A selective image encryption scheme using LICC hyperchaotic system

IET Image ProcessingVolume 16, Issue 12 p. 3342-3358 ORIGINAL RESEARCHOpen Access A selective image encryption scheme using LICC hyperchaotic system Chengjing Wei, Chengjing Wei orcid.org/0000-0002-9413-1639 School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, Guangxi, China Key Laboratory of Data Analysis and Computation, Laboratory of Guangxi Colleges and Universities, Guilin, Guangxi, ChinaSearch for more papers by this authorGuodong Li, Corresponding Author Guodong Li [email protected] orcid.org/0000-0003-1275-2982 School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, Guangxi, China Key Laboratory of Data Analysis and Computation, Laboratory of Guangxi Colleges and Universities, Guilin, Guangxi, China Correspondence Guodong Li, School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China. Email: [email protected]Search for more papers by this author Chengjing Wei, Chengjing Wei orcid.org/0000-0002-9413-1639 School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, Guangxi, China Key Laboratory of Data Analysis and Computation, Laboratory of Guangxi Colleges and Universities, Guilin, Guangxi, ChinaSearch for more papers by this authorGuodong Li, Corresponding Author Guodong Li [email protected] orcid.org/0000-0003-1275-2982 School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, Guangxi, China Key Laboratory of Data Analysis and Computation, Laboratory of Guangxi Colleges and Universities, Guilin, Guangxi, China Correspondence Guodong Li, School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China. Email: [email protected]Search for more papers by this author First published: 30 June 2022 https://doi.org/10.1049/ipr2.12569AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this paper, an improved multi-dimensional hyperchaotic system derived from the logistic map and the ICMIC based on closed-loop coupling (LICC hyperchaotic system) is proposed and it is used in the design of selective image encryption. The performance analysis shows that the LICC hyperchaotic system with a strong coupling degree has parameters whose value can be set more flexibly and can generate a wider and more uniform distribution of chaotic sequences. Then selective image encryption is proposed. Firstly, an adaptive pseudo-random sequence generator based on hash function is designed to ensure the difference of sequences used in different encryption steps. Second, a novel bit-level scrambling is designed to increase the encryption speed. Finally, an efficient rule to select image blocks with weak security is proposed and the second step of the encryption is performed at the selected blocks of the image to ensure security. The encryption performance and security analysis show that the proposed encryption algorithm based on an improved hyperchaotic system is secure enough to resist several types of attacks. 1 INTRODUCTION In the era of the continuous breakthrough of Internet technology and more and more frequent information transmission, various forms of information can be transmitted on the network, which dramatically facilitates our life. In the transmission process of daily routine, information is often vulnerable to illegal theft, tampering, and dissemination, leading to data leakage. Especially in some fields like military, aerospace, medical, and other areas, there is often a need to transmit important, high-definition pictures. It is necessary to design appropriate encryption schemes that can transfer these confidential images safely while ensuring their quality. Chaotic systems are widely used in the design of image encryption schemes due to their characteristics of determinacy, unpredictability, extreme initial value sensitivity, and ergodicity. Therefore, the study of the theory of chaotic systems has been developed. Many researchers have improved several kinds of one-dimensional (1D) chaotic maps. Midoun et al. used the fraction of cosine over sine to design a new 1D chaotic system (1-DFCS) [1]. Mansouri et al. improved the Sine map and constructed a one-dimensional sine-powered map [2]. They also designed a new 1D chaotic map amplifier that can enhance the chaotic behaviour of one-dimensional chaotic maps which are input into the amplifier [3]. The improvement of this kind of chaotic map can improve the chaotic state, and the calculation speed is fast. However, the structure of one-dimensional chaotic maps is simple. The chaotic sequences are often not random enough, and the small number of parameters leads to a small keyspace and is vulnerable to brute force attack. High-dimensional chaotic systems have more complex dynamic characteristics. Chen et al. applied the theory of fractional order to the discrete three-neuron Hopfield neural network to enhance the chaotic behaviour [4]. Li et al. used multi-frequency-frequency conversion sinusoidal to improve the chaotic system of the 6D cellular neural network [5]. A memristor was added into the generalized Lü system with multi-scroll by Yan et al. [6]. Due to the complex dynamic characteristics of the high-dimensional chaotic system equations, it is not easy to design an improved system. And the improved system is often more complicated, and it has brought inconvenience to the calculation. In the design of image encryption algorithms, researchers have designed a variety of encryption schemes according to the characteristics of image information storage. Encryption schemes usually consist of diffusion and scrambling. In some designs of diffusion step, DNA coding, compressive sensing etc are combined with changing the value of pixels [7-10]. The rule of DNA coding is limited and cannot resist the brute force attack if not well designed. Algorithms using compressive sensing are usually robust, but it is easy to cause information loss in the decryption process if the encryption scheme is improper. In scrambling steps, Zhao et al. improved the Joseph traversal to scramble the pixels of images [11]. Wang et al. proposed a new scrambling using ladder transformation [12]. The scrambling method on the bit level can scramble the locations of pixels, and at the same time, change the value of pixels. In the work of Wang et al. [12], the binary sequences of the plain image are scrambled by designing a special DNA operation rule. Dong et al. [13] proposed a globally scrambling algorithm on the bit level. However, the number of cycles required for a bit-level scrambling method usually takes more time. By now, there are few novel bit-scrambling methods with high efficiency. To improve the efficiency of diffusion, many researchers have contributed to selective encryptions. Khan et al. [14] proposed a selection rule based on the CC index for further encryption steps. Inspiring by the natural up-down water wave phenomena, Firous et al. [15] designed a selective encryption technique using the concept of water waves to well encrypt the image. Cun et al. [7] proposed a selective image encryption method based on the local graph structure (LGS) algorithm. These novel selective encryptions provide a new direction for diffusion design. Based on unique selection rules, the uncertainty of the encryption scheme can be improved. Furthermore, a reasonable selection can find the sections of the image with weak security. The discrete chaotic maps are frequently improved and applied for image encryptions. However, one-dimensional chaotic maps are limited to their insufficient complexity, and fewer changeable parameters, which can lead to the vulnerability of encrypted images to resist attacks. Most of the existing high complexity and high-dimensional chaotic systems are based on continuous systems. Until now, the research on the improvement of discrete multi-dimensional chaotic systems and their application in encryption is insufficient. Motivated by the previous work mentioned above, a more efficient algorithm based on a multi-dimensional discrete hyperchaotic system with high complexity and wider ranges of parameters is designed. To enhance the efficiency and security of high-definition image encryption, an improved multi-dimensional hyperchaotic system combining the Logistic map and the ICMIC map through the closed-loop coupling method (LICC) is proposed. The equations generating pseudo-random sequences are relatively simple, and the calculation speed is fast as it does not need to solve the differential equations. According to the needs, it can flexibly change its value of parameters, initial values, the dimension of the system, the number of iterations of the sequence, etc. without affecting its chaotic effect. At the same time, to improve the bit level scrambling method, a bit-block scrambling algorithm is proposed, which can rapidly scramble the position of bits. Then, to eliminate the statistical characteristics within the bit-blocks, a selective encryption scheme based on a novel selection rule is designed. By conducting the selection, the sections containing more information about the original image will be chosen for further encryption, which can improve the speed of the encryption under the premise of security. The following parts of the paper are presented as follows. In Section 2, the LICC hyperchaotic system is proposed. And its performance is presented. The selective image encryption algorithm is described in Section 3. The security of the proposed encryption scheme is shown in Section 4. And the conclusion of the paper is given in Section 5. 2 DESIGN OF LICC HYPERCHAOTIC SYSTEM A hyperchaotic map using the closed-loop coupling method An m-dimensional controlled discrete-time system using the closed-loop coupling method proposed by Liu et al. [16] can be described as X ( i + 1 ) = B A f [ x m ( i ) , X ( i + 1 ) , ω ] , \begin{equation}X(i + 1) = BAf[{x_m}(i),X(i + 1),\omega ],\end{equation} (1)where X ( i ) = [ x 1 ( i ) , x 2 ( i ) , … , x m ( i ) ] T ( m ≥ 2 ) , $X(i) = {[{x_1}(i),{x_2}(i), \ldots ,{x_m}(i)]^T}(m \ge 2),$ A = 0 0 … 0 a a 0 … 0 0 0 a ⋱ 0 0 ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 … a 0 , $A = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{5}{c}@{}} 0&0& \ldots &0&a\\ a&0& \ldots &0&0\\ 0&a& \ddots &0&0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0&0& \ldots &a&0 \end{array} } \right],$ B = D i a g { g [ x 1 ( i ) , c ] , g [ x 2 ( i ) , c ] , … , g [ x n ( i ) , c ] } . $B = Diag\{ g[{x_1}(i),c],g[{x_2}(i),c], \ldots ,g[{x_n}(i),c]\} .$ f [ x m ( i ) , X ( i + 1 ) , ω ] = f 1 [ x 1 ( i + 1 ) , ω ] f 2 [ x 2 ( i + 1 ) , ω ] ⋮ f m − 1 [ x m − 1 ( i + 1 ) , ω ] f m [ x m ( i ) , ω ] $f[{x_m}(i),X(i + 1),\omega ] = \left[ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{f_1}[{x_1}(i + 1),\omega ]}\\[6pt] {{f_2}[{x_2}(i + 1),\omega ]}\\[6pt] \vdots \\[6pt] {{f_{m - 1}}[{x_{m - 1}}(i + 1),\omega ]}\\[6pt] {{f_m}[{x_m}(i),\omega ]} \end{array} } \right]$ is an uniformly bounded non-linear feedback controller. If the Sine map and the ICMIC function are applied respectively, the system can be described as x 1 ( i + 1 ) = a sin [ ω x m ( i ) ] sin [ c / x 1 ( i ) ] x 2 ( i + 1 ) = a sin [ ω x 1 ( i + 1 ) ] sin [ c / x 2 ( i ) ] ⋮ x m ( i + 1 ) = a sin [ ω x m − 1 ( i + 1 ) ] sin [ c / x m ( i ) ] \begin{equation}\left\{ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{x_1}(i + 1) = a\sin [\omega {x_m}(i)]\sin [c/{x_1}(i)]}\\[6pt] {{x_2}(i + 1) = a\sin [\omega {x_1}(i + 1)]\sin [c/{x_2}(i)]}\\[6pt] \vdots \\[6pt] {{x_m}(i + 1) = a\sin [\omega {x_{m - 1}}(i + 1)]\sin [c/{x_m}(i)]} \end{array} } \right.\end{equation} (2) A three-dimensional LICC hyperchaotic system Although the chaotic system constructed by combining the sine map and the ICMIC map through the closed-loop coupling method is hyperchaotic, the attractor in Figure 1 shows that the chaotic sequence cannot traverse the whole range of D = { ( x , y ) | x ∈ [ − 1 , 1 ] , y ∈ [ − 1 , 1 ] } $D{\rm{ = }}\{ (x,y)| {x \in [ - 1,1],y \in [ - 1,1]} \} $ . Therefore, the system is improved in this paper so that the attractor of it can traverse all the D regions. And the chaotic system has higher Lyapunov exponents. FIGURE 1Open in figure viewerPowerPoint Attractor of l 2D hyperchaotic system proposed in [16] Still choose the ICMIC function as in matrix B. Select the Logistic map as a uniformly bounded non-linear feedback controller and combine it with the Cosine map to ensure that sequence values range from [ − 1, 1]. A 3D improved system can be shown as x ( i + 1 ) = cos c 1 ay ( i ) ( 1 − y ( i ) ) + 1 az ( i ) ( 1 − z ( i ) ) sin b x ( i ) y ( i + 1 ) = cos c 1 ax ( i ) ( 1 − x ( i ) ) + 1 az ( i ) ( 1 − z ( i ) ) sin b y ( i ) z ( i + 1 ) = cos c 1 ax ( i ) ( 1 − x ( i ) ) + 1 ay ( i ) ( 1 − y ( i ) ) sin b z ( i ) \begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{l}{x(i+1)=\cos \left[c\left(\frac{1}{\textit{ay}(i)(1-y(i))}+\frac{1}{\textit{az}(i)(1-z(i))}\right)\right]\sin \left[\frac{b}{x(i)}\right]}\\[14pt] {y(i+1)=\cos \left[c\left(\frac{1}{\textit{ax}(i)(1-x(i))}+\frac{1}{\textit{az}(i)(1-z(i))}\right)\right]\sin \left[\frac{b}{y(i)}\right]}\\[14pt] {z(i+1)=\cos \left[c\left(\frac{1}{\textit{ax}(i)(1-x(i))}+\frac{1}{\textit{ay}(i)(1-y(i))}\right)\right]\sin \left[\frac{b}{z(i)}\right]}\end{array} \right. \end{equation} (3) In system (3), the value of one of the sequences is feedback controlled by the values of the other two sequences, which increases the coupling degree between the chaotic system sequences. Performances analysis of the 3D LICC hyperchaotic system 2.3.1 The phase diagrams The phase diagrams of the chaotic system can directly reflect the distribution of chaotic time series. Figure 2 shows the phase diagram of the LICC hyperchaotic system with a = 3.9 , b = π , c = π , $a = 3.9,\,b = \pi ,\,c = \pi ,$ and ( x 0 , y 0 , z 0 ) = ( 0.3 , 0.4 , 0.5 ) $({x_0},{y_0},{z_0}) = (0.3,0.4,0.5)$ . The number of iterations is 50,000. The chaotic sequences can traverse all regions of D = { ( x , y ) | x ∈ [ − 1 , 1 ] , y ∈ [ − 1 , 1 ] } $D{\rm{ = }}\{ (x,y)| {x \in [ - 1,1],y \in [ - 1,1]} \} $ , and their distribution is relatively uniform. FIGURE 2Open in figure viewerPowerPoint Attractors of 3D LICC hyperchaotic system. (a) x–y. (b) y–z. (c) z–x 2.3.2 The Lyapunov exponents The Lyapunov exponents of a chaotic system show an expanding or contracting nature in several directions in the phase space [17]. A positive Lyapunov exponent represents expanding nature, and the system is chaotic. If there are two or more positive Lyapunov exponents, the system is hyperchaotic [18]. Calculate the Lyapunov exponents of the system (3) with a = 3.9 , b = π , c = π $a = 3.9,b = \pi ,c = \pi $ , ( x 0 , y 0 , z 0 ) = ( 0.3 , 0.4 , 0.5 ) $({x_0},{y_0},{z_0}) = (0.3,0.4,0.5)$ using the QR method and obtain three LEs, λ 1 = 4 . 50097 , λ 2 = 3 . 1980 9 , λ 3 = 2 . 22305 ${\lambda _1} = {\rm{4}}{\rm{.50097}},{\lambda _2} = {\rm{3}}{\rm{.1980}}9,{\lambda _3} = {\rm{2}}{\rm{.22305}}$ . Set the initial values to be ( x 0 , y 0 , z 0 ) = ( 0.3 , 0.4 , 0.5 ) $({x_0},{y_0},{z_0}) = (0.3,0.4,0.5)$ and substitute them into the LICC system and iterate 1000 times. Calculate the values of LEs varying with the change of parameters a, b, and c, respectively. The result is shown in Figure 3. According to Figure 3a, when b = π , c = π $b = \pi ,c = \pi $ the value of LEs decreases as parameter a increases and keeps stable as a increases to some level, with a relatively high result of λ 1 = 8 . 07463 , λ 2 = 6 . 55487 , λ 3 = 4 . 98367 ${\lambda _1} = {\rm{8}}{\rm{.07463}},{\lambda _2} = {\rm{6}}{\rm{.55487}},{\lambda _3} = {\rm{4}}{\rm{.98367}}$ while a = 0.01 $a = 0.01$ , which shows that the chaotic system performs better while the parameter a is smaller. FIGURE 3Open in figure viewerPowerPoint Lyapunov exponents of 3D LICC hyperchaotic system. (a) Change of LEs with parameter a when N, c = π $c=\pi $ . (b) Change of LEs with parameter b when a = 0.1 $a=0.1$ , c = π $c=\pi $ . (c) Change of LEs with parameter c when a = 0.1 $a=0.1$ , b = π $b=\pi $ The Lyapunov exponents of another two hyperchaotic systems are calculated in MATLAB to prove that the LE values of the proposed system are larger. Table 1 shows the comparison result. In the table, all the LEs are obtained in the condition of ( x 0 , y 0 , z 0 ) = ( 0.1 , 0.1 , 0.1 ) $({x_0},{y_0},{z_0}) = (0.1,0.1,0.1)$ . TABLE 1. The comparison result of Lyapunov exponents System name Parameters LEs LICC hyperchaotic system a = 1 $a=1$ , b = 3 $b=3$ , c = π $c=\pi $ 3.8993 3.4379 3.4378 SF-SIMM a = 1 $a=1$ , ω = π $\omega =\pi $ , c = 3 $c=3$ 3.7881 3.1385 −0.0309 Mod-Sys [19] a = 1 $a=1$ , ω = π $\omega =\pi $ , c = 3 $c=3$ 3.7492 3.0370 2.2349 2D-SCMM [20] a = 1 $a=1$ , ω = 2 π $\omega =2\pi $ , φ = 0 $\varphi =0$ , c = 100 $c=100$ 2.7716 2.4071 1.5937 2.3.3 Bifurcation diagram analysis Correspondingly, the bifurcation diagram can also show the change of the chaotic sequence with the change of the parameters. As shown in Figure 4, the distribution of chaotic sequence varying with the change of parameters a, b, and c is plotted respectively. Each parameter changes with a step size of 0.01. The length of chaotic sequences shown in the diagrams is 100. It can be seen in Figure 4 that the pseudo-random sequences can still be evenly distributed in [−1, 1] while parameters are changed, which shows that the system can stably maintain chaos when the parameters change. FIGURE 4Open in figure viewerPowerPoint Bifurcation diagram of 3D LICC hyperchaotic system. (a) Bifurcation with parameter a when b = π $b=\pi $ , c = π $c=\pi $ . (b) Bifurcation with parameter b when a = 0.1 $a=0.1$ , c = π $c=\pi $ . (c) Bifurcation with parameter c when a = 0.1 $a=0.1$ , b = π $b=\pi $ 2.3.4 The complexity analysis The permutation entropy (PE) can measure the complexity of a sequence. For the simplicity and rapidity of the calculation and the robustness of non-linear monotonous transformations [16], the PE becomes a proper index to measure the complexity of chaotic sequences. For a one-dimensional sequence x with length N, reconstruct the sequence into a new sequence with n-dimensional and lag time of τ X i = [ x ( i ) , x ( i + τ ) , … , x ( i + ( n − 1 ) τ ) ] T \begin{equation}{X_i} = {[x(i),x(i + \tau ), \ldots ,x(i + (n - 1)\tau )]^T}\end{equation} (4) By reconstructing the sequence through formula (4), the dynamic evolution with time can be evenly considered in the measurement of entropy. Then PE is defined as [21] H ( n ) = − ∑ i = 1 n ! P ( π i ) log 2 P ( π i ) \begin{equation}H(n) = - \sum_{i = 1}^{n!} {P({\pi _i}){{\log }_2}P({\pi _i})} \end{equation} (5)where P ( π i ) $P({\pi _i})$ represents the occurrence probability of the i-th type of permutation of n different numbers. It can be found that H(n) gets the maximum log 2 ( n ! ) ${\log _2}(n!)$ when all the probabilities of permutations are the same. Thus, the larger the H ( n ) $H(n)$ , the more random the sequence. To normalize the index, the following formula is used to get the PE. P E = H ( n ) / log 2 ( n ! ) \begin{equation}PE = H(n)/{\log _2}(n!)\end{equation} (6) The PE varying with three parameters respectively are calculated. And the following figures are obtained. As shown in Figure 5, when the parameter values change, PE changes slightly and stays at a level close to 1, indicating that the sequences all have high complexity and do not change significantly with the change of parameters. FIGURE 5Open in figure viewerPowerPoint PE of improved chaotic system. (a) Change of PE with parameter a when b = π $b=\pi $ , c = π $c=\pi $ . (b) Change of PE with parameter b when a = 0.1 $a=0.1$ , c = π $c=\pi $ . (c) Change of PE with parameter c when a = 0.1 $a=0.1$ , b = π $b=\pi $ The PE of another two hyperchaotic systems is calculated in MATLAB. To show the complexity and the robustness of the LICC hyperchaotic system, all the PE values are calculated when one of the parameters changes within a range. And the maximum, minimum, and average PE values are obtained. Table 2 shows the comparison result. In the table, all the PEs are obtained in the condition of ( x 0 , y 0 , z 0 ) = ( 0.1 , 0.1 , 0.1 ) $({x_0},{y_0},{z_0}) = (0.1,0.1,0.1)$ . TABLE 2. The comparison result of permutation entropy System name Parameters LEs Maximum Minimum Average LICC hyperchaotic system a = ( 0 , 5 ) $a=(0,5)$ , b = 3 $b=3$ , c = π $c=\pi $ 0.9932 0.0233 0.9447 SF-SIMM a = ( 0 , 1 ) $a=(0,1)$ , ω = π $\omega =\pi $ , c = 3 $c=3$ 0 c = 3 $c=3$ .9913 0.0381 0.6110 Mod-Sys [19] a = ( 0 , 1 ) $a=(0,1)$ , ω = π $\omega =\pi $ , c = 3 $c=3$ 0.9927 0.0071 0.6198 2D-SCMM [20] a = ( 0 , 1 ) $a=(0,1)$ , ω = 2 π $\omega =2\pi $ , φ = 0 $\varphi =0$ , c = 100 $c=100$ 0.9929 0 0.7929 In the design of a pseudo-random sequence generator, to ensure the output pseudorandom sequence has better randomness, the applicable range of the parameter values of the chaotic system needs to be determined precisely to avoid generating a non-chaotic or poor chaotic sequence. Spectral entropy (SE) is another measurement for the complexity of sequences which reflects the randomness of the sequence from the perspective of the frequency domain. The SE can reflect the complexity of the whole sequence, and it was proved that the SE is robust to different lengths of the sequence [22]. The value range of SE is between [0, 1]. The closer the value is to 1, the more stable the energy spectrum is and the better the randomness of the sequence is. The calculation of SE is described as [22] S E = − ∑ k = 0 L / 2 − 1 P k log P k ; P k = X ′ ( k ) 2 ∑ k = 0 L / 2 − 1 X ′ ( k ) 2 ; X ′ ( k ) = ∑ n = 0 L − 1 x ( n ) e − j 2 π L n k , k = 0 , 1 , … , L − 1 . \begin{equation} \def\eqcellsep{&}\begin{array}{l} \hspace*{12pt}SE = - \sum_{k = 0}^{L/2 - 1} {{P_k}\log {P_k}} ;\\[6pt] \hspace*{15pt} {P_k} = \frac{{{{\left| {X^{\prime}(k)} \right|}^2}}}{{\sum_{k = 0}^{L/2 - 1} {{{\left| {X^{\prime}(k)} \right|}^2}} }};\\[10pt] X^{\prime}(k) = \sum_{n = 0}^{L - 1} {x(n){e^{ - j\frac{{2\pi }}{L}nk}}} ,k = 0,1, \ldots ,L - 1. \end{array} \end{equation} (7)where x ( n ) , n = 0 , 1 , … , L − 1 $x(n),n = 0,1, \ldots ,L - 1$ is the observed sequence. Take 300 values for each parameter, set the initial values ( x 0 , y 0 , z 0 ) = ( 0.3 , 0.4 , 0.5 ) $({x_0},{y_0},{z_0}) = (0.3,0.4,0.5)$ , and calculate the SE for LICC changing with two parameters shown in Figure 6. FIGURE 6Open in figure viewerPowerPoint SE of LICC system changing with two parameters. (a) Change with a and b. (b) Change with b and c. (c) Change with a and c As shown in Figure 6a, there are some white areas in the range of 0 < b < 1 $0<b<1$ and a > 1 $a>1$ , indicating that the randomness of the sequence taken within this parameter range is low. In Figure 6b, there is a small area with light colour when b ≤ 0.0334 $b \le 0.0334$ . Figure 6c shows that dark red is distributed in the whole range of a , c ∈ [ 0 , 5 ] $a,c \in [0,5]$ . According to the SE analysis, the complexity of LICC is stable at a high level with the change of parameters. In the calculation of the pseudo-random sequence, the sequence with high randomicity can be obtained by avoiding gettings parameter in the range of 0 < b < 1 $0<b<1$ . 2.3.5 The NIST SP800-22 test The NIST SP800-22 test is a standard test for randomness of a sequence proposed by the National Institute of Standards and Technology. It consists of 15 statistical tests. By calculating the P value and the confidence interval of the pass rate of each statistical test, the randomness of the sequence can be determined [23]. To test the randomness of sequences generated by the LICC system, 100 sequences generated by the system with different parameters are selected. Since the pseudo-random sequences need to be converted to numbers in the range of [0, 255] in the design of image encryption, the 100 sequences are first converted to numbers in the range of [0, 255] and converted to a binary form before the NIST test is conducted. The test result is shown in Table 3. TABLE 3. The result of NIST test Statistical test P value Proportion Result Frequency 0.851383 99/100 Pass Block frequency 0.719747 100/100 Pass Cumulative sums 0.726136 98–99/100 Pass Run 0.574903 99/100 Pass Longest run 0.224821 99/100 Pass Rank 0.437274 100/100 Pass FFT 0.162606 99/100 Pass Non-overlapping template 0.501714 95–100 /100 Pass Overlapping template 0.275709 99/100 Pass Universal 0.474986 100/100 Pass Approximate entropy 0.419021 97/100 Pass Random excursions 0.447162 62–63/63 Pass Random excursions variant 0.483764 60–63/63 Pass Serial 0.366416 99/100 Pass Linear complexity 0.055361 100/100 Pass In Table 3, for the statistical tests with more than one P value and proportions, the averages of P value, the maximums, and minimums of proportion are shown. The complete NIST results are given in the supplementary document. The significance level α can be set to be a value in the range of [0.001, 0.01] [24]. In this paper, the value of α is set to be 0.005. If the P value ≥ α $\ge \alpha $ , then the null hypothesis that the sequence is random is accepted. The minimum pass rate for each statistical test except the random excursion (variant) test is approximately = 96 for a sample size of 100 binary sequences. The minimum pass rate for the random excursion (variant) test is approximately = 60 for a sample size of 63 binary sequences. Table 3 gives the final test results in the fourth column, showing that the sequences generated by the LICC system can pass the NIST SP800-22 test. 3 SELECTIVE IMAGE ENCRYPTION ALGORITHM Adaptive pseudo-random sequence generator based on the LICC hyperchaotic system In many existing works, the Hash function is usually used to generate the message digest of the original image as the basis for calculating the parameters of the chaotic system, which can generate the dynamic key to adapt to a different image. However, it is important to consider the effect of different parameters on the sequence complexity. The design of the pseudo-random sequence generator proposed in this paper strictly controls the range of parameters, extending the range of parameters as far as possible under the premise of guaranteeing the good randomness of sequences. The adaptive hyperchaotic sequences are calculated as follows: (1) Calculate SHA-256 value of the plaintext image, and transform the value into 256 binary numbers as K = { k 1 , k 2 , … , k 256 } , ( k i ∈ { 0 , 1 } , i = 1 , 2 , … , 256 ) . $K{\rm{ = }}\{ {{k_1},{k_2}, \ldots ,{k_{256}}} \},({k_i} \in \{ {0,1} \},i = 1,2, \ldots ,256).$ (2) Generate five 4 × 8 matrices K 1 i ( i = 1 , 2 , … , 5 ) ${K_{1i}}(i = 1,2, \ldots ,5)$ and six 2 × 8 matrices K 2 i ( i = 1 , 2 , … , 6 ) ${K_{2i}}(i = 1,2, \ldots ,6)$ from K using formula as follows: K 1 i = k ( i − 1 ) × 32 + 1 k ( i − 1 ) × 32 + 2 … k ( i − 1 ) × 32 + 8 k ( i − 1 ) × 32 + 9 k ( i − 1 ) × 32 + 10 … k ( i − 1 ) × 32 + 16 ⋮ ⋮ ⋱ ⋮ k ( i − 1 ) × 32 + 25 k ( i − 1 ) × 32 + 26 … k ( i − 1 ) × 32 + 32 , i = 1 , 2 , … , 5 . K 2 i = k ( i − 1 ) × 16 + 161 k ( i − 1 ) × 16 + 162 … k ( i − 1 ) × 16 + 168 k ( i − 1 ) × 16 + 169 k ( i − 1 ) × 16 + 170 … k ( i − 1 ) × 16 + 176 , i = 1 , 2 , … , 6 . \begin{eqnarray} && {{K}_{1i}}= \left[ { \def\eqcellsep{&}\begin{array}{cccc} {{k}_{(i-1)\times 32+1}} & {{k}_{(i-1)\times 32+2}} & \ldots & {{k}_{(i-1)\times 32+8}} \\ {{k}_{(i-1)\times 32+9}} & {{k}_{(i-1)\times 32+10}} & \ldots & {{k}_{(i-1)\times 32+16}} \\ \vdots & \vdots & \ddots & \vdots \\ {{k}_{(i-1)\times 32+25}} & {{k}_{(i-1)\times 32+26}} & \ldots & {{k}_{(i-1)\times 32+32}} \end{array} } \right],\nonumber \\ && \quad i=1,2,\ldots ,5. \nonumber \\ && {{K}_{2i}}=\left[ \def\eqcellsep{&}\begin{array}{cccc} {{k}_{(i-1)\times 16+161}} & {{k}_{(i-1)\times 16+162}} & \ldots & {{k}_{(i-1)\times 16+168}} \\ {{k}_