# Cartoon Image Encryption Algorithm by a Fractional-Order Memristive Hyperchaos

Authors: Shu-ying Wang; Jian-feng Zhao; Xiao-yan Wang; Li-tao Zhang
Journal: Journal of Electrical and Computer Engineering
 (2019)

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## Abstract

Based on the Adomian decomposition method and Lyapunov stability theory, this paper constructs a fractional-order memristive hyperchaos. Then, the 0–1 test analysis is applied to detect random nature of chaotic sequences exhibited by the fractional-order systems. Comparing with the corresponding integer-order hyperchaotic system, the fractional-order hyperchaos possesses higher complexity. Finally, an image encryption algorithm is proposed based on the fractional-order memristive hyperchaos. Security and performance analysis indicates that the proposed chaos-based image encryption algorithm is highly resistant to statistical attacks.


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## Body

## 1. Introduction

With the rapid development of Internet and information technology, multimedia become an important mode of interaction. The widespread use of digital image, making the security problem of image data become more and more prominent, the research on digital image security is of great theoretical value and practical significance. Chaotic systems are extremely sensitive to initial conditions, the orbital unpredictability, and the internal randomness [1–3]. Therefore, it becomes one of the key research objects of modern cryptography to introduce chaos theory into the field of image security [4–6].Memristor is a kind of resistance and in possession of the function of memory. Since 2008, memristor has obtained people’s close concern, which played an active promoting role in the development of circuit and system theory [7, 8]. Chaotic systems designed by memristor have complex dynamic characters, and thus they have wide application prospect in secure communication and image encryption [9–14].The rest of this paper is organized as follows: the fractional-order memristive hyperchaotic system is presented in Section2. The memristive hyperchaotic system shows complicate nonlinear dynamics behavior, and therefore it is difficult to predict trajectories. The proposed image encryption algorithm is described in Section 3. Block permutation and diffusion strategy of plain text processing are used in order to enhance the security of the encryption algorithm. In Section 4, different kinds of images are encrypted as examples by using the designed algorithm, and numerical experiment results are investigated. It is robust against chosen/known plain image attack with just one permutation-diffusion round. Finally, the conclusions are drawn in Section 5.


## 2. Memristive Hyperchaos

Memristive chaos circuit can create a new signal generator, and therefore, it becomes a new research field. A memristive hyperchaos [15] is proposed as follows:(1)dq1x1dtq1=35x2−x1,dq2x2dtq2=95−4cx1+cx2−x1x3+k⋅Wx4x1,dq3x3dtq3=x1x2−3x3,dq4x4dtq4=x1,where k is the coefficient related with memristor Wx4=−a+b⋅x4, x1,x2,x3,andx4 are the state variables, a,b,c,d,andf are the intrinsic parameters, and qii=1,2,3,4 are the Caputo derivatives of fractional order. It is interesting that system (1) has an infinite set of equilibria, including infinite stable equilibria and unstable equilibria, so it shows more plentiful dynamic behaviors.The Adomian decomposition algorithm needs less computation time as well as memory resources, but it yields more accurate results. When solving the integer-order differential system, it is even more accurate than the Runge–Kutta algorithm. Thus, the Adomian decomposition algorithm is adopted in this paper to analyze fractional-order memristive hyperchaos and generate hyperchaotic sequences. Based on the conformable Adomian decomposition method [16], analytical solution of system (1) is expressed with seven terms by the formula(2)x˜it=∑j=06cij⋅t−t0j⋅qiΓj⋅qi+1i=1,2,3,4.The initial conditions are assumed asxi0=xit0,i=1,2,3,4. We select ci0=xi0, ci1=dqixi/dtqit=t0, thus xi1=ci1⋅t−t0qi/Γqi+1i=1,2,3,4 is obtained. In order to obtain other corresponding coefficients, according to fractional calculus property and the nonlinear term decomposition formula, the coefficients of the other five terms can be deduced, respectively,(3)c12=35c21−c11,c22=95−4c−akc11+cc21−c11c30+c10c31+kbc11c40+c10c41,c32=−3c31+c11c20+c10c21,c42=c11,c13=35c22−c12,c23=95−4c−akc12+cc22−c12c30+c10c32+c11c31Γ32+kbc12c40+c10c42+c11c41Γ32,c33=−3c32+c12c20+c10c22+c11c21Γ33,c43=c12,c14=35c23−c13,c24=95−4c−akc13+cc23−c13c30+c10c33+c12c31+c11c32Γ42+kbc13c40+c10c43+c12c41+c11c42Γ42,c34=−3c33+c13c20+c10c23+c12c21+c11c22Γ43,c44=c13,c15=35c24−c14,c25=95−4c−akc14+cc24−c14c30+c10c34+c13c31+c11c33Γ52+c12c32Γ42Γ52Γ32+kbc14c40+c10c44+c13c41+c11c43Γ52+c12c42Γ42Γ52Γ32,c35=−3c34+c14c20+c10c24+c13c21+c11c23Γ53+c12c22Γ43Γ53Γ33,c45=c14,c16=35c25−c15,c26=95−4c−akc15+cc25−c15c30+c10c35+c14c31+c11c34Γ62+c13c32+c13c32Γ52Γ62Γ32+kbc15c40+c10c45+c14c41+c11c44Γ62+c13c42+c12c43Γ52Γ62Γ32,c36=−3c35+c15c20+c10c25+c14c21+c11c24Γ63+c13c22+c12c23Γ52Γ62Γ32,c46=c15,where Γ3j=Γ2qj+1/Γ2qj+1, Γ4j=Γ3qj+1/Γqj+1⋅Γ2qj+1, Γ5j=Γ4qj+1/Γqj+1⋅Γ3qj+1, and Γ6j=Γ5qj+1/Γqj+1⋅Γ4qj+1.Now, consider the case of the derivational fractional ordersqi=0.89i=1,2,3,4 and parameters a=15,b=0.02,c=−10,k=1 in system (1). And also, take initial conditions as x10,x20,x30,x40=1,0,1,0. As shown in Figure 1, the fractional-order memristive hyperchaos has richful dynamic evolution behaviors.Figure 1
Fractional-order chaotic attractor: (a) (x1-x2-x3) plane; (b) (x1-x2-x4) plane.
(a)
(b)Then, the “0–1 test” of fractional-order memristive hyperchaos is conducted by the translation componentsp,s shown in Figure 2, where p and s are determined by observation data, which means that they are independent of original data and system dimension. Considering sequence x1t obtained from fractional-order memristive hyperchaos, numerical simulation displays unbounded trajectory similar to Brownian motion, which shows that the fractional-order memristive hyperchaos has rich dynamic characteristics. The 0–1 test provides a theoretical and experimental basis for application of fractional-order hyperchaos in information security and secure communication.Figure 2
0–1 test of improper fractional-order memristive hyperchaos.


## 3. Proposed Image Encryption

This section presents the image encryption scheme in the framework of symmetric key cipher architecture. For simplicity, we employ gray-scale images with the size ofL=M×N. Based on fractional-order memristive hyperchaos, the proposed encryption algorithm is used for gray image, color image, and binary image. Our algorithm is based on fully layered encryption technique to provide better security. We perform color transformation to separate the RGB color in its component matrix.The fractional-order memristive chaotic system is converted into four one-dimensional chaotic sequences. To eliminate some harmful effect of chaos transient process, we make one thousand times interaction in advance. It is necessary to make the following pretreatment for chaotic sequences:(4)Kji=10mxji−roundxjij=1,2,3,4.Suppose that the positive integerm=12. As shown in Figure 3, a vast majority of sequence K1 fluctuate around zero and locate in the interval [−0.003, 0.003], which indicates that the sequence has a high degree of randomness.Figure 3
Self-correlations of sequenceK1.Divide plain imageP into sections Ajj=1,2,3,4. Pixel value substitution is made according to the following encryption equation:(5)Cji=Aji+KjL+1−imod256⊕Cji−1+Kjimod256,where Cj0=0. Then, the encrypted sequence Cjj=1,2,3,4 is translated into M/2×N/2 two-dimensional matrix, and the encrypted image is Ci=C1i,C2i;C3i,C4i. Decryption is the inverse process of encryption.


## 4. Simulation Results and Security Analysis

An encryption algorithm should be robust against any statistical attacks. In this section, the results and security of encryption performance are evaluated by the histogram, the information entropy, and the correlation of two adjacent pixels in the encrypted.


### 4.1. Histogram Analysis

Histogram analysis of plain image and encrypted image has been performed to validate the algorithm. When some data of plain image or histogram are captured, the statistical attack will be very effective and highly performed. Figures4(b), 5(b), and 6(b) illustrate the histograms of cartoon images “Snow White” (Figure 4(a)), “Mickey Mouse” (Figure 5(a)), and “Binary image” (Figure 6(a)). Histograms of their encrypted images are flat as shown in Figures 4(f)–4(h), 5(d), and 6(d), respectively. It is noticeable that the designed algorithm results in uniform distributions of cipher images, which can resist cipher-only attack.Figure 4
Histogram of color Snow White image and its encrypted image: (a) plain image; (b) histogram of red component of plain image; (c) histogram of green component of plain image; (d) histogram of blue component of plain image; (e) encrypted image; (f) histogram of red component of encrypted image; (g) histogram of green component of encrypted image; (h) histogram of blue component of encrypted image.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 5
Histogram of triangle gray Mickey Mouse image and its encrypted image: (a) Mickey Mouse image; (b) histogram of (a); (c) encrypted image; (d) histogram of (c).
(a)
(b)
(c)
(d)Figure 6
Histogram of binary image and its encrypted image: (a) binary image; (b) histogram of (a); (c) encrypted image; (d) histogram of (c).
(a)
(b)
(c)
(d)


### 4.2. Correlation Coefficient Analysis

In simulation, we randomly select four thousand pixel pairs of horizontal, vertical, diagonal, and counterdiagonal adjacent positions in each direction from plain images and their corresponding cipher images, respectively. To test correlation coefficients of adjacent pixels between the plain images and encrypted images, correlation coefficients are determined by the following formula:(6)Rxy=Convx,yDx⋅Dy,where(7)Ex=1N∑i=1Nxi,Dx=1N∑i=1Nxi−Ex2,Convx,y=1N∑i=1Nxi−Exyi−Ey.For the gray Mickey image, distributions of four directions are shown with different markers in Figure7. The numeric representations of correlation coefficients are calculated and listed in Table 1. The correlation coefficients in plain images are all greater than 0.9, but they are smaller than 0.05 in corresponding encrypted images. Table 1 indicates that the performance of the proposed algorithm is better. Therefore, the proposed encryption algorithm dramatically randomized the pixels.Figure 7
Correlation coefficients of plain and encrypted rectangle image Snow White.Table 1
Correlation coefficients of plain image and encrypted image.
Image
Horizontal (0°)
Vertical (90°)
Diagonal (135°)
Counterdiagonal (45°)
Information entropies
Color image (Snow White)
R
Plain image
0.956870
0.969474
0.928532
0.920907
7.403101
Encrypted image
0.062626
0.058355
0.043477
0.044810
7.701237
G
Plain image
0.944415
0.960116
0.911802
0.902076
6.975635
Encrypted image
0.068252
0.082912
0.049494
0.067867
7.719597
B
Plain image
0.918471
0.943933
0.849832
0.902076
7.395466
Encrypted image
0.048840
0.046259
0.048311
0.051689
7.677057
Gray image (Mickey Mouse)
Plain image
0.962524
0.963978
0.935959
0.942013
4.079404
Encrypted image
7.503378 × 10−4
3.060559 × 10−4
3.191240 × 10−4
8.427805 × 10−5
7.990418
Binary image
Plain image
0.943381
0.965594
0.923663
0.926494
0.661291
Encrypted image
−4.794829 × 10−6
1.172771 × 10−4
−7.071503 × 10−4
0.002179
7.988849


### 4.3. Information Entropy Analysis

The conception of “information entropy” is proposed by Shannon. Information entropy is expressed byHx=−∑i=1npxilog2pxi, where pxi represents probability of symbol xi. In addition, 0≤pxi≤1i=0,1,2,…,n and ∑i=1npxi=1. Information entropy is the whole average uncertainty and statistical properties of overall information sources. Suppose that the source emits n=28 symbols with equal probability, we obtain the largest entropy Hmax=log2n=8. Information entropies of various encrypted images are listed in Table 1, which are close to the theoretical value 8. This means that information leakage in the encryption process is negligible, and the encryption system is secure against entropy attack.


## 4.1. Histogram Analysis

Histogram analysis of plain image and encrypted image has been performed to validate the algorithm. When some data of plain image or histogram are captured, the statistical attack will be very effective and highly performed. Figures4(b), 5(b), and 6(b) illustrate the histograms of cartoon images “Snow White” (Figure 4(a)), “Mickey Mouse” (Figure 5(a)), and “Binary image” (Figure 6(a)). Histograms of their encrypted images are flat as shown in Figures 4(f)–4(h), 5(d), and 6(d), respectively. It is noticeable that the designed algorithm results in uniform distributions of cipher images, which can resist cipher-only attack.Figure 4
Histogram of color Snow White image and its encrypted image: (a) plain image; (b) histogram of red component of plain image; (c) histogram of green component of plain image; (d) histogram of blue component of plain image; (e) encrypted image; (f) histogram of red component of encrypted image; (g) histogram of green component of encrypted image; (h) histogram of blue component of encrypted image.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 5
Histogram of triangle gray Mickey Mouse image and its encrypted image: (a) Mickey Mouse image; (b) histogram of (a); (c) encrypted image; (d) histogram of (c).
(a)
(b)
(c)
(d)Figure 6
Histogram of binary image and its encrypted image: (a) binary image; (b) histogram of (a); (c) encrypted image; (d) histogram of (c).
(a)
(b)
(c)
(d)


## 4.2. Correlation Coefficient Analysis

In simulation, we randomly select four thousand pixel pairs of horizontal, vertical, diagonal, and counterdiagonal adjacent positions in each direction from plain images and their corresponding cipher images, respectively. To test correlation coefficients of adjacent pixels between the plain images and encrypted images, correlation coefficients are determined by the following formula:(6)Rxy=Convx,yDx⋅Dy,where(7)Ex=1N∑i=1Nxi,Dx=1N∑i=1Nxi−Ex2,Convx,y=1N∑i=1Nxi−Exyi−Ey.For the gray Mickey image, distributions of four directions are shown with different markers in Figure7. The numeric representations of correlation coefficients are calculated and listed in Table 1. The correlation coefficients in plain images are all greater than 0.9, but they are smaller than 0.05 in corresponding encrypted images. Table 1 indicates that the performance of the proposed algorithm is better. Therefore, the proposed encryption algorithm dramatically randomized the pixels.Figure 7
Correlation coefficients of plain and encrypted rectangle image Snow White.Table 1
Correlation coefficients of plain image and encrypted image.
Image
Horizontal (0°)
Vertical (90°)
Diagonal (135°)
Counterdiagonal (45°)
Information entropies
Color image (Snow White)
R
Plain image
0.956870
0.969474
0.928532
0.920907
7.403101
Encrypted image
0.062626
0.058355
0.043477
0.044810
7.701237
G
Plain image
0.944415
0.960116
0.911802
0.902076
6.975635
Encrypted image
0.068252
0.082912
0.049494
0.067867
7.719597
B
Plain image
0.918471
0.943933
0.849832
0.902076
7.395466
Encrypted image
0.048840
0.046259
0.048311
0.051689
7.677057
Gray image (Mickey Mouse)
Plain image
0.962524
0.963978
0.935959
0.942013
4.079404
Encrypted image
7.503378 × 10−4
3.060559 × 10−4
3.191240 × 10−4
8.427805 × 10−5
7.990418
Binary image
Plain image
0.943381
0.965594
0.923663
0.926494
0.661291
Encrypted image
−4.794829 × 10−6
1.172771 × 10−4
−7.071503 × 10−4
0.002179
7.988849


## 4.3. Information Entropy Analysis

The conception of “information entropy” is proposed by Shannon. Information entropy is expressed byHx=−∑i=1npxilog2pxi, where pxi represents probability of symbol xi. In addition, 0≤pxi≤1i=0,1,2,…,n and ∑i=1npxi=1. Information entropy is the whole average uncertainty and statistical properties of overall information sources. Suppose that the source emits n=28 symbols with equal probability, we obtain the largest entropy Hmax=log2n=8. Information entropies of various encrypted images are listed in Table 1, which are close to the theoretical value 8. This means that information leakage in the encryption process is negligible, and the encryption system is secure against entropy attack.


## 5. Conclusion

In this study, the Adomian decomposition method is adopted to solve fractional-order memristive hyperchaos. Rich dynamic characteristics of the conformable fractional-order hyperchaotic system are shown by using the 0–1 test. A novel scheme for image encryption of digital images is proposed based on the hyperchaotic system, and it has been validated for color image, gray image, binary image, and so on. Numerical simulations demonstrate that the designed scheme not only maintains the larger secret space but also has better cryptographic performances. The algorithm is analyzed by encrypted image, histogram, correlation coefficients, and so on, therefore effectively ensuring a secure image communication. Effectiveness of the proposed algorithm is fully evaluated by numerical experiments of histogram, correlation coefficients, and information entropy.



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