Image Encryption Algorithm Based on a Novel Wide-Range Discrete Hyperchaotic Map

Abstract

Existing hyperchaotic systems suffer from a small parameter range and small key space. Therefore, we propose herein a novel wide-range discrete hyperchaotic map(3D-SCC) based on the mathematical model of the Sine map. Dynamic numerical analysis shows that this map has a wide-range of parameters, high sensitive, high sensitivity of sequences and good ergodicity, which proves that the system is well suited to the field of communication encryption. Moreover, this paper proposes an image encryption algorithm based on a dynamic cycle shift scramble algorithm and image-sensitive function. First, the image feature is extracted by the image-sensitive function to input into the chaos map. Then, the plaintext image is decomposed by an integer wavelet, and the low-frequency part is scrambled by a dynamic cyclic shifting algorithm. The shuffled low-frequency part and high-frequency parts are reconstructed by wavelet, and the chaotic matrix image is bitwise XOR with it to obtain the final ciphertext. The experimental results show that the average NPCR is 99.6024%, the average UACI is 33.4630%, and the average local Shannon entropy is 7.9029, indicating that the statistical properties of the ciphertext are closer to the ideal value. The anti-attack test shows that the algorithm can effectively resist cutting attacks and noise attacks. Therefore, the algorithm has great application value in the field of image encryption.

1. Introduction

1.1. Background

The chaotic system is a non-linear system in which chaotic motion exists. Chaotic motion is a never-repeating, reversible, non-periodic motion generated by a deterministic system with sensitive dependence on initial conditions [1]. Chaotic systems have the characteristics of singular attractor, ergodicity and internal randomness [2]. Lyapunov Exponent (LE) is the rate of the exponential separation with the time of initially close trajectories, which can judge the appearance of chaotic movement [3]. Chaotic systems have the characteristics of positive Lyapunov exponents. Simple chaotic systems have one positive Lyapunov exponent, while hyperchaotic systems have two or more positive Lyapunov exponents, which mean that the hyperchaotic systems have more complex dynamical behavior and represent more complex topologies [4].

The types of chaotic systems can be classified as continuous chaotic systems and discrete chaotic systems. Continuous chaotic systems are chaotic systems based on the observation of time series, where the state of the system is time-dependent, such as the Lorenz system [5] and Chua’s circuit [6]. Unlike continuous dynamical systems, discrete dynamical systems are special types of dynamical systems whose instantaneous states are described by discrete variables [7], such as Logistic map [8], Henon map [9] and the Sine map [10].

Since Mr. Lorenz proposed Lorenz chaos in 1963 [11], chaos systems have attracted the attention of scholars, and many chaos systems have been proposed. For one-dimensional chaos, there are typical mappings such as Logistic mapping, Tent mapping and Sine mapping. However, although one-dimensional chaos has high computational efficiency, it also has the problems of small parameter range, low complexity and low ergodicity. Therefore, more scholars pay attention to high-dimensional chaos and try to construct more complex chaos with hyperchaotic behavior [12,13,14,15]. Zhongyun Hua et al. proposed a two-dimensional chaotic 2D-SLMM combining Logistic mapping and Sine mapping, which has a wider range of parameters, better ergodicity, hyperchaotic characteristics and relatively low implementation cost [12]. Gholamin, P. et al. proposed a new three-dimensional continuous chaos system, which has a new type of attractor but does not have hyperchaotic behavior [13]. Hongxiang Zhao et al. proposed a high-complexity hyperchaotic system by improving the Henon map [14]. At present, there are few studies on the construction of high-dimensional discrete chaos. This paper proposes a novel discrete three-dimensional hyperchaotic map based on the mathematical model of a Sine map.

With the rapid development of the Internet and computer technology, vast amounts of information are transmitted, stored and shared every day. In recent years, people have paid more and more attention to information security. The mainstream methods to ensure image security are digital steganography, digital watermarking and image encryption. Digital steganography protects image security by hiding secret messages in normal messages [16,17,18]. Digital watermarking protects image security by hiding one or more secret marks in the host media [19,20,21]. Image encryption makes the image a meaningless snowflake image through pixel scrambling and diffusion [22,23,24,25]. This article mainly studies image encryption. In cryptography, confusion and diffusion are two properties of the operation of a secure cipher [10]. In image encryption algorithms, confusion is the operation of changing the pixels value of plaintext. Diffusion is the operation of changing the location of the pixels of plaintext. The ability of the chaos system is to generate pseudo-random sequences with unpredictability, ergodicity and sensitivity, and these characteristics are closely related to the confusion and diffusion principles of modern cryptography. Therefore, chaos systems are often used in image encryption algorithms [11,26,27,28,29,30,31,32,33,34,35].

At present, some encryption algorithms only perform encryption in the time domain, and there is a risk of being deciphered. Therefore, scholars have studied many encryption algorithms performing in both frequency domain and time domain [23,28,31,36,37,38]. Feng-Ping An et al. proposed an encryption algorithm based on adaptive wavelet chaos [36]. This algorithm has a relatively strong anti-attack ability, but the decrypted image is distorted to a certain degree. Shakir H.R et al. proposed an image encryption method based on selective AES coding of wavelet transform and chaotic pixel shuffling [37]. The algorithm is simple and efficient. However, image encryption in the frequency domain may affect the quality of the decrypted image [13,36].

1.2. Motivations and Contributions

For image encryption algorithms based on chaotic mapping, the security depends mainly on the chaotic mapping complexity and the algorithm [39]. However, some existing chaotic mappings have some drawbacks, such as small chaotic range, low complexity and less complex dynamical behaviour. Some of the existing encryption algorithms also have some shortcomings, such as poor flexibility and weak resistance to attacks. Therefore, the main motivations of the work lie in:

• The high-dimensional discrete chaotic map with a large parameter range of hyperchaotic behaviour is designed for an image encryption algorithm, which is rare with existing algorithms.
• Design a dynamic scrambling algorithm that is more difficult to crack than traditional scrambling algorithms.
• Design image-sensitive algorithms for use in an encryption algorithm, which enhance the link between plaintext and ciphertext to resist both differential and plaintext attacks.
• This encryption algorithm should be an image algorithm that performs simultaneous diffusion or obfuscation operations in the frequency and time domains to improve security and efficiency.
• The image encryption algorithm should have the advantage of clear decrypted images, resistance to noise and clipping attacks and the flexibility to encrypt square or rectangular images.

For the above motivations, this paper proposes an image encryption algorithm based on integer wavelet transform and the new discrete hyperchaotic chaotic map. First, it used the proposed sensitivity function to generate the initial value of the novel discrete hyperchaotic map. Then, it performs integer wavelet decomposition on plaintext image, scrambles the low-frequency part by the proposed dynamic cyclic shift scrambling algorithm and performs wavelet reconstruction on the scrambled low-frequency part and high-frequency parts. Finally, it performs bitwise XOR between the reconstructed image and the chaotic matrix to obtain the final ciphertext image.

Section 2 proposes a novel discrete hyperchaotic map and conducts a series of tests on it. Section 3 proposes the sensitive function based on the convolution, the dynamic circle shifting algorithm and the encryption algorithm based on the above algorithm. Section 4 carries out the simulation experiments and security analysis to the ciphertext. Section 5 summarizes and discusses the full text.

2. Mathematical Model and Dynamic Analysis of the Novel Discrete Hyperchaotic Map

The Sine map [5] is a one-dimensional chaos map, which is defined as:

$$x_{i+1}=ssin\left(\pi x_i\right),$$

(1)

where parameter $s\in$ [0, 1]. The Sine map is chaotic when $s\in$ [0.87, 1].

The Sine map has a small range of parameters, and its sequence is not complex enough, so it cannot meet the requirements of communication encryption. Therefore, inspired by the Sine map, this paper presents a three-dimensional chaotic map. The detailed steps of generating the three-dimensional map are explained as follows. In the first step, two new dimensions are added to the Sine map. To increase the diversity of frequencies, two dimensions are added as cos functions, as Equation (2) shown.

$$\left\{\begin{array}{c}\hfill x_{i+1}=s_{1}sin\left(\pi x_i\right)\\ \hfill y_{i+1}=s_{2}cos\left(\pi y_i\right)\\ \hfill z_{i+1}=s_{3}cos\left(\pi z_i\right)\end{array}\right.$$

(2)

where $s_1,s_2$ and $s_3$ are the control parameters.

In Equation (2), these three dimensions are unrelated to each other. Therefore, the three sequences are coupled in the next step. The method of coupling in this paper is to design a perturbation function associated with three dimensions as a perturbation term that is added to the input term in the trigonometric function. Let this perturbation term be g(h,w,z), and the new equation is Equation (3) as shown.

$$\left\{\begin{array}{c}\hfill x_{i+1}=s_{1}sin(\pi x_i+g(y,x,z))\\ \hfill y_{i+1}=s_{2}cos(\pi y_i+g(z,y,x))\\ \hfill z_{i+1}=s_{3}cos(\pi z_i+g(x,y,z))\end{array}\right.$$

(3)

The next step requires adjusting the sequence range and expanding the parameter space. To make the sequence more evenly distributed and have better ergodicity, let the gain term of the trigonometric function be equal to 1, i.e., $s_1=s_2=s_3=1$. At this point, the parameter space needs to be increased, so the gain item in the trigonometric input becomes an adjustable parameter. The new equation, Equation (4), is shown below.

$$\left\{\begin{array}{c}\hfill x_{i+1}=sin(a{x}_i+g(y,x,z))\\ \hfill y_{i+1}=cos(b{y}_i+g(z,y,x))\\ \hfill z_{i+1}=cos(c{z}_i+g(x,y,z))\end{array}\right.$$

(4)

where $a,b,c$ are the control parameters.

The next step is to discuss the construction of the perturbation function. The output value of the set perturbation function should not be much larger than the value of the sequence $x,y,z$. Otherwise, the perturbation term becomes the main determinant of the output value of the sequence. The output values for different input terms should fluctuate up and down in some number, i.e., the output terms are only slightly different. Based on the above conditions, this paper sets up the perturbation function $g(h,w,z)$ as shown in Equation (5).

$$g(h,w,z)=\frac{h^2}{w\times z+ϵ}$$

(5)

where $ϵ$ is the numbers close to 0. Since the value range of the sequences $x,y,z$ is [−1, 1], the output value of the perturbation term fluctuates up and down at 1. Adding a number close to 0 to the denominator is to prevent the denominator from being 0. Substitute Equation (5) into Equation (4) and obtain the new equation as:

$$\left\{\begin{array}{c}\hfill x_{i+1}=sin[{(a{x}_i+{\displaystyle \frac{y_i^2}{x_i{z}_i+ϵ}})}^h]\\ \hfill y_{i+1}=cos[{(b{y}_i+{\displaystyle \frac{z_i^2}{x_i{y}_i+ϵ}})}^h]\\ \hfill z_{i+1}=cos[{(c{z}_i+{\displaystyle \frac{x_i^2}{y_i{z}_i+ϵ}})}^h]\end{array}\right.$$

(6)

where $a,b,c,h$ are the control parameters, and $a,b,c\in [0,{10}^7]$, $h\in [1,2,3,\dots ,10]$. This paper calls it 3D-SCC for short. It is easy to know that the 3D-SCC is bounded, and the output range is [−1, 1]. Through a trajectory analysis, the Lyapunov exponent, the bifurcation diagram, and a complexity analysis, The 3D-SCC has good chaotic characteristics, such as infinite parameter range, high sensitivity and high complexity.

2.1. Phase Diagrams

The phase diagrams of the 3D-SCC with the parameter $a=b=c=5,h=3$ and the initial value $(x_0,y_0,z_0)=(0.1,0.1,0.1)$ are shown in Figure 1. It can be seen that the chaotic sequences are almost uniformly distributed in the [−1, 1] space, meaning that the 3D-SCC map has good ergodicity.

Histograms of the 3D-SCC map and the Sine map are exhibited in Figure 2. Two sequences of length 15000 produced by the 3D-SCC map and the Sine map, respectively, are normalised to [0, 1]. Then, we calculate the frequency of each interval to obtain the histogram. As can be observed, the frequencies of the intervals [0, 0.05] and (0.95, 1] are higher than those of other intervals in the histogram of the 3D-SCC map. However, beyond these, the frequencies of other intervals are very close, which can be considered as almost uniformly distributed. However, in the histogram of the Sine map, there are intervals of zero frequency. Therefore, the 3D-SCC map has better ergodicity.

2.2. Lyapunov Exponent and Bifurcation Diagrams

The Lyapunov exponent (LE) and bifurcation diagrams(BD) are the important indicators to verify the dynamic state of chaos [3].

The LE is the rate of the exponential separation with time of initially close trajectories. In this paper, the Jacobian matrix method [40] is used to calculate the Lyapunov exponent of the chaos map in the following situations. Figure 3 shows the LEs and corresponding BD of the Sine map with the initial conditions $x_0=0.1$ and $s\in [0,1]$. We can notice that the LEs of the Sine map are positive only when $s\in [0.87,1]$, and a wide range of the window period appears in the bifurcation diagram. Figure 3 exhibits the LEs and corresponding BDs of the 3D-SCC map, with the initial conditions $(x_0,y_0,z_0)=(0.1,0.1,0.1)$ and $a,b,c\in [0,20],h=3$. We can notice that the LE1 and the LE2 are always positive, and there are no window periods in BDs, meaning that the 3D-SCC has a wide range of hyperchaotic behavior in the parameter space. Two parameters are fixed at 2. Figure 4 shows the LEs and corresponding BDs as the remaining parameter changes from 0 to 20. And two parameters are fixed at 100. Figure 5 shows the LEs and corresponding BDs as the remaining parameter changes from 0 to ${10}^7$. Interestingly, as one of the system parameters increases, the LE value of the chaotic map increases. Moreover, the three LE values are positive with an infinite range of parameters, which is not common in other discrete chaotic map models. It shows that the system has a high degree of chaos and has good robustness.

2.3. Complexity Analysis

In the application of cryptography, the better the randomness of the chaotic time series, the higher the security of the cryptographic system. This paper uses approximate entropy (ApEn) [41] to test the complexity of the chaotic sequence of the system.

The ApEn complexity graph is used to measure the complexity of the system sequence, as Figure 6 shows. The color bars in Figure 6 represent the ApEn complexity value for x sequence of the 3D-SCC with corresponding parameters. The red region corresponds to higher complexity. The ApEn values are all greater than 1.8907, and the highest is 2.0411. The ApEn values are relatively stable, with a large area of red distributed on the top layer, indicating that the AE values are maintained at a high value and only fluctuate in a small range. Overall, the 3D-SCC map can output highly complex sequences within a wide range of parameters.

Figure 7 calculates the approximate entropy of other discrete chaos maps and compares them with the proposed map. It can be seen from Figure 6 that the AE values of the 3D-SCC are the largest within the parameter range, and all are stable at [1.893, 2.013]. Compared with other discrete chaotic sequences, the 3D-SCC system has high complexity with a wide parameter range. To summarize, this map is an excellent pseudo-random password generator, which is very suitable for the field of communication security.

3. Design of the Image Encryption Based on the Novel Discrete Hyperchaotic Map

This section proposes an image encryption based on the novel discrete hyperchaotic map, an image-sensitive function based on convolution and a dynamic cycle shift scramble algorithm. We explain the image-sensitive function and the dynamic cycle shift scramble algorithm and give detailed steps of the encryption algorithm below. Figure 8 shows the encryption algorithm flowchart.

3.1. Image-Sensitive Function Based on Convolution

The image sensitivity function is a function that is highly sensitive to image information. Even if the information of two images is slightly different, the output value is also completely different. This paper proposes an image-sensitive function based on a convolution operation to extract features. The calculation of the feature matrix uses each pixel value, the mean of all pixels value and the standard deviation of all pixels of the image, which is highly sensitive to the image.

The processed mean value and the processed standard deviation as initial values and parameters are input into the Sin-Tent chaos system to generate a chaotic sequence, and this sequence is used to construct a matrix as a convolution kernel. Then, the convolution kernel is used to perform a two-dimensional discrete convolution operation with the image to obtain an image feature matrix.

Sin-Tent chaos is defined as follows [34].

$$x_{n+1}=\left\{\begin{array}{cc}\hfill (\frac{r{x}_n}{2}+(4-r)sin(\frac{\pi x_n}{4}))mod1,& x_n<\frac{1}{2}\hfill \\ \hfill (\frac{r(1-x_n)}{2}+(4-r)sin(\frac{\pi x_n}{4}))mod1,& x_n<\frac{1}{2}\hfill \end{array}\right.,r\in (0,4]$$

(7)

Suppose the image is $P_{M\times N}$, and the specific steps of extracting features are as follows.

Step 1 Calculate the parameter r and the initial value $x_1$ as follows:

$$\left\{\begin{array}{cc}\hfill r& =mod(std\left(P(:)\right)\times {10}^3,4)+1\hfill \\ \hfill x_1& =mod(mean\left(P(:)\right)\times {10}^3,0.99)+1\hfill \end{array}\right.$$

(8)

where $std(.)$ is the function calculating the standard deviation of data, and $mean(.)$ is the function calculating the mean of the data.

Step 2 The parameter r and the initial value $x_1$ are input into Sin-Tent chaos, and a chaotic sequence of length $(M/m_1\times N/n_1)$ is obtained, as Equation (7) shows. The chaotic sequence is constructed into a matrix of $(M/m_1)\times (N/n_1)$ as the convolution kernel $W_{m1\times n1}$.

Step 3 The horizontal sliding step length of the convolution operation is $n_1$, and the vertical sliding step length is $m_1$. Namely, the image is divided into $(M/m_1)\times (N/n_1)$ non-overlapping blocks. The convolution kernel W performs convolution operation with each block; finally, a feature matrix S of $m_1\times n_1$ is obtained, as Equation (9) shows. $P_{ij}$ is the image block in the $i$-th row and $j$-th column of the image P.

$$S_{ij}=\sum _{u=1}^{M/m_1}\sum _{v=1}^{N/n_1}{p}_{ij}(u,v)W(i-u+1,i-v+1)$$

(9)

3.2. Dynamic Cycle Shift Scramble Algorithm

This section proposes a dynamic cycle shift scramble algorithm. The algorithm uses chaotic sequences to select the scrambling parameters for each time. The scrambling parameters include selecting a row or column, the number of rows/columns, and the cyclic shift step. Suppose the image to be scrambled is P. The size is $M\ast N(M<N)$. The total times of shifts are $num$. The chaotic sequences are $c1,c2,c3$ and their length are $num$. Figure 9 shows an example of the algorithm.The specific algorithm steps are as follows. The algorithm program is shown in Algorithm 1. $circshift$ ($one$-$dimensionalvectora,integern$) means circulating the one-dimensional vector a to the left by n bits; $ceil(.)$ is the round-up function.

Step 1 Use Equation (6) to process the chaotic sequence $c1,c2,c3$:

$$\left\{\begin{array}{cc}\hfill c{1}^{\prime }=& mod(mod(floor(c1\ast {10}^{10}),ceil(N/M)\ast 2),2)\hfill \\ \hfill c{2}^{\prime }=& mod(floor(c2\ast {10}^{10}),M-1)\hfill \\ \hfill c{3}^{\prime }=& mod(floor(c3\ast {10}^{10}),N-1)\hfill \end{array}\right.$$

(10)

where $mod(.)$ is the modulus function, and $ceil(.)$ is a function that rounds to positive infinity, and Set $i=1$.

Step 2 If $c{1}^{\prime }\left(i\right)=0$, select column scrambling and go to Step4. Otherwise, select the row scrambling and go to Step5.

Step 3 Select the $c{3}^{\prime }\left(i\right)th$ column and perform the cyclic shift on this column with a step size of $ceil\left(c{3}^{\prime }\right(num-i+1)/2)$.

Step 4 Select the $c{2}^{\prime }\left(i\right)th$ row and perform the cyclic shift on this row with a step size of $ceil(c{2}^{\prime }(num-i+1)/2)$.

Step 5 If $i=num$, the algorithm ends. Otherwise, $i=i+1$, go to Step2.

Algorithm 1: Dynamic cycle shift scramble algorithm.

Procedure DCS($P,c1,c2,c3,num$)

Input: image P, chaos sequences $c1,c2,c3,num$

Output: Scrambled image $P^{\prime }$

1: $[M,N]=size\left(P\right)$

2: $num=M\ast 50;$

3: $c{1}^{\prime }=mod(mod(floor(c1\ast {10}^{10}),ceil(N/M)\ast 2),2);$

% (the sequence of selecting scrambled rows or columns)

4: $c{2}^{\prime }=mod(floor(c2\ast {10}^{10}),M-1)+1;$

% (the sequence of selecting the scrambling step size)

5: $c{3}^{\prime }=mod(floor(c3\ast {10}^{10}),N-1)+1;$

% (the sequence of selecting the scrambling step size)

6: for i = 1 to $num$

7:    if $c{1}^{\prime }\left(i\right)==0$

8:      $P^{\prime }(:,c{1}^{\prime }(i\left)\right)=circshift\left(P\right(:,c{3}^{\prime }\left(i\right)),ceil(c{3}^{\prime }(num-i+1)/2\left)\right)$

% (Select the $c{3}^{\prime }\left(i\right)$ column and cyclically shift $ceil\left(c{3}^{\prime }\right(num-i+1)/2$ bits)

9:    else

10:      $P^{\prime }\left(c{2}^{\prime }\right(i),:)=circshift\left(P\right(:,c{2}^{\prime }\left(i\right)),ceil(c{2}^{\prime }(num-i+1)/2\left)\right)$

% (Select the $c{2}^{\prime }\left(i\right)$ column and cyclically shift $ceil\left(c{3}^{\prime }\right(num-i+1)/2$ bits)

11:    end if

12: end for

3.3. Image Encryption Steps

Using the hyperchaotic map 3D-SCC, this section proposes an image encryption algorithm based on the dynamic cycle shift scramble algorithm, the image-sensitive function and the integer wavelet transform. Suppose the plaintext image is $P_{M\times N}$, the gray image, and the specific encryption steps are as follows. If the plaintext image is a colour image, each of the three channels will be encrypted using the following algorithm, and then the ciphertexts of the three channels will be combined into the ciphertext of the plaintext image.

Step 1 According to the steps in Section 2.2, setting $m_1$ = 8 and $n_1=2$ (if M is not a multiple of 8, and N is not a multiple of 2, the condition can be achieved by adding 0), generating the feature matrix $S_{8\times 2}$ of the plaintext image. $S_{8\times 2}$ is the key, and we set the private key $\{key1,key2\}$.

Step 2 Calculate $S{a}_{1\times 2},S{b}_{1\times 2},S{c}_{1\times 2},Sx{0}_{1\times 2},Sy{0}_{1\times 2},Sz{0}_{1\times 2},S{h}_{1\times 2},set1,set2$ as follows:

$$\left\{\begin{array}{cc}\hfill Sa=& mod\left(S\right(1,:),100),Sb=mod\left(S\right(2,:),100),\hfill \\ \hfill Sc=& mod\left(S\right(3,:),100),Sh=floor\left(mod\right(s(7,:),10\left)\right)+1,\hfill \\ \hfill Sx0=& mod\left(S\right(4,:),1),Sy0=modS(5,:),1),Sz0=mod(S(6,:),1)\hfill \\ \hfill set1=& mod\left(S\right(8,1)+100,100)\hfill \\ \hfill set2=& mod\left(S\right(8,2)+100,100)\hfill \end{array}\right.$$

(11)

Step 3 Perform integer haar wavelet transform on the plaintext image to obtain the low-frequency part $LL$, and the high-frequency parts $LH,HL,HH$. Only encrypt $LL$ by using $S{a}_{1\times 2},S{b}_{1\times 2},S{c}_{1\times 2},Sx{0}_{1\times 2},Sy{0}_{1\times 2},Sz{0}_{1\times 2},S{h}_{1\times 2},set1$ and the dynamic cycle shift scramble algorithm to obtain the scrambled low-frequency part $L{L}^{\prime }$. The algorithm programs are shown in Algorithms 2 and 3.

Algorithm 2: 3D-SCC.

Procedure 3D-SCC ($a,b,c,h,x1,y1,z1,m$)

Input: the parameters $a,b,c,h,$ the initial value $x1,y1,z1,$ the length of sequence m.

Output: sequences $x,y,z$

1: $x\left(1\right)=x1$

2: $y\left(1\right)=y1$

3: $z\left(1\right)=z1$

4: for i = 2 to m

5: $x\left(i\right)=sin{((a\ast x(i-1)+y(i-1)\ast y(i-1)/(x(i-1)\ast z(i-1)+10(-10)))}^h)$

6: $y\left(i\right)=cos{((b\ast y(i-1)+z(i-1)\ast z(i-1)/(x(i-1)\ast y(i-1)+10(-10)))}^h)$

7: $z\left(i\right)=cos{((c\ast z(i-1)+x(i-1)\ast x(i-1)/(y(i-1)\ast z(i-1)+10(-10)))}^h)$

8: end for

Algorithm 3: Scrambling part.

Input: $LL,M,N,set1,S{a}_{1\times 2},S{b}_{1\times 2},S{c}_{1\times 2},Sx{0}_{1\times 2},Sy{0}_{1\times 2},Sz{0}_{1\times 2},S{h}_{1\times 2}$.

Output: the scrambled part $L{L}^{\prime }$

1: $[c1,c2,c3]$

$=3D-SCC\left(Sa\right(1),Sb(1),Sc(1),Sh(1),Sx0(1),Sy0(1),Sz0(1),M/4\ast N/4+set1)$

% (According to algorithm 2)

2: $L{L}^{\prime }=DCS(LL,c1(set1+1:end),c2(set1+1:end),c3(set1+1:end))$

% (According to algorithm 1)

Step 4 Perform integer wavelet reconstruction on $L{L}^{\prime },LH,HL,HH$ to obtain the semi-encrypted image $CP$.

Step 5 Input $sa1,sb1,sc1,sx01,sy01,sz01,sh1$ into the 3D-SCC chaos map and process the obtained chaotic sequence into a chaotic matrix. The semi-encrypted image $CP$ is divided into four non-overlapping image blocks. Four different chaotic matrices are used to perform bitwise XOR operations with the four image blocks, respectively, to obtain the final ciphertext image $EP$. The algorithm program is shown in Algorithm 4.

Algorithm 4: Diffusion part.

Input: the semi-encrypted image $CP,M,N,set2,S{a}_{1\times 2},S{b}_{1\times 2},S{c}_{1\times 2},Sx{0}_{1\times 2},Sy{0}_{1\times 2},Sz{0}_{1\times 2},S{h}_{1\times 2}$.

Onput: the final encrypted image $EP$.

1: $m2=M/2$

2: $n2=N/2$

3: $[x,y,z]=3D-SCC\left(Sa\right(2),Sb(2),Sc(2),Sh(2),Sx0(2),Sy0(2),Sz0(2),M\ast N+set2)$

4: $x1=reshape(floor\left(mod(x(set1+1:end)\ast {10}^{1}0,256)\right),m2,n2)$

5: $y1=reshape(floor\left(mod(y(set2+1:end)\ast {10}^{1}0,256)\right),m2,n2)$

6: $z1=reshape(floor\left(mod(z(set4+1:end)\ast {10}^{1}0,256)\right),m2,n2)$

7: $EP(1:m2,1:n2)=bitxor\left(CP\right(1:m2,1:n2),x1)$

8: $EP(1:m2,n2+1:b)=bitxor\left(CP\right(1:m2,n2+1:b),y1)$

9: $EP(m2+1:end,1:n1)=bitxor\left(CP\right(m2+1:end,1:n1),z1)$

10: $EP(m2+1:end,n2+1:end)=bitxor\left(CP\right(m2+1:end,n2+1:end),bitxor(y1,z1\left)\right)$

4. Simulation Experiments

To test the effectiveness of the algorithm, this paper used the USC-SIPI[dataset] images dataset and Lena image (512 × 512) for experimentation. The following will make a statistical analysis, an anti-differential attack test and an anti-attack test on the ciphertext to illustrate the security of the ciphertext. This paper used MATLAB 2020a to conduct simulation experiment on a personal computer with an Inter(R) Core(TM) i5-1035G1 CPU @ 3.60 GHz and 16.00 GB memory, and the operating system was a Microsoft Windows 10.

4.1. Ciphertext of Scrambling Algorithm

The traditional scrambling algorithm by chaotic sequence is to process the chaotic sequence in ascending or descending order to obtain a chaotic subscript sequence and use the subscript sequence to scramble the rows or columns of the image. The ciphertext obtained by this algorithm has obvious horizontal stripes, vertical stripes or lattice textures, which can be easily deciphered, as shown in Figure 10a–c. Based on this problem, we propose a dynamic scrambling algorithm that alternately scrambles between rows and columns. The ciphertext after scrambling is shown in Figure 10d, and no texture can be seen, which is better than traditional algorithms.

4.2. Histogram

The histogram is an important statistical characteristic of the image. The 3D histogram describes the gray level of each position of the image, including both the gray level numerical information and the position information. Figure 11, Figure 12 and Figure 13 show the plaintexts, corresponding ciphertexts and their histograms. They include gray and color, square and rectangle images. The plaintext types include grayscale and colour maps, and sizes include rectangular and square, which illustrate the flexibility of the algorithm. It can be seen that the plaintext histograms have obvious statistical characteristics, while the ciphertexts conceal the statistical characteristics of the plaintexts, which verify the effectiveness of the encryption algorithm.

4.3. Correlation Coefficient of Adjacent Pixels

The correlation of adjacent pixels is one of the important characteristics of an image. In plaintext images, neighboring pixels have a high correlation. An effective encrypted image should generate ciphertext with a low correlation between adjacent pixels. The correlation of adjacent pixels in the three directions of the ciphertext is shown in

Table 1. Correlation coefficients of cipher images by other scheme.

Image	Scheme	Horizontal	Vertical	Diagonal	Average
Lena	this paper	0.001348	0.00016	0.002236	0.001248000
	[42]	−0.00205	−0.00509	−0.00398	0.003706667
5.1.12	this paper	0.000357	−0.000128	−0.00374	0.001407667
	[42]	0.00224	−0.00209	0.00102	0.001783333
boat.512	this paper	0.001582	0.000183	0.000286	0.000683667
	[42]	−0.00019	−0.00329	−0.000044	0.001174667
5.3.01	this paper	−0.00011	−0.000036	−0.001721	0.000622333
	[42]	−0.00037	0.00156	0.0000639	0.000664633

. Bolded datas in the table indicate better results. It can be seen that the correlation coefficients of adjacent pixels are all close to 0, which eliminates the statistical characteristics of the plaintext. Compared with other schemes, the correlation between adjacent pixels of the ciphertext of our algorithm is closer to 0. In Figure 14, most of the adjacent pixels of the plaintext are concentrated on the diagonal, which shows that the adjacent pixels of the plaintext are relatively correlated. The adjacent pixels of the ciphertext are evenly distributed, and there is no statistical law, which verifies the effectiveness of the encryption algorithm.

4.4. Information Entropy

Information entropy reflects the amount of information contained in an image and is a statistical form of image features. The Shannon entropy [10] is calculated as follows.

$$H\left(m\right)=\sum _{i=0}^{255}p\left(m_i\right)log\frac{1}{p\left(m_i\right)}$$

(12)

where m is the image, and $p\left(m_i\right)$ is the probability of $m_i$ gray value in the image. The ideal value of the Shannon entropy for the 256 gray level image is 8. Recently, Wu et al. proposed the local Shannon entropy(LoSE) to fairly evaluate the randomness [43]. For an image m, it randomly chooses k non-overlapping image blocks with $T_B$ pixels in the image P, and LoSE is calculated as follows:

$${\overline{H}}_{k,T_B}\left(m\right)=\sum _{i=1}^k\frac{H\left(m{b}_i\right)}{k}$$

(13)

According to [43], the LoSE is better to fall in the interval $[h_l^{\ast },h_r^{\ast }]$. Let $(k,T_B,\alpha )=(20,1963,0.05)$ and $[h_l^{\ast },h_r^{\ast }]=[7.901901,7.903037]$.

Table 2. LoSEs of different cipher images encrypted by different schemes.

Image	[44]	[22]	[45]	Our
5.1.09	7.902572	7.902488	7.902967	7.903489
5.1.10	7.900289	7.901408	7.902436	7.903029
5.1.11	7.903740	7.904619	7.904837	7.906463
5.1.12	7.902851	7.900101	7.890333	7.90536
5.1.13	7.900933	7.902922	7.903234	7.902903
5.1.14	7.905610	7.902073	7.902911	7.90531
5.2.08	7.902309	7.902984	7.9035000	7.904298
5.2.09	7.904542	7.902610	7.902604	7.902574
5.2.10	7.901099	7.902295	7.902812	7.902946
5.3.01	7.903069	7.904512	7.902591	7.902073
5.3.02	7.902729	7.901861	7.902850	7.902474
7.1.01	7.903035	7.901204	7.903333	7.902694
7.1.02	7.902222	7.902573	7.901752	7.902734
7.1.03	7.902076	7.902113	7.904041	7.902889
7.1.04	7.901261	7.902004	7.901952	7.902316
7.1.05	7.902481	7.902155	7.902065	7.902303
7.1.06	7.902062	7.902419	7.902507	7.901997
7.1.07	7.902413	7.902489	7.902027	7.901730
7.1.08	7.902846	7.901884	7.902023	7.902573
7.1.09	7.905016	7.905070	7.902044	7.901978
7.1.10	7.902937	7.902061	7.902021	7.902089
7.2.01	7.902964	7.902284	7.902051	7.903141
boat.512	7.905313	7.902848	7.902769	7.90236
gray21.512	7.902398	7.902648	7.902200	7.902637
ruler.512	7.898843	7.901700	7.902178	7.902199
Pass/All	14/25	16/25	18/25	18/25
Mean	7.902624363	7.902453	7.90216154	7.90298236
Std	0.001517617	0.001054775	0.002567945	0.001175332

shows LoSEs of cipher images encrypted by different schemes. Bolded datas in the table indicate better results. Compared with other algorithms, our scheme has the highest pass rate, the highest mean value and the lowest standard deviation, which indicates the cipher images encrypted by our scheme are random-like.

4.5. Anti-Differential Attack

The differential attack is to use the same encryption algorithm to encrypt the plaintext images before and after the change by making small changes to the plaintext image and to compare the encrypted results to achieve the purpose of deciphering the algorithm. The number of pixels changes rate (NPCR) and the number average changing intensity (UACI) are used to evaluate the sensitivity of the ciphertext image of the algorithm to the plaintext image [46]. The calculation formulas of NPCR and UACI are shown in Equations (14) and (15).

$$NPCR={\displaystyle \frac{{\sum }_{i,j}D(i,j)}{M\times N}}\times 100\%,D(i,j)=\left\{\begin{array}{c}\hfill 0,C_1(i,j)\ne C_2(i,j)\\ \hfill 1,C_1(i,j)=C_2(i,j)\end{array}\right\}$$

(14)

$$UACI={\displaystyle \frac{|C_1(i,j)-C_2(i,j)|}{255\times M\times N}}\times 100\%$$

(15)

where $C_1$ and $C_2$ are two ciphertext images encrypted with only slightly different images.

In [46], Wu et al. proposed a statistical hypothesis of NPCR and UACI. We set $\alpha =0.05$ and the $N_{0.05}^{\ast }$ and $(U_{0.05}^{\ast -},U_{0.05}^{\ast +})$ of different sizes image as shown in

Table 3. The $N_{0.05}^{\ast }$ and $(U_{0.05}^{\ast -},U_{0.05}^{\ast +})$ of different sizes image.

Size	${\mathit{N}}_{0.05}^{\ast }$	$({\mathit{U}}_{0.05}^{\ast -},{\mathit{U}}_{0.05}^{\ast +})$
256 × 256	99.57%	[33.2824%, 33.6447%]
512 × 512	99.59%	[33.3730%, 33.5541%]
1028 × 1028	99.60%	[33.4183%, 33.5088%]

. We used 25 grayscale images from the USC-SIPI database for experiments. We randomly modified a pixel value and calculated the NPCR and UACI. In addition, each image was tested 100 times. The results are shown in Figure 15.

Table 4. NPCR of different cipher images encrypted by different schemes.

NPCR (%)	[47]	[22]	[44]	Our	Pass Rate
5.1.09	99.6064	99.7528	99.5975	99.6109	96
5.1.10	99.6154	99.8062	99.6184	99.6108	96
5.1.11	99.6244	99.7253	99.6064	99.6071	97
5.1.12	99.5703	99.7955	99.6661	99.6088	95
5.1.13	99.6109	99.7803	99.6005	99.6087	95
5.1.14	99.6364	99.8016	99.5915	99.6097	93
5.2.08	99.5870	99.8493	99.6128	99.6082	99
5.2.09	99.6260	99.8257	99.6022	99.6091	93
5.2.10	99.6124	99.7627	99.6271	99.6081	95
5.3.01	99.5931	99.8585	99.6012	99.6092	95
5.3.02	99.6128	99.6954	99.6005	99.6092	97
7.1.01	99.5992	99.7768	99.6158	99.6097	95
7.1.02	99.6075	99.8138	99.6316	99.6078	97
7.1.03	99.6079	99.8074	99.5973	99.6106	95
7.1.04	99.5988	99.8947	99.6075	99.6091	95
7.1.05	99.6170	99.7383	99.6237	99.6088	94
7.1.06	99.6272	99.7414	99.6173	99.6104	95
7.1.07	99.5931	99.7612	99.6429	99.6085	97
7.1.08	99.6094	99.8077	99.9629	99.6102	95
7.1.09	99.6162	99.6616	99.6105	99.6111	99
7.1.10	99.6045	99.7879	99.6154	99.6080	95
7.2.01	99.6156	99.7927	99.6099	99.6099	93
boat.512	99.6154	99.8913	99.5954	99.6086	93
gray21.512	99.6022	99.8131	99.6041	99.6114	98
ruler.512	99.6120	99.6414	99.5936	99.6118	98
Mean	99.608844	99.783304	99.626084	99.60943231
Std	0.013922017	0.061815138	0.072156304	0.001228304

and

Table 5. UACI of different cipher images encrypted by different schemes.

Image Name	[47]	[22]	[44]	Our	Pass Rate
5.1.09	33.4456	33.4856	33.2879	33.4285	96
5.1.10	33.4946	33.5371	33.5309	33.4864	97
5.1.11	33.5541	33.4718	33.4305	33.4505	97
5.1.12	33.4302	33.3971	33.4458	33.5024	92
5.1.13	33.4438	33.4921	33.4509	33.4384	96
5.1.14	33.4655	33.3135	33.528	33.4211	96
5.2.08	33.4008	33.4903	33.492	33.4974	87
5.2.09	33.4804	33.4358	33.5279	33.4791	97
5.2.10	33.4563	33.528	33.4087	33.4311	89
5.3.01	33.4585	33.4763	33.4775	33.4595	93
5.3.02	33.4605	33.476	33.4993	33.4728	95
7.1.01	33.5037	33.4215	33.4945	33.453	93
7.1.02	33.4237	33.4998	33.5126	33.466	95
7.1.03	33.4291	33.504	33.4546	33.4607	92
7.1.04	33.4739	33.4481	33.5024	33.4413	93
7.1.05	33.4362	33.4627	33.4838	33.4755	93
7.1.06	33.3954	33.4839	33.4615	33.4859	92
7.1.07	33.4073	33.4499	33.5115	33.4498	95
7.1.08	33.4332	33.465	33.4534	33.4614	96
7.1.09	33.4177	33.4024	33.414	33.4747	97
7.1.10	33.4344	33.3796	33.4766	33.457	95
7.2.01	33.4556	33.4761	33.4651	33.4606	97
boat.512	33.4654	33.448	33.4625	33.4889	95
gray21.512	33.4608	33.5062	33.5159	33.4653	98
ruler.512	33.4262	33.3741	33.4415	33.4699	98
Mean	33.450116	33.456996	33.469172	33.46308266
Std	0.021134797	0.052105538	0.051387795	0.034869694

show the NPCR and UACI results of different schemes. Bolded datas in the table indicate better results. It can be observed that the scheme of this paper has a high pass rate and is closer to the ideal value than other schemes, which indicates our scheme exhibits resistance against differential attacks.

4.6. Rubust Test

The attacked ciphertext image was decrypted to test the anti-attack of the algorithm. The ciphertext images were attacked by cutting 12.5%, 25%, add 5%, 10%, 15%, 20% Salt and Pepper noise, add 5%, 10%, 15% Gaussian noise and Poisson noise, respectively. The ciphertext images after the cropping attack and the corresponding decrypted images are shown in Figure 16. The ciphertext images after noise attack and the corresponding decrypted images are shown in Figure 17. Although the decrypted image is slightly noisy, the content is clear and identifiable, which shows that the algorithm has a certain degree of resistance to attack. The peak signal-to-noise ratio (PSNR) is used to measure the quality of the decrypted image, and the calculation formula of PSNR is shown in Equation (16). The higher the value of PSNR is, the smaller the degree of distortion. The PSNR value of the decrypted image is shown in Figure 18. As can be observed, the PSNR value of Salt and Pepper noise attack is bigger than that of the Gaussian noise attack, which means the encryption algorithms are more resistant to Salt and Pepper noise attacks. It can be concluded that the proposed algorithm has a strong ability to resist noise and cropping attacks.

$$PSNR=10\times lg(\frac{{255}^2}{\frac{1}{10}\times {\sum }_{m=1}^M{\sum }_{n=1}^N[f(i,j)-g(i,j)]})$$

(16)

4.7. Key Space

The key space is the number of possibilities for all combinations of encryption system keys, which is determined by the length of the key. When the key is the decimal bit, the precision is 15 and the length is $\theta$. There are ${10}^{\theta \ast 15}$ kinds of key combination possibilities, i.e., the size of the key space is ${10}^{\theta \ast 15}$. From the relationship between key space and key length, the longer the key length, the larger the key space. The longer the key length is, the larger the key space. In theory, the longer the key length of the encryption system, the more secure the encryption system is, but in reality, it is impossible to set an infinite key length, i.e., it is impossible to have infinite key space, and in fact, there is no need for infinite key space. The literature [48] suggests that when the key space is greater than $2^{100}$, the key can effectively resist exhaustive attacks. The key space of this paper is ${10}^{18\ast 15}$, or $2^{135}$, which is sufficient to resist brute force and exhaustive attacks. In terms of the key space of the system alone, the algorithm proposed in this paper has good security performance.

4.8. Known and Chosen Plaintext Attack

The known plaintext attack is a mode of attack in which the attacker owns a piece of plaintext and the corresponding ciphertext. The goal of the attacker is to construct an equivalent decryption algorithm or derive an equivalent key for the cryptograph to crack the entire ciphertext [49]. Selective plaintext attacks are performed by constructing a special plaintext image and inputting it into encryption algorithms, using ciphertext pairs and a deciphering algorithm to obtain an equivalent key [49].

For a linear encryption system, the encryption system can be easily broken when an attacker uses a known or chosen plaintext attack on the encryption system. In the encryption algorithm proposed in this paper, the key is strongly correlated with the size and pixel value of the plaintext. In addition, the encryption methods used in this paper for confusion in the wavelet domain and diffusion using xor operation are both non-linear processes, making it difficult for an attacker to obtain the correct key. A plaintext image with all 0 pixels is fed into the encryption algorithm, and the corresponding ciphertext is a snowflake image with no statistical features. When an attacker adopts a chosen plaintext attack on the image encryption algorithm based on integer haar wavelet transform and dynamic confusion, he will undoubtedly only obtain a disorganized image and will not be able to obtain the original image information from the image. Thus, the proposed image encryption algorithm based on integer haar wavelet transform and dynamic scramble can withstand both known plaintext attacks and chosen plaintext attacks.

5. Conclusions

This paper proposes a novel wide-range discrete hyperchaotic map (3D-SCC), the mathematical model of the Sine map, which is based on one sine function and two cosine functions. Through trajectory analysis, the Lyapunov exponent spectrum and approximate entropy analysis, we verified that the proposed map is in the hyperchaotic state with a wide range of parameters, has enough complex sequences and good ergodicity. To improve the security of the encryption algorithm, we designed an image-sensitive function based on convolution to better resist differential attacks and the dynamic cycle shift scrambling algorithm to better scramble. In addition, we performed integer wavelet transform on the plaintext and only scrambled the low-frequency part to improve computational efficiency. Simulated experiments show that the algorithm has excellent performances against attacks, in particular statistical attacks, differential attacks, cut attacks and noisy attacks. The algorithm in this paper is safe, efficient and anti-attack and has practical application value.