 5.1978846064 −1500.63494764
S9 3902904.93013 2599.57438182 −5.197883763 −1501.36505241
S10 0.00126415720 0.00042138576 0.0012641572 0.000000059188
Table 3.  

Equilibrium points of formula (5) when x > C6.

Fig. 2.  

Poincare map of (a) x = 0 and (b) y = 0.

Fig. 3.  

Lyapunov exponents of the system with a = 3, b = 1, c = 9, e = 0.2, m = 1.

Fig. 4.  

Power spectrum of the system.

Fig. 5.  

Time domain waveform of (a) ty and (b) tw.

Fig. 6.  

Hardware circuit implementation of new hyper-chaotic system.

Fig. 7.  

SPICE simulation results in (a) xy, (b) yz, (c) xz, and (d) xw planes of the system.

Fig. 8.  

Block diagram of proposed image encryption.

Fig. 9.  

Plain image of (a) Lena, (b) Cameraman, and (c) Girl; first encrypted image of (d) Lena, (e) Cameraman, and (f) Girl.

Fig. 10.  

Second encrypted image of (a) Lena, (b) Cameraman, and (c) Girl; decrypted image of (d) Lena, (e) Cameraman, and (f) Girl.

Fig. 11.  

Gray histogram of plaintext of (a) Lena, (b) Cameraman, (c) Girl; gray histogram of ciphered image of (d) Lena, (e) Cameraman, and (f) Girl.

Fig. 12.  

Correlation distribution of (a) plaintext and (b) ciphertext image in horizontal, vertical, and diagonal direction of Lena.

Fig. 13.  

Correlation distribution of (a) plaintext and (b) ciphertext image in horizontal, vertical, and diagonal direction of Cameraman.

Fig. 14.  

Correlation distribution of (a) plaintext and (b) ciphertext image in horizontal, vertical, and diagonal direction of Girl.

Test image Plain image Cipher image
Horizontal Vertical Diagonal Horizontal Vertical Diagonal
Lena (proposed) 0.99570 0.98470 0.99080 −0.00570 0.00160 0.00210
Lena (Ref. [17]) 0.97190 0.94260 0.91990 0.01020 0.00670 0.00520
Lena (Ref. [20]) 0.96457 0.97864 0.95098 −0.02457 −0.02264 −0.01930
Cameraman 0.99290 0.98040 0.98810 0.00590 0.00800 −0.00170
Girl 0.99620 0.98970 0.99270 0.00830 0.00520 −0.00460
Table 4.  

Correlation coefficients of plain image and cipher image.

Test image Plaintext image Ciphertext image
Lena (proposed) 7.39820 7.96920
Lena (Ref. [17]) 7.99740
Lena (Ref. [20]) 7.43087 7.98859
Cameraman 7.00060 7.96820
Girl 7.49570 7.98780
Table 5.  

Information entropy of test images.

Test image NPCR/% UACI/% BACI/%
Lena 99.62 33.42 26.73
Cameraman 99.59 33.54 26.79
Girl 99.60 33.43 26.76
Table 6.  

Values of NPCR, UACI, and BACI of different images after modifying one pixel value.

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