A Novel Chaos-Based Image Encryption Using Magic Square Scrambling and Octree Diffusing

Abstract

Digital chaotic maps have been widely used in the fields of cryptography owing to their dynamic characteristics, however, some unfavorable security properties arise when they operate on devices with limited precision. Thus, enhancing the properties of chaotic maps are beneficial to the improvement of chaos-based encryption algorithms. In this paper, a scheme to integrate a one-dimensional Logistic map by perturbation parameters with a delayed coupling method and feedback control is proposed and further deepens the randomness by selectively shifting the position of the chaotic sequence. Then, through a number of simulation experiments, the results demonstrate that the two-dimensional chaotic map treated by this mode exhibits better chaotic characteristics, including a larger chaos range and higher complexity. In addition, a new image encryption algorithm is designed based on these modified chaotic sequences, in which magic square theorem is incorporated to exchange pixel positions, and the octree principle is invoked to achieve pixel bit shifting. Several simulation experiments present findings that the image encryption algorithm contains a high level of security, and can compete with other encryption algorithms.

1. Introduction

1.1. Background

With the development of scientific innovations and modern technologies such as the Internet, various forms of multimedia carriers have gradually emerged, among which digital images have seen wide usage in politics, economy, national defense, education, and other fields attributing to their advantages of visual visibility. Thus, methods to secretly and securely transmit and store digital images, particularly those containing private information, has become a central issue among current research fields. Image encryption presents an effective pattern for people to send or receive information securely over a network safely. Specifically, images filled with plain-text information are reconstructed as noise-like data, making it difficult for attackers to recover the original data. In the past decades, in order to meet security demands, researchers have proposed various technologies for image encryption [1,2,3,4]. Zhang, S. and Gao, T. [1] transformed images into a DNA sequence format and hyper-image format for shuffling and replacement, and designed a chaotic pseudo-random number generator to participate in the above operations, subsequently achieving secure encryption. Devi, R. S. et al. [2] utilized security force algorithm, multiple cyclic shift and chaotic displacement techniques to encrypt segmented images, which can effectively resist statistical attacks. Yong, W. et al. [3] proposed a cellular automata-based image encryption model that supports parallel computation, and the encryption algorithm based on this model is considered to be stochastic and sensitive. Xiang, H. and Liu, L. [4] generated improved chaotic sequences by suppressing the dynamic degradation of the chaotic map, and the image encryption was carried out by using a chaotic sequence, rendering the algorithm more secure. Among the existing methods, chaos-based schemes are considered to have great potential over other encryption schemes for image encryption due to their pseudo-randomness, ergodicity and extreme sensitivity to initial values, overcoming the weaknesses of large data volumes and highly correlated image data.

Thus far, many solutions to enlarge chaotic properties have been proposed and their practicability has gradually been verified [5,6,7,8,9,10,11]. Tang, J. et al. [5] introduces delay-coupled state perturbation control parameters and state variables to enhance the dynamics of chaotic systems. Liu, L. et al. [6] presented a new double-perturbed mode to ameliorate the dynamics of baker map, including perturbation of state variables and system parameters. Liu, B. et al. [7] used state variables to interfere with the control parameters of another map to couple two chaotic maps. Hoang, T. M. [8] proposed a regressive chaotic mapping model (PCM-VNI) with a different number of iterations, where perturbations are applied to the state variables and control parameters at the bit level. Researchers of [5,6,7,8] all adopt the method of perturbing the control parameters and state variables to improve the chaotic characteristics, but the state spaces of their fixed structures are not destroyed, thus there remains room for improving the dynamical characteristics. Khlebodarova, T. M. et al. [9] proposed the use of feedback control functions to improve the dynamics of chaotic and hyper-chaotic systems. Xiang, H. et al. [10] simultaneously perturbed the state variables and control parameters of the chaotic map by itself, and the feedback control function was used to improve the chaos degradation. Roberta, Hansen. et al. [11] proposed a new alternative to the ordinary feedback function control to stabilize the unstable periodic orbits of chaotic and hyper-chaotic dynamical systems via modulation of appropriate control parameters. The feedback function describes use of the state function to control the state variables of the digital chaotic map, thus destroying the original state space. However, using this method alone does not significantly improve the performance of the chaotic map, rather it works better when combined with other methods.

According to the above description, to overcome the weaknesses in the improvement process of chaotic systems, this paper proposes a coupled two-dimensional map of the double disturbance control parameters. Firstly, we use the previous variable of the current state variable and the current variable of another subsystem to interfere with the control function to extend the range of chaos parameters. Secondly, a sinusoidal function is introduced to play the role of feedback control so that the input of the current graph affects its output, which increases the complexity of the system. After modification, the chaotic systems proposed in this paper not only complement the shortcomings of one-dimensional maps but additionally enhance the dynamics of two-dimensional chaotic systems. Moreover, to further improve the unpredictability of chaotic sequences used for image encryption, a bit-level exchange operation is introduced to enhance the chaotic features. The chaotic sequences generated by a modified chaotic system are subject to a selective swap operation, which causes a change in the chaotic sequence value subsequently affecting the next sequence value in the same manner as the system input. To the best of our knowledge, fractional bit selection swap operations on chaotic sequences to enhance the unpredictability of the sequences have yet to be studied. By utilizing this improved scheme, the ability to resist violent attacks and statistical attacks is enhanced.

To verify the effectiveness of the modified chaotic system, a novel image encryption algorithm is provided. Although the advanced chaotic sequences render the chaos-based image encryption algorithms more secure, to make the image encryption algorithm unassailable, we add two mathematical concepts: magic square and octree, which correspond to two encryption programs proposed by Shannon: confusion and diffusion. In the confusion algorithm, the pixels are shuffled row by row and column by column according to the generated chaotic sequences and magic square matrix. In the diffusion algorithm, the bits in a pixel are rearranged by using chaotic sequences and octree layout, and bit XOR operation is used to vary the pixel values. Here, using magic square and octree to harden chaos-based encryption methods, even if the saboteurs decipher the sequences, render the solving of the correct plain-text a difficult process.

The rest of this paper is organized as follows. An improved chaotic sequence and its chaotic characteristics analysis are provided in Section 2. Moreover, some numerical simulations of an example are provided in this section. In Section 3, a novel image encryption algorithm based on modified sequences is proposed. Several security experimental analyses are presented in Section 4. Lastly, Section 5 forms a conclusion for the whole paper.

1.2. Related Works

1.2.1. Improved One-Dimensional Chaotic Maps Applied to Image Encryption Algorithm

L, Shouliang. et al. [12] proposed a 1D logical mapping DLCL combined with a time-delay state and coupling method, and contributed a novel color image encryption algorithm based on this DLCL map. In this scheme, the unpredictability of chaotic sequences is improved by extending the bond space of the 1D Logistic map, and a new system with a wider chaotic range, better ergodicity, and larger maximum LE is obtained. Elmanfaloty, R . A. et al. [13] introduced a 1D differential chaotic map with five variables without initial conditions, analyzed its chaotic characteristics, and applied it to an image encryption algorithm. Its unique feature is that adding four parameters to withstand brute-force attacks, although this map remains simple. Wang, X. et al. [14] introduced a new chaotic map. The text image is randomly scrambled and encrypted by 1DSLE, which is a new map engendered by embedding the Sine map into the Logistic map. This scheme preserved the advantage of a 1D map, that is, its ease of implementation on hardware, and compensated for the defect of the small key space.

Generally speaking, most of the improved methods of a 1D chaotic map focus on increasing the number of parameters, such as control parameters or initial conditions, to increase the key space. However, the dynamics of a 1D chaotic map remain limited by its low complexity.

1.2.2. Improved High-Dimensional Chaotic Maps Applied to Image Encryption Algorithm

Haq, T. et al. [15] presented a 4D mixed chaotic system, which was used in the diffusion phase of the color image encryption algorithm. Among them, this four-dimensional chaotic map was a nonlinear combination of the Sine map and Tinkerbell map, showing rich chaotic behaviors. Tong, X. J. et al. [16] designed a perturbed high-dimensional chaotic system according to the definition of topological conjugation, which mainly uses perturbation to enlarge the cycle of the chaotic map, so as to receive four chaotic sequences with high complexity. Then, this perturbed chaotic system is used in the confusion-diffusion operation in the encryption algorithm. Kaur, M. et al. [17] presented a major technique that optimizes the initial parameters of the 5-D chaotic map by using the non-dominant sorting genetic algorithm of local chaotic search. Image encryption mainly uses this optimized five-dimensional system to diffuse the image sub-bands after dual-tree complex wavelet transform (DTCWT), and finally obtains the encrypted image by inverse DTCWT.

In general, the purpose of introducing an improved high-dimensional map into image encryption technology is to render encryption more reliable by using its more complex dynamic characteristics. Nevertheless, the high computational cost of a high-dimensional chaotic system remains inconducive to the speed of image encryption.

1.2.3. Improved Two-Dimensional Chaotic Maps Applied to Image Encryption Algorithm

At present, image encryption algorithms based on improved two-dimensional maps emerge endlessly. The fundamental basis explaining this phenomenon is that 2D chaotic maps contain not only more complex chaotic characteristics than one-dimensional chaotic maps, but also their computational cost is lower than that of high-dimensional maps.

Gao, X. [18] proposed two one-dimensional maps, sin (hπ/x) and rsin (πx), which are combined to transform it into a 2D hyper-chaotic system, and the chaotic performance of this map is proved by attractor locus, 0-1 test, bifurcation graph, and permutation entropy. Then, based on this system, chaotic sequences are used to disturb the image through row and column shifts to change the pixel value, and finally the ciphertext image is obtained. Kumar, V. et al. [19] proposed a color image encryption algorithm combining Lorentz map, Logistic map, and DNA cryptography. Among them, the Logistic map is used to generate a 2D chaotic map by modifying the initial sequence with hamming distance values of three channels, so as to strengthen the plain-text correlation of encryption technology. Monda, B. et al. [20] presented a novel 2D sine–cosine chaotic map and designed an image encryption scheme with a high confusion and blur ability. Zhu, H. et al. [21] used a Logistic map to modulate a Sine map, and combined the results of modulation and Sine map to obtain a 2D coupled map (LSMCL), which applied an image encryption algorithm with two round permutation and diffusion operation. Yang, chao. et al. [22] expressed a 2D multi-fold chaotic map (2D-MCCM), and on this basis, a chaotic S-box based on 2D-MCCM was constructed. Meanwhile, S-Box is used in image encryption technology to improve security and efficiency. Toktas, A. et al. [23] introduced a 2D chaotic map of multi-objective optimization strategy based on an artificial bee colony algorithm, which is used in image encryption programs.

1.3. Preliminaries

This section presents some related preliminaries which are used in this paper.

1.3.1. Logistic Map and Its Modification

The classic Logistic map is widely used in various scientific fields, including applications of image encryption. The 1D Logistic map with formula can be written as follows:

$$x_{i+1}=f\left(x_i,a\right)=a\times x_i\left(1-x_i\right)$$

(1)

where a is the control parameter, chaos states will occur when a is selected in an appropriate value. However, chaos states exist only within a small range of parameters, which makes the original Logistic map vulnerable to attacks by unauthorized access.

Delayed coupling method: the parameter perturbation function h is realized, which uses state variables x_i and y_i to disturb the control parameter a.

$$\left\{\begin{array}{c}{h}_1\left(x_i\right)=a+\left(4-a\right)\times x_{i-1}\times y_i\\ h_2\left(y_i\right)=a+\left(4-a\right)\times y_{i-1}\times x_i\end{array}\right.$$

(2)

where h₁ and h₂ are the control parameter functions of x and y dimensions in the new model, respectively. h₁ is acted on by the previous state variable x_i−1 and the current state variable y_i. Meanwhile, h₂ is disturbed by state variables y_i−1 and x_i. Naturally, two 1D Logistic maps are coupled through above-controlling parameter functions.

Feedback function control: we construct the feedback functions g₁ and g₂ of the perturbation state variables x_i and y_i.

$$\left\{\begin{array}{c}{g}_1\left(x_i\right)=sin\left(\pi \times x_i\right)\\ g_2\left(x_i\right)=sin\left(\pi \times x_i\right)\end{array}\right.$$

(3)

The feedback function aims to self-perturb the chaotic sequence by taking part of the output of chaotic mapping as input, which can be composed of linear or nonlinear functions. In this paper, a simple, low-power sinusoidal function is adopted to destroy the state space, thereby enhancing chaos properties.

1.3.2. Meaning of Magic Squares

An n-order magic square is a square of n × n filled with n² numbers, where the sum of the elements in each column is equal to the sum of the elements in each row. The 16-order magic square used in this article is shown below (

Table 1. The 16-order magic square.

256	2	3	253	252	6	7	249	248	10	11	245	244	14	15	241
17	239	238	20	21	235	234	24	25	231	230	28	29	227	226	32
33	223	222	36	37	219	218	40	41	215	214	44	45	211	210	48
208	50	51	205	204	54	55	201	200	58	59	197	196	62	63	193
192	66	67	189	188	70	71	185	184	74	75	181	180	78	79	177
81	175	174	84	85	171	170	88	89	167	166	92	93	163	162	96
97	159	158	100	101	155	154	104	105	151	150	108	109	147	146	112
144	114	115	141	140	118	119	137	136	122	123	133	132	126	127	129
128	130	131	125	124	134	135	121	120	138	139	117	116	142	143	113
145	111	110	148	149	107	106	152	153	103	102	156	157	99	98	160
161	95	94	164	165	91	90	168	169	87	86	172	173	83	82	176
80	178	179	77	76	182	183	73	72	186	187	69	68	190	191	65
64	194	195	61	60	198	199	57	56	202	203	53	52	206	207	49
209	47	46	212	213	43	42	216	217	39	38	220	221	35	34	224
225	31	30	228	229	27	26	232	233	23	22	236	237	19	18	240
16	242	243	13	12	246	247	9	8	250	251	5	4	254	255	1

).

1.3.3. Meaning of Octrees

The definition of an octree details that if the tree is not empty, there will be exactly eight or zero children of any node in the tree. Putting the application scenario on the image pixel matrix, and removing the pixel values at the boundary positions, results in 8 neighboring pixel points at each pixel position. Figure 1 is an example of a pixel matrix with size 6 × 8, where we can see that the pixels not in the edge position all have 8 neighboring pixel values.

2. The Proposed Chaotic Map and Its Performance

2.1. Model of Improved Logistic Maps

In this paper, we propose the 2D chaotic map with the concepts of delay, coupling, and linear feedback, which is dependent on the original Logistic. The model can be expressed as follows:

$$\left\{\begin{array}{l}{x}_{i+1}=\left(\left(a+\left(4-a\right)\times x_{i-1}\times y_i\right)\times x_i\times \left(1-x_i\right)+sin\left(\pi \times x_i\right)\right) mod 1\\ y_{i+1}=\left(\left(a+\left(4-a\right)\times y_{i-1}\times x_i\right)\times y_i\times \left(1-y_i\right)+sin\left(\pi \times y_i\right)\right) mod 1\end{array}\right.$$

(4)

Note that values of output states variables x_i+1 and y_i+1 need to be in the range of (0, 1), in this case, we employ the mod operation.

After that, two ideal chaotic sequences with certain complexity emerge from this map. Nevertheless, for the image encryption algorithm utilizing chaos sequence, the more unpredictable the sequence is, the better its anti-attack ability has. As mentioned in Wu, Y. et al. [24], it is considered that the extended control parameter is a profitable method to defend sequence attacks. On this basis, a new defense method is added in this paper, that is, disposal of the generated chaotic sequences by selective swapping, which makes it more effective in the field of image encryption. The steps are as follows:

○

Step 1: Assume that the resulting chaotic sequence generated by Equation (4) is {X} = {x₁, x₂, x₃, …, x_M×N}, each of the three adjacent values act as a group, that is m_i = {x_i, x_i+1, x_i+2}, where i = 1, 2, 3, …, M × N − 2, and operate each group one by one.

○

Step 2: Each value in m_i is taken to 15 decimal places and separately stored in three arrays by doing the following, that is, from k = 1 to k = 15:

$$\left\{\begin{array}{l}{\left\{S_1\right\}}_k=mod\left(floor \left(x_i\times {10}^k\right),10\right)\\ {\left\{S_2\right\}}_k=mod\left(floor\left(x_{i+1}\times {10}^k\right),10\right)\\ {\left\{S_3\right\}}_k=mod\left(floor\left(x_{i+2}\times {10}^k\right),10\right)\end{array}\right.$$

(5)

where floor() is a down rounding function. mod 10 means to divide the value by 10 to obtain the remainder, {S₁} is an array from x_i, {S₂} is an array from x_i+1, {S₃} is an array from x_i+2.

○

Step 3: Each group m_i is operated in Step 2, then results in three arrays n_i = {{S₁}, {S₂}, {S₃}}, respectively. Where i = 1, 2, 3, …, M × N − 2.

○

Step 4: For n_i, traverse the arrays {S₂} and {S₃} one by one from j = 1 to j = length (S₂). When {S₂}_j is equal to 0 but {S₃}_j is not equal to 0, let the q be equal to{S₃}j, and then exchanging {S₁}_j with {S₁}_q. Where length() is a function to obtain the length of the array.

○

Step 5: Repeat the above operation until {X} is completely covered, and the y-dimension is the same, no more tautology here.

Through the simple operations above, the chaotic sequences generated by Equation (4) will be renovated. In this way, even if the saboteur cracked the original sequences, any algorithms using the chaotic series as a pseudo-random numbers generator will remain secure. In fact, this method is valid for any chaotic sequence. Moreover, the mathematics model constructed above is universal for all digital maps and can be extended to coupled multiple chaotic systems. The n-dimensional chaotic map can be shown as:

$$\left\{\begin{array}{c}{x}_{i+1}{}^{(1)}=f_1\left(x_i,h_1\left(x_i^{(2)},x_i^{(3)},\cdots ,x_i^{(N)},x_{i-1}{}^{(1)}\right),g_1\left(x_i\right)\right)\\ x_{i+1}{}^{(2)}=f_2\left(x_i,h_2\left(x_i^{(1)},x_i^{(3)},\cdots ,x_i^{(N)},x_{i-1}{}^{(2)}\right),g_2\left(x_i\right)\right)\\ ⋮\\ x_{i+1}{}^{(N)}=f_N\left(x_i,h_N\left(x_i^{(1)},x_i^{(2)},\cdots ,x_i^{(N-1)},x_{i-1}{}^{(N)}\right),g_N\left(x_i\right)\right)\end{array}\right.$$

(6)

In this model, f represents any arbitrary chaotic map, h denotes the parameter control function equivalent to the output of a in Equation (1), and g is the feedback function with x_i as the input. Given that functions h and g are diverse, thus some inevitable conditions must be satisfied for the system to possess chaotic properties: the numerical range of h_i should belong to the chaos control parameter domain, the input (x_i, x_i−1) and the output x_i+1 need to be limited to the state space (0, 1).

2.2. Performance Analysis of the Improved Map

Next, we set a = 3.99, x₁ = 0.65477880678 for Equations (1) and (4), and set x₂ = 0.56784938, y₁ = 0.3456789, y₂ = 0.2346784 for Equation (4). All of them are used for the following simulation experiments in this article unless otherwise stated.

2.2.1. Trajectories and Phase Diagrams

This section describes the dynamic representation of the improved sequences, in terms of phase spaces and trajectories to demonstrate the effectiveness of our method. As is commonly known, a satisfactory chaotic system is considered to possess the aperiodic trajectory and the phase diagram without distinct structural features.

Here, the x-and y-dimensional trajectories generated by the novel scheme are depicted in Figure 2a,b, respectively. From Figure 2, after about 500 cycles of iterations, both trajectories remain random and neither of them has periodic cycles, which can represent good randomness. Figure 3a,b plot the x-and y-dimensional phase spaces of new chaotic sequences, respectively, Figure 3c portrays the phase diagram of Equation (1). As shown in Figure 3, the improved chaotic phase space breaks the inverted U-shaped structure of the classical Logistic map, and its distribution is full of the whole phase space. This result confirms that the proposed chaos system exhibits ideal dynamics characteristics.

2.2.2. Complexity Analysis

ApEn and PE Approximate entropy (ApEn) is one of the nonlinear kinetic parameters used to measure the complexity of time series, as mentioned in Pincus, S. M. [25]. For time series, the larger the ApEn value, the higher the orbital complexity generated by chaos. Meanwhile, permutation entropy (PE) is another valid factor, which acts similarly to ApEn, and calculates entropy by displacement. Compared with ApEn, PE is available for any form of sequences depending on its simple calculation and strong anti-noise robustness. In this experiment, ApEn and PE values of the modified and original chaotic scheme with different parameter settings are shown in Figure 4 and Figure 5, respectively.

From Figure 4, we can observe that the modified ApEn value hovers around 1.75, whereas that of the Logistic map is approximately 0.5. Larger ApEn values directly indicate that the improved sequences will barely display an observable regularity. As seen in Figure 5, the modified PE values are infinitely close to ideal value 1, far greater than that of Equation (1), which emphatically proves the superiority of our method in enhancing the complexity of the chaotic sequence.

Lyapunov exponent analysis As a widely recognized measuring method, the Lyapunov index is taken to describe the presence of chaotic behavior in a map. When LE > 0, the chaos trajectories under two different input conditions will diverge exponentially through iteration, thus showing that chaos is sensitive to initial values. That is to say, chaos must have a positive Lyapunov exponent and the larger an LE value is, the more prominent the chaotic features are. Figure 6 reveals the comparison of the Lyapunov exponents of the original sequence and modified sequence. In this test, we set a ∈ [2, 4], it can be observed that the sequence generated by our scheme possesses a positive LE value within the entire parameter range. Moreover, the LE value is larger than that of the original map.

2.2.3. Auto-Correlation Analysis

The auto-correlation function is accessed to characterize the degree of correlation between the states of a random process itself at any two different moments. For an ideal chaotic sequence, the auto-correlation should be fast attenuation along with the state interval, similar to the δ-function. The auto-correlation analysis of improved sequences is shown in Figure 7, where it can be observed that the self-relevance is the highest when the interval is 0, and in other cases, it rapidly drops to 0. The shape of its curve is the same as that of the δ-function, which proves there is almost no relation between any two variables in the output sequence, and can be considered as good statistical independence.

2.2.4. Sensitivity Analysis

This part aims to estimate whether the chaotic map has the property of being extremely sensitive to initial conditions. In this section, the control parameter a is varied by 2⁻¹⁴ to receive the trajectories of two chaotic sequences, as shown in Figure 8a. Similarly, by changing the initial values x₁, x₂, y₁, and y₂ by 2⁻¹⁴, the multiple trails will be engendered, see Figure 8b–e, respectively. We can observe that with a slight input deviation, the results are significantly different, marking that the modified chaotic series has unpredictable long-run evolutionary behavior and satisfactory sensitivity.

2.2.5. Bifurcation Analysis

The bifurcation diagram visually shows the change of system from a non-chaotic region to a chaotic state at a certain parameter value. According to Figure 9a, only within the range of [3.5699, 4] does the Logistic map exhibit chaotic behavior, and its periodic window appears at 3.78. However, it can be seen from Figure 9b that the modified system not only has a wider chaotic parameter domain (parameter range is [2.5, 4]) but also does not exist non-chaotic field, which implies our scheme is stable in expressing chaotic behavior.

3. A Novel Image Encryption Algorithm Based on Improved Map

In order to verify the practicability of improved sequences, a novel image encryption algorithm is proposed in this paper. Chaotic sequences are mainly applied in two aspects: (1) Combing with the magic square to realize shuffling; (2) Using with the octree for substitution.

3.1. Shuffling Algorithm

Assume that the size of image A is M × N, {x_i} is a chaotic sequence with length M × N generated by Section 2.1, and {B} is the generated magic square of order √M, all of which are used to shuffle the pixels of the image. The shuffling algorithm can be described as follows.

○

Step 1: Converting the sequence {x_i} into integer sequences by using the following equations.

$$I\left(x_i\right)=1+\left(floor\left(\left|x_i\right|\times {10}^5\right)mod N\right)$$

(7)

where floor(x_i) denotes the largest integer which is less than x_i.

○

Step 2: Transforming {I$\left(x_i\right)$} after Step 1 into a sequence matrix of the same size as {A}, that is, {$I\left(x_i\right)$} with length 1 × (M × N) is reshaped to {X}with length M × N.

$$\begin{gathered} \left\{I\left(x_i\right)\right\}=\left\{x_1,x_2,x_3,x_4,\cdots ,x_{M\times N}\right\} \\ \Downarrow \\ {X}_{ij}=\left(\begin{array}{ccccc}{X}_{11}& X_{12}& X_{13}& \cdots & X_{1N}\\ X_{21}& X_{22}& X_{23}& \cdots & X_{2N}\\ X_{31}& X_{32}& X_{33}& \cdots & X_{3N}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ X_{M1}& X_{M2}& X_{M3}& \cdots & X_{MN}\end{array}\right) \\ \left(X_{11}=x_1,X_{12}=x_2,X_{13}=x_1,\cdots ,X_{MN}=x_{M\times N}\right) \end{gathered}$$

○

Step 3: Reconstructing the magic square matrix {B} into an array {b} of 1 row and √M columns.

$$\begin{gathered} B_{ij}=\left(\begin{array}{ccccc}{B}_{11}& B_{12}& B_{13}& \cdots & B_{1\sqrt{M}}\\ B_{21}& B_{22}& B_{23}& \cdots & B_{2\sqrt{M}}\\ B_{31}& B_{32}& B_{33}& \cdots & B_{3\sqrt{M}}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ B_{\sqrt{M}1}& B_{\sqrt{M}2}& B_{\sqrt{M}3}& ⋮& B_{\sqrt{M}\sqrt{M}}\end{array}\right) \\ \Downarrow \\ \left\{b\right\}=\left\{b_1,b_2,b_3,\cdots ,b_M\right\} \left(b_1=B_{11},b_2=B_{12},b_3=B_{13},\cdots ,b_M=B_{\sqrt{M}\sqrt{M}}\right) \end{gathered}$$

○

Step 4: Row shuffling: Transforming the positions of pixels in the ith row according to {b} and {X}_ij by using the following equations.

$$temp=a_{ij}$$

(8)

$$a_{ij}=a_{ij\prime }$$

(9)

$$a_{ij\prime }=temp$$

(10)

$$j^{\prime }={\left\{X\right\}}_{ik}$$

(11)

$$j={\left\{b\right\}}_k$$

(12)

where i = 1, 2, …, M, k = 1, 2, 3, …, N, a is the original pixels. temp is used to temporarily store the pixels to be exchanged.

○

Step 5: Column shuffling: Similarly, transforming the positions of pixels in the jth column according to {b}_j and {X}_ij by using the following equations.

$$temp=a_{ij}$$

(13)

$$a_{ij}=a_{i\prime j}$$

(14)

$$a_{i\prime j}=temp$$

(15)

$$i^{\prime }={\left\{X\right\}}_{kj}$$

(16)

$$i={\left\{b\right\}}_k$$

(17)

where j = 1, 2, …, M, k = 1, 2, 3, …, N, a is the original pixels. temp is used to temporarily store the pixels to be exchanged.

The flowchart of the shuffling process can be briefly depicted in Figure 10. Factually, the order of row shuffling and column shuffling processes can be exchanged or staggered.

3.2. Substitution Algorithm

The substitution algorithm can be divided into two levels, one is bit transformation in a pixel value, and the other is bit XOR operation. The bit change operation is inspired by the octree, which means that all the internal pixels have 8 adjacent pixels, excluding the peripheral pixels of the image matrix. In this part, we use {y_i} generated by Section 2.1 and pixel position represented by the octree to carry out the bit pixels substitution, reusing {x_i} and {y_i} in the XOR operation. The detailed steps of the substitution algorithm are summarized as follows.

○

Step 1: Similar to Step 2 in Section 3.1, transforming {y_i} with length 1 × (M × N) into a matrix {Y}of size M × N.

○

Step 2: Taking the value of {Y}_ij to 8 decimal places, that is {Y}_ij,k, where k = 1, 2, …, 8, i = 2, 3, …, M − 1 and j = 2, 3, …, N − 1.

○

Step 3: Identifying the position of 8 pixels adjacent to the pixel a_ij, where a_ij comes from image A′. Note that since the definition of octree does not include peripheral pixels, so i, j ≠ 1, i ≠ M and j ≠ N. The details are as follows (

Table 2. Adjacent pixel position annotation.

a1 = a(i−1)(j−1)	a8 = a(i−1)j	a7 = a(i−1)(j+1)
a2 = ai(j−1)	aij	a6 = ai(j+1)
a3 = a(i+1)(j−1)	a4 = a(i+1)j	a5 = a(I + 1)(j+1)

).

○

Step 4: Converting the pixel a_ij into 8 bits representation a_ij,l, where l = 1, 2, …, 8 and a_ij,l = 0 or 1. The bit transformation process in pixel a_ij can be described as follows.

$$a_{ij,l}^{\prime }=a_{ij,l\prime }$$

(18)

$$a_{ij,l\prime }^{\prime }=a_{ij,l}$$

(19)

$$l={\left\{Y\right\}}_{ij,k}$$

(20)

$$l^{\prime }=1+a_{{\left\{Y\right\}}_{ij,k}}mod 8$$

(21)

Here, k = 1, 2, …, 8, i = 2, 3, …, M − 1 and j = 2, 3, …, N − 1. a′_ij denotes the pixel value of i row and j column after bit transformation. mod 8 means to convert a value to a number between [0, 7].

○

Step 5: Normalizing the chaotic matrix {Y} processed by Step 1 to (0.255), thus becoming matrix {Y₁}.

$$Y_1=\left(255\times Y\right)$$

(22)

○

Step 6: The encrypted image A′ ₁ can be obtained by performing the bit XOR operation with {Y₁}, where A′₂ comes from the image A′ that has undergone above diffusion operation.

$${A^{\prime }}_1={A^{\prime }}_2\oplus Y_1$$

(23)

○

Step 7: Transforming the chaotic sequence {x_i} generated by Equation (4) to matrix {X₁} of size M × N, and normalizing to (0.255) to {X₂}.

$$X_2=\left(255\times X_1\right)$$

(24)

○

Step 8: The final encrypted image A″ can be obtained by performing the bit XOR operation with {X₂}:

$$A^{″}={A^{\prime }}_1\oplus X_2$$

(25)

The flowchart of the substitution process can be depicted in Figure 11.

3.3. The Image Encryption Algorithm Based on Improved Chaotic Sequences

Combining the shuffling and substitution algorithms, the integrated encryption schemes can be summarized as follows.

○

Step 1: Read the plain text image A. Assume its size to be M × N.

○

Step 2: Calculate the fractional portion p of the pixel average.

$$p={\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=N}^N{a}_{ij}∕M\times N-floor \left({\displaystyle \sum }_{i=1}^M{\displaystyle \sum }_{j=N}^N{a}_{ij}∕M\times N\right)$$

(26)

○

Step 3: Select randomly parameter a and initial values x₁, x₂, y₁ and y₂ in the chaotic region.

○

Step 4: Generate two chaotic sequences {x_i}, {y_i}, according to Section 2.1 by using the main control parameter a, and four initial conditions x′₁, x′₂, y′₁ and y′₂.

$$\left\{\begin{array}{l}{x}_1^{\prime }=\left(x_1\times p\right)\times \left(mod 1\right)\\ x_2^{\prime }=\left(x_2\times p\right)\times \left(mod 1\right)\\ y_1^{\prime }=\left(y_1\times p\right)\times \left(mod 1\right)\\ y_2^{\prime }=\left(y_2\times p\right)\times \left(mod 1\right)\end{array}\right.$$

(27)

○

Step 5: Shuffle the image A according to the shuffling algorithm by using sequence {x_i} to obtain A′. Here, shuffle the rows first, and then do the same for the columns.

○

Step 6: Substitute the shuffled image A′ according to the substitution algorithm by using sequences {x_i} and {y_i}.

○

Step 7: Save as the encrypted image A″.

The flowchart of the proposed image encryption algorithm can be depicted in Figure 12.

3.4. The Image Decryption Algorithm Based on Improved Chaotic Sequences

The decryption process, as opposed to the encryption process, is described in detail here.

• Step 1: Read the encrypted image A″.
• Step 2: Obtain the key sent by the encryption party, namely, parameter a, initial values x₁, x₂, y₁ and y₂ and plain-text average pixel value p.
• Step 3: Generate two chaotic sequences {x_i}, {y_i}, according to Section 2.1 by using the main control parameter a, and four initial conditions x′₁, x′₂, y′₁ and y′₂.

  $$\left\{\begin{array}{l}{x}_1^{\prime }=\left(x_1\times p\right)\times \left(mod 1\right)\\ x_2^{\prime }=\left(x_2\times p\right)\times \left(mod 1\right)\\ y_1^{\prime }=\left(y_1\times p\right)\times \left(mod 1\right)\\ y_2^{\prime }=\left(y_2\times p\right)\times \left(mod 1\right)\end{array}\right.$$

  (28)
• Step 4: A″ performs XOR with {X₂} obtained in Step 7 of Section 3.2, and then XOR with {Y₁} obtained in Step 5, then an encrypted image A′₂ with magic square scrambling and octree diffusion is obtained.

  $${A^{\prime }}_1=A^{𠌣}\oplus X_2$$

  (29)

  $${A^{\prime }}_2={A^{\prime }}_1\oplus Y_1$$

  (30)
• Step 5: Input the image A′₂ after step 4 as the undiffused image in the octree replacement operation in Section 3.2, substitute the shuffled image A′₂ according to the octree substitution algorithm by using sequences {x_i} and {y_i}. Finally, the obtained encrypted image A′ is the scrambled image A′ after reverse diffusion.
• Step 6: Shuffle the image A′ according to the shuffling algorithm by using sequence {x_i} to obtain A which is inversely scrambled. Here, encryption is scrambled in the order of rows first and then columns. Therefore, when decrypting, the reverse scrambling shall be carried out in the order of columns first and then rows.
• Step 7: Save as the decrypted image A.

4. Experimental Tests and Security Analysis

In this section, the grayscale Baboon image, Horse image, Flower image and Cameraman image with size 256 × 256 are used as the plain-text images, and the conditions in these experimental tests are selected as a = 3.99, x₁ = 0.65477880678, x₂ = 0.56784938, y₁ = 0.3456789 and y₂ = 0.2346784.

4.1. Encryption and Decryption Experiments

This encryption and decryption were performed with four images as an example, and the results are shown in Figure 13. Figure 13e–h shows that the information of encrypted images is completely hidden, causing the images to be unrecognized. For these four different images, the features of the encrypted images are almost indistinguishable, a phenomenon that is further demonstrated by the ensuing security analysis. Therefore, this image encryption algorithm is effective and can be extended to all different classes of images. Figure 13i–l imply that the plain-text images can be decrypted accurately with the correct keys, which indicates that our decryption scheme also performs correctly.

4.2. Histogram Test

Histogram describes the distribution of the image pixels from the perspective of the internal gray level. In an ideal encrypted image, the histogram should be smooth and the pixel values evenly distributed, which can prevent the image information from being leaked. The histograms of the original images are plotted in Figure 14a–d, both showing a distinct inhomogeneous structure. While after encryption, the histogram of the encrypted image is well-distributed and relatively smooth, as shown in Figure 14e–h.

Next, we used the variance to test the homogeneity of the histogram of the encrypted images. Ideally, the histogram variance of the encrypted image is much smaller than that of the original image, and the smaller the value is, the higher the uniformity of the gray value is, which means that the proposed scheme is better. The formula for variance is expressed as:

$$Variance\left(H\right)=\frac{1}{n^2}{\displaystyle \sum }_{i=1}^n{\displaystyle \sum }_{j=1}^n\frac{{\left(h_i-h_j\right)}^2}{2}$$

(31)

where H = {h₁, h₂, h₃, …, h_n} is the array of the histogram values, and h_i and h_j are the numbers of pixels whose grey values are equal to i and j, respectively.

Table 3. Variance comparison results.

Algorithm	Images	Variance
		Original	Encrypted	Decrypted
Proposed	Baboon	57,916	255.3569	57,916
	Horse	189,049	223.5373	189,049
	Flower	1,969,923	275.9059	1,969,923
	Cameraman	124,091	286.8784	124,091
	Baboon	57,916	255.3569	57,916
Patro, K. A. K .et al. [26]	Cameraman	110,970	250.4141	110,970
Patro, K. A. K. et al. [27]	Cameraman	110,970	259.1328	110,970
Tsafack, N. et al. [28]	Baboon	753,712.8784	909.4980	-
Patro, K. A. K. et al. [29]	Cameraman	110,970	334.5391	-
Patro, K. A. K. et al. [30]	Cameraman	110,970	298.1875	110,970

shows the histogram variance results for the five test images that have gone through our scheme and gives the results in comparison with some other schemes. From Table 3, we can observe that histogram variances of the “Cameraman” image, and the ”Baboon” image encrypted by our scheme are close to the schemes of Patro, K. A. K. et al. [26], Patro, K. A. K. et al. [27], Tsafack, N. et al. [28], Patro, K .A. K and Acharya, B. [29], Patro, K. A. K and Acharya, B. [30]. From the above test results, we can prove that our method is available in resisting statistical attacks.

4.3. Chi-Square Test Analysis

The Chi-square test can further verify the homogeneity of the image grayscale values. In general, the smaller the obtained Chi-square value is, the higher the gray uniformity is. The equation of the Chi-square test can be expressed as follows:

$${\chi }_{test}^2={\displaystyle \sum }_{i=0}^{256}\frac{{\left(f_i-n{p}_i\right)}^2}{n{p}_i}$$

(32)

where f_i and np_i are the expected and observed frequencies, respectively. The expression for np_i is,

$$n{p}_i=\frac{M\times N}{256}$$

(33)

where M × N is the image size. The values of the Chi-square test results for the histograms of the images processed by our encryption scheme are shown in

Table 4. The Chi-square test results.

Algorithms	Images	χ2 Test	Testing Results
			χ22550.05 = 293.2478	χ22550.01 = 310.457
Proposed	Baboon	254.3594	Pass	Pass
	Horse	222.6641	Pass	Pass
	Flower	274.8281	Pass	Pass
	Cameraman	285.7578	Pass	Pass
Patro, K.A. K. et al. [31]	Baboon	225.3438	Pass
Yang, F. et al. [32]	Baboon (512 × 512)	265.4	Pass

. From Table 4, it can be concluded that the hypothesis is tested, both at 5% and 1% level of significance. In addition, the Chi-square value of the “Baboon” image differs very little from those of the Patro, K. and Acharva, B. [31] and Yang, F. et al. [32], which indicates that our scheme is quite competitive in terms of uniform grayscale values.

4.4. Correlation Analysis

For a satisfactory encryption scheme, there should be no correlation between the adjacent pixels of the encryption result. However, correlation generally exists in the unprocessed image, thus eliminating the correlated relation of neighboring pixels is one of the basic requirements of image encryption. The adjacent pixels of four plain-text images and encrypted images in horizontal, vertical, and diagonal directions are displayed in Figure 15, Figure 16, Figure 17 and Figure 18, which are the Baboon image, the Horse image, the Flower image, and the Cameraman image, respectively. It can be seen that (a–c) of Figure 15, Figure 16, Figure 17 and Figure 18 all indicate that pixel pairs with strong correlation exist in the plain-text image and are concentrated in the vicinity of the diagonal. While (d–f) of Figure 15, Figure 16, Figure 17 and Figure 18 show that the distribution states of adjacent pixels in the encrypted image are random in space. The above results demonstrate that the encryption scheme has achieved the effect of disturbing the relevance of neighboring pixels. Furthermore, in order to further quantify the correlation test result, the correlation coefficient Cor can be calculated by this formula:

$$Cor=\frac{Q{{\displaystyle \sum }}_{i=1}^Q\left(x_i\times y_i\right)-{{\displaystyle \sum }}_{i=1}^Q{x}_i\times {{\displaystyle \sum }}_{i=1}^Q{y}_i}{\sqrt{\left(Q{{\displaystyle \sum }}_{i=1}^Q{x}_i{}^2-{\left({{\displaystyle \sum }}_{i=1}^Q{x}_i\right)}^2\right)\times \left(Q{{\displaystyle \sum }}_{i=1}^Q{y}_i{}^2-{\left({{\displaystyle \sum }}_{i=1}^Q{y}_i\right)}^2\right)}}$$

(34)

where x_i and y_i are two pixel sequences with length Q. The value of Cor will be in the interval [0, 1]. A value of Cor close to 0 indicates that the pixel pairs in the image are not correlated. Conversely, a value of Cor close to 1 indicates that there is a strong correlation between adjacent pixels.

Table 5. Correlation coefficient analysis.

Images	Horizontal	Vertical	Diagonal
Baboon image	0.8502	0.8174	0.7646
Encrypted Baboon	0.0002	0.0038	−0.0007
Horse image	0.6425	0.6682	0.5179
Encrypted Horse	0.0027	−0.0051	−0.0030
Flower image	0.8862	0.9613	0.8772
Encrypted Flower	0.0025	0.0025	0.0013
Cameraman image	0.9333	0.9569	0.9052
Encrypted Cameraman	0.0055	−0.0065	−0.0024
Patro, K. A. K. et al. [26] Cameraman	−0.0013	0.0017	0.0070
Patro, K. A. K. et al. [27] Baboon	0.0029	0.0003	−0.0020
Patro, K. A. K. et al. [31] Baboon	0.0003	−0.0002	−0.0017

compares the Cor of various schemes, and it can be seen that the Cor of the encrypted image processed by our encryption scheme is closer to 0 in all directions, which interprets this encryption algorithm as competitive in this sense.

4.5. Key Space Analysis

Preventing attackers from using brute force attacks to decipher requires the certain space of the key, which is at least greater than 2¹²⁸. In this encryption scheme, 5 tunable parameters, including the parameters a, and initial keys x₁, x₂, y₁, y₂, are selected as the secret keys. Thus, we can approximately estimate the key space as 10^r × 10^r × 10^r × 10^r × 10^r = (10^r)⁵, and set the largest accuracy r = 14 to compare the results with other encryption algorithms. Then, the key space is approximately equal to (10¹⁴)⁵ ≈ 2²³³, which is much larger than 2¹²⁸, as shown in

Table 6. Key space.

Algorithms	Key Space
Proposed	2233
Tang, J. et al. [5]	2231
Zhang, Y. P. et al. [33]	2.6 × 1023
Li, R. Z. et al. [34]	2200
Liu, Y. et al. [35]	2186
Chen, C. et al. [36]	2135
Liao, X. et al. [37]	2194
Patro, K. A. K. et al. [38]	1.2446 × 2327

. According to the key spaces in other articles in Table 6, it can be proved that our algorithm has a larger key space. Such a result means that our scheme has a high ability to resist brute-force attacks. In addition, the nonlinear chaotic system is used to generate the input sequence of the encryption system in this algorithm. Due to the inherent nonlinearity, unpredictability and pseudo-randomness of the chaotic system, it is sufficient to resist linear attacks.

4.6. Key Sensitivity Analysis

The key sensitivity is reflected in the fact that a small change in the key, whether in encryption or decryption, will result in a considerable deviation from the correct result. Figure 19 shows the decrypted images of the above four encrypted images with only 10⁻¹⁰ differences in the secret keys a, x₁, x₂, y₁, and y₂, respectively. The results exhibit that all decrypted images are noisy and unrecognizable, indicating that the decryption process of the scheme is sensitive to the keys. Such appearances can also be described by the numerical experiment of calculating the mean square error (MSE):

$$MSE=\frac{1}{Q}{\displaystyle \sum }_{i=1}^Q{\left(u_i-z_i\right)}^2$$

(35)

where Q is the size of the image, u_i and z_i are the two images for comparison. The MSE diagrams of control parameter and initial values during encryption and decryption are shown in Figure 20 and Figure 21, respectively. As can be seen from Figure 20 and Figure 21, if the key is slightly different, the image after encryption or decryption will have a wide variation. Both results will conclude that in either case, our algorithm has excellent key sensitivity.

4.7. Information Entropy Analysis

Information entropy is instrumental in pointing out the quantity of information carried by the encrypted image, and explaining whether the image is random-like. Its formula P(X) can be described as follows:

$$P\left(X\right)=-{\displaystyle \sum }_{i=1}^{Q}h\left(x_i\right)lo{g}_{2}h\left(x_i\right)$$

(36)

where Q is the cardinal number of symbols of information source X, h(x_i) denotes the probability of symbol x_i in the information source X. If P(X) of the encrypted image is infinitely close to 8, it means that it carries less privacy information, which can be considered as a random and ideal image.

Table 7. Information entropy.

	Original Image	Encrypted Image
Baboon	7.2436	7.9972
Horse	6.5645	7.9976
Flower	5.5145	7.9970
Cameraman	7.0084	7.9968
Yang, Y. et al. [39] Cameraman	-	7.9970
Kaur, G. et al. [40] Baboon	7.7067	7.9852
Kaur, G. et al. [40] Cameraman	7.0097	7.9846

enumerates the values of P(X) under diverse algorithms, including our proposed and existing schemes. From Table 7, we observe that the information entropy of all four encrypted images manufactured by our proposed encryption algorithm is close to the ideal value 8, proving that our encryption algorithm has strong competitiveness in this sense.

4.8. Local Shannon Entropy Analysis

The local Shannon entropy reflects the distribution of randomness within each local region of the image, and its central technology is to divide the image into non-overlapping image blocks and find the mean value of their information entropy. The local entropy calculation in this test is according to the steps in Wu, Y. et al. [41], and the results are listed in

Table 8. Local entropy.

	Original Image	Encrypted Image
Baboon	6.787927	7.9020763
Flower	5.159931	7.902534
Horse	6.294101	7.901957
Cameraman	5.166450	7.901997
Kaur, G. et al. [40] Cameraman	-	7.8822
Kaur, G. et al. [40] Baboon	-	7.8945
El-Latif, A. A. A. et al. [42] Rustic	5.44114	7.90284

. From Table 8, it can be seen that the entropy values are close to those calculated in Kaur, G. et al. [40] and El-Latif, A. A. A. et al. [42], which indicates that the present scheme is acceptable in local entropy testing.

4.9. Resistance to Differential Attack Analysis

Differential attack is a general way to attack the algorithms with the correlation between encrypted images. By constantly changing the plain-text images, the attacker has a high probability of cracking the scheme. Thus, as a secure encryption algorithm, it should have the ability to be highly sensitive to plain-text. The sensitivity can be visually shown by calculating the number of pixels change rate (NPCR) and unified average changing intensity (UACI):

$$NPCR=\frac{{{\displaystyle \sum }}_{i,j}Q\left(i,j\right)}{M\times N}\times 100\%$$

(37)

$$UACI=\frac{1}{M\times N}\left({\displaystyle \sum }_{i,j}\frac{\left|D\left(i,j\right)-D^{\prime }\left(i,j\right)\right|}{255}\right)\times 100\%$$

(38)

where D and D′ are two images with size M × N, and the ideal values of NPCR and UACI are 0.9961 and 0.3346, respectively.

In this test, we randomly change the pixels by only 1 bit and then analyze the two encrypted images by using NPCR and UACI, whose results are shown in

Table 9. The values of NPCR and UACI.

Algorithms	NPCR	UACI
Baboon	0.9963	0.3344
Horse	0.9961	0.3347
Flower	0.9964	0.3365
Cameraman	0.9962	0.3344
Toktas, A. and Erkan, U. [23] Cameraman	0.9961	0.3347
Patro, K.A. K. et al. [27] Cameraman	0.9962	0.3344
Yang, Y. et al. [39] Cameraman	0.9963	0.3341
Patro, K. A. K. et al. [43] Baboon (512 × 512)	0.9961	0.3347
Sravanthi, D. et al. [44] Cameraman	0.9961	0.3342

. According to Table 9, both values of these four images are close to the ideal values, which indicates that the encryption technique is highly sensitive to plain-text images. Meanwhile, it can be seen that the UACI and NPCR values of Baboon image obtained by the method proposed in this paper are approximately similar to the values of [23,27,39,43,44]. These results manifest that the encryption algorithm is comparable to other algorithms in terms of sensitivity to plain-text images.

4.10. Computational Complexity and Speed Analysis

4.10.1. Computational Complexity Analysis

The algorithm performs three main operations on grayscale images. These operations are (i) chaotic sequence generation operation, (ii) phantom square dislocation operation, and (iii) octree diffusion operation. The computational complexity of performing these operations is as follows.

(i)

In the chaotic sequence generation operation, as shown in Section 2.1, the two Logistic chaotic maps are reshaped into a new two-dimensional chaotic system by coupling, delay, and feedback control operations, while the chaotic sequences are processed by selective swapping. Therefore, the computational complexity of the proposed method for generating a new chaotic sequence is O(M × N × B), where M × N is the size of the image, and B is the number of decimal places of the retained sequence values.

(ii)

In the magic square permutation operation, a 16-order magic square is generated and the values of this matrix are used to permute each row step by step. Subsequently, the permutation operation is performed for each column, so the computational complexity is O(M × N) + O(M × N) = 2 × O(M × N), M is the number of pixels swapped in each row and N is the number of pixels swapped in each column.

(iii)

In the octree diffusion operation, the pixel points at the boundary are removed and the other pixel values are swapped on bit bits according to the diffusion rules in Section 3.2. The computational complexity is O(8 × (M − 2) × (N − 2)), where M − 2 is the number of pixels swapped per row and N − 2 is the number of pixels swapped per column.

In conclusion, the overall computational complexity of the proposed method in this paper is O(M × N), and is similar to that of other image encryption algorithms [31,45].

4.10.2. Speed Analysis

Speed is also an important measure to evaluate the practicability of an encryption algorithm. In this test, the encryption and decryption algorithms are processed by MATLAB 2019 on a computer with 2.80 GHz CPU and 8 GB memory. The encryption speed of different algorithms is shown in

Table 10. Speed tests.

Algorithms	Images	Unit (s)	Speed (Mb/s)
Proposed	Baboon	0.6087	0.8214
	Horse	0.6033	0.8287
	Flower	0.5957	0.8393
	Cameraman	0.6021	0.8304
Khan, J. S. and Ahmad, J. [46]	Flower	0.55	-
Ayoup, A. M. et al. [47]	Flower	7.21	-

. These results show that our algorithm is more efficient and acceptable for practical applications.

4.11. Robustness Analysis

The encryption algorithm with better robustness will maintain the ability to recover the image if the cipher image is ruined by noise or data loss. In this test, using Salt and Pepper noise and slicing attack with different intensities to interfere with encrypted images, and then acquire decrypted images, see Figure 22. From Figure 22, we can know that regardless of what damage the encrypted images suffer, such as data loss of 5% and 10% or noise attack of 1% and 3%, the decrypted images can basically restore the “Baboon” image. These prove that the image encryption algorithm proposed in this paper possesses the property of robustness to defend against noise attack and data loss.

5. Conclusions

To enhance the security of chaos-based image encryption algorithms, this paper proposes a method realized on parameter disturbance and nonlinear feedback to improve the original Logistic map. The delay and the coupling states are combined to perturb the parameters, while the nonlinear feedback adopts the sine function. Then, using selective switching to manage the chaotic sequences generated by the improved map, which renders the chaotic sequence not only subject to the chaotic system but also controlled by the selective exchange operation. In this paper, we take 1D Logistic as an example to carry out simulation experiments, and the results show that the chaotic sequence generated by this method shows significant improvement in terms of dynamic characteristics. Furthermore, a novel image encryption algorithm is realized, which combines chaotic sequence with magic square and octree to achieve secure encryption results. Following this, several experiments, including histogram analysis, correlation analysis, key sensitivity analysis, key space analysis, robustness analysis, resistance to differential attack, and information entropy analysis, are employed to prove that the encryption algorithm has a high-level of security, and is quite competitive to other chaos-based encryption algorithms.

The key contribution of this paper is to provide a coupling two-dimensional map of the double disturbance control parameters with improved dynamics, which uses a new selective exchange approach to enhance the complexity of the sequences. Meanwhile, a novel encryption scheme with high security is proposed. Curiously, this paper extends the concept of octree to the pixel-bit level of an image, presenting a new idea for diffusion operations. Considering the practicality of chaotic systems, the proposed chaotic system is only computer-simulated in this paper, and can be easily realized with real circuits since the chaotic model consisted of some simple operations. In our future work, the chaotic system will be realized with real circuits. Furthermore, we will further improve the security level of the chaos-based image encryption algorithms.