# A Hidden Chaotic Attractor with an Independent Amplitude-Frequency Controller

Authors: Yousuf Islam; Chunbiao Li; Yicheng Jiang; Xu Ma; Akif Akgul
Journal: Complexity
 (2022)

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## Abstract

In this paper, a three-dimensional chaotic system with a line equilibrium is studied, in which a single nonbifurcation parameter is used to control the amplitude and frequency. A variety of chaotic signals can be modified using the amplitude-frequency control switch. The realization of circuit simulation based on multisim further verifies the theoretical analysis. Finally, the method for encrypting color images is tested, and the process performance is valued. It shows that the novel chaotic oscillation has a promising application prospect in image encryption.


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## Body

## 1. Introduction

Since chaos was first modeled by Edward Norton Lorenz in 1963 [1], great attention has been attracted for the reason that in such deterministic systems, chaos refers to the presence of seemingly random irregular motion [2]. In fact, chaos is a universal phenomenon in nonlinear systems and an inherent property of nonlinear dynamic systems. Chaos is a fundamental concept in nonlinear systems, and it is frequently used to characterize phenomena including bifurcation and periodic motion [3, 4]. It is found that a chaotic system will display bifurcation under specific parameters and that periodic and aperiodic motion can become entangled. Many famous 3-D chaotic systems have been proposed, including Arneodo systems, Sprott systems, Chen systems, Lv-Chen systems, Cai systems, and T systems [5–8]. A chaotic circuit is a significant research topic and an important representation of chaos. Chua proposed the first chaotic circuit in 1984 [9–12], trying to bridge the gap between chaos theory and chaotic circuits. People have performed significant research on chaos theory and produced numerous novel breakthroughs in recent years as a result of the wide application of chaos [13–17]. For information encryption, chaotic sequence signals have a significant application value. Chaos has been extensively applied for many aspects due to the intricate relationship between chaos and cryptography, and significant advancement has been performed in this area. Because of the irregularities and unpredictability of chaotic signals, chaos and corresponding fundamental systems have received a lot of attention [18–24]. A high-dimensional system’s chaotic dynamics seem to be more complex, and sometimes, they show hyperchaos for more than two positive Lyapunov exponents [25, 26]. Therefore, it has been proven that chaotic signals can contribute to enhancing the security of chaos-based communication as well as digital encryption. Furthermore, a hidden chaotic attractor is an important phenomenon found ten years ago. The hidden oscillation has received a lot of attention because of its potential threat and possible applications. Finding hidden chaotic attractors in nonlinear dynamical systems has become a major issue in nonlinear dynamical system research [27–29]. Many hidden attractors have been found in memristive systems [30] and hyperchaotic systems [31]. However, there is not much attention on hidden attractors with amplitude-frequency control. Recently, Wang et al. [32] studied the amplitude control and encryption application of chaotic signal, and they found that modular circuit cells with systematically configured parameters are useful for implementing multipiecewise Chua’s diode. Wang et al. [33, 34] also studied the hidden oscillations in Chua’s circuit and modified Sprott-A systems, where all the basins of attraction are not intersected with any equilibrium point indicating hidden attractor.In this paper, a chaotic system with a line of equilibrium points is focused on, where the attractor stands in the region with negativey, and thus, the basin of attraction does not intersect with all equilibrium points indicating hidden attractor. Furthermore, the amplitude and frequency of hidden oscillation can be controlled by a single knob. Circuit implementation shows the convenience of amplitude-frequency control of the chaotic signal. Image encryption proves the merits of this system. The remaining of this paper is organized as follows. In Section 2, the system model is elaborated. In Section 3, the dynamics of the system are analyzed. In Section 4, the system is implemented in a simulated circuit. Finally, a chaotic system is applied to image encryption. The conclusions are presented in the last section.


## 2. A Novel Hidden Attractor

In reference [35], the 3-dimensional chaotic system is articulated as(1)x˙=cy+y2−ayz,y˙=−z2+byz,z˙=xy.The system state variables arex, y, and z, while the constants a, b, and c are the real coefficients, with dots representing a derivative of time t. When a = 0.9, b = 1, c = 1, and IC = (2, −2, 2), the system (1) exhibits chaos in the region with negative y, as shown in Figure 1. There are six terms in this new system, with two nonlinear items. The Lyapunov exponents are L1= 0.1314,L2=0, and L3= −0.8453.Figure 1
Chaotic attractor of system (1) witha = 0.9, b = 1, and c = 1 under initial condition IC = (2, − 2, 2): (a) (x)-(y), (b) (x)-(z), (c) (y)-(z), and (d) (x)-(y)-(z).
(a)(b)(c)(d)


## 3. Dynamical Analysis

### 3.1. Equilibrium Points

Letx˙=y˙=z˙=0. The equilibrium points of the system can be calculated as(2)cy+y2−ayz=0,−z2+byz=0,xy=0,(3)EO=x,0,0,E1=0,−c,0,E2=0,cab−1,bcab−1,where a = 0.9, b = 1, and c = 1. Solving this equation, the dynamical system (1) has three nontrivial equilibrium points in (2) which is independent of the value of the parameters a, b, and c.(4)J=0c+2y−az−ay0bzby−2zyx0.AsJ−λI=0, the characteristic equation is(5)detJ−λI=−aλy2+2ayz2+bcy2+bλ2z+bλxy+2by3−2cyz−λ3−2λxz−4y2z=0.In the Jacobian matrix of system (1), the equilibrium pointE is defined as(6)JEO=JE1=JE2=0c+2y−az−ay0bzby−2zyx0.The eigenvalues of the matrix at the equilibrium point can be determined as follows:(7)EO:λ1=0;λ2=0;λ3=0,E1:λ1=−0.7113;λ2=0.3556+1.1311i;λ3=0.3556−1.1311i,E2:λ1=0.9911;λ2=−5.4956+8.4079i,λ3=−5.4956−8.4079i.The eigenvalues of the system are obtained whilea = 0.9, b = 1, and c = 1. The obtained all eigenvalues are given in Equation (7). The equilibrium E10,−c,0 is a saddle-focus point of index-2; therefore, this equilibrium point E1 is unstable, and we can see in E2 that λ1 is a positive real number, and λ2 and λ3 are a pair of complex conjugate eigenvalues with a negative real number. The equilibrium E20,c/ab−1,bc/ab−1 is a saddle-focus point of index-1; therefore, this equilibrium point E2 is also unstable.


### 3.2. Dissipativity Analysis

Inferenceµ is a region in the horizontal surface A3, and Vt is set to be the volume of µ(t).Here, we obtain(8)V˙t=∫μt∞∇×Fdxdydz.Therefore, the dissipativity of the proposed chaotic system is(9)∇×F=∂x˙x+∂y˙y+∂z˙z,=∂cy+y2−ayzx+∂−z2+byzy+∂xyz=0+bz+0=z=γ,where F is the 3-dimensional chaotic system, and b and z are the real parameters. The above equation is rewritten as(10)V˙t=∫μt∞γdxdydz=γVt.Therefore, we can obtainVt = eγV0; if ∇×F<0, then system (1) is dissipative and the state of the system is bounded by the state of the system (when a = 0.9, b = 1, and c = 1).


### 3.3. Amplitude and Frequent Control

In system (1), parameterc is a single knob for amplitude and frequency control. Let x⟶mx, y⟶my, z⟶mz, t⟶t/m (m>0); then, system (1) turns to be(11)x˙=cmy+y2−ayz,y˙=−z2+byz,z˙=xy,indicating that the parameterc can control the amplitude and frequency of all variables x, y, and z, as shown in Figure 2. In Figure 3(a), when the linear coefficient c is increased, system (1) keeps the chaotic state of all the time and the chaotic area continue to increase. The rescaled amplitude and frequency can also be proved by Lyapunov exponents, as shown in Figure 3(b).Figure 2
Rescaled attractor of system (1) witha = 0.9, b = 1, and initial values (2, −2, 2) under the parameter c: (a) (x)-(y) and (b) signal x (t).
(a)(b)Figure 3
Dynamical evolution of system (1) witha = 0.9, b = 1, and initial values (2, − 2, 2) when c is in [1, 5]: (a) bifurcation diagram and (b) Lyapunov exponents.
(a)(b)


## 3.1. Equilibrium Points

Letx˙=y˙=z˙=0. The equilibrium points of the system can be calculated as(2)cy+y2−ayz=0,−z2+byz=0,xy=0,(3)EO=x,0,0,E1=0,−c,0,E2=0,cab−1,bcab−1,where a = 0.9, b = 1, and c = 1. Solving this equation, the dynamical system (1) has three nontrivial equilibrium points in (2) which is independent of the value of the parameters a, b, and c.(4)J=0c+2y−az−ay0bzby−2zyx0.AsJ−λI=0, the characteristic equation is(5)detJ−λI=−aλy2+2ayz2+bcy2+bλ2z+bλxy+2by3−2cyz−λ3−2λxz−4y2z=0.In the Jacobian matrix of system (1), the equilibrium pointE is defined as(6)JEO=JE1=JE2=0c+2y−az−ay0bzby−2zyx0.The eigenvalues of the matrix at the equilibrium point can be determined as follows:(7)EO:λ1=0;λ2=0;λ3=0,E1:λ1=−0.7113;λ2=0.3556+1.1311i;λ3=0.3556−1.1311i,E2:λ1=0.9911;λ2=−5.4956+8.4079i,λ3=−5.4956−8.4079i.The eigenvalues of the system are obtained whilea = 0.9, b = 1, and c = 1. The obtained all eigenvalues are given in Equation (7). The equilibrium E10,−c,0 is a saddle-focus point of index-2; therefore, this equilibrium point E1 is unstable, and we can see in E2 that λ1 is a positive real number, and λ2 and λ3 are a pair of complex conjugate eigenvalues with a negative real number. The equilibrium E20,c/ab−1,bc/ab−1 is a saddle-focus point of index-1; therefore, this equilibrium point E2 is also unstable.


## 3.2. Dissipativity Analysis

Inferenceµ is a region in the horizontal surface A3, and Vt is set to be the volume of µ(t).Here, we obtain(8)V˙t=∫μt∞∇×Fdxdydz.Therefore, the dissipativity of the proposed chaotic system is(9)∇×F=∂x˙x+∂y˙y+∂z˙z,=∂cy+y2−ayzx+∂−z2+byzy+∂xyz=0+bz+0=z=γ,where F is the 3-dimensional chaotic system, and b and z are the real parameters. The above equation is rewritten as(10)V˙t=∫μt∞γdxdydz=γVt.Therefore, we can obtainVt = eγV0; if ∇×F<0, then system (1) is dissipative and the state of the system is bounded by the state of the system (when a = 0.9, b = 1, and c = 1).


## 3.3. Amplitude and Frequent Control

In system (1), parameterc is a single knob for amplitude and frequency control. Let x⟶mx, y⟶my, z⟶mz, t⟶t/m (m>0); then, system (1) turns to be(11)x˙=cmy+y2−ayz,y˙=−z2+byz,z˙=xy,indicating that the parameterc can control the amplitude and frequency of all variables x, y, and z, as shown in Figure 2. In Figure 3(a), when the linear coefficient c is increased, system (1) keeps the chaotic state of all the time and the chaotic area continue to increase. The rescaled amplitude and frequency can also be proved by Lyapunov exponents, as shown in Figure 3(b).Figure 2
Rescaled attractor of system (1) witha = 0.9, b = 1, and initial values (2, −2, 2) under the parameter c: (a) (x)-(y) and (b) signal x (t).
(a)(b)Figure 3
Dynamical evolution of system (1) witha = 0.9, b = 1, and initial values (2, − 2, 2) when c is in [1, 5]: (a) bifurcation diagram and (b) Lyapunov exponents.
(a)(b)


## 4. Circuit Implementation

Circuit verification is also an essential step in the implementation of the proposed chaotic system to ensure its correctness. Meanwhile, the key issue in the circuit is how to implement a circuit expression for a 3D chaotic system by converting and transforming it into a realistic circuit using electronic devices such as capacitors and resistors. To authenticate the efficiency of our proposed 3-dimensional chaotic system, the circuit implementation is designed and simulated with the NI multisim circuit simulation software, and the simulation results are detailed in this section.For this circuit construction, it is transformed to be(12)x˙=1R1C1y+1R2C1y2−1R3C1yz,y˙=−1R4C2z2+1R5C2yz,z˙=1R8C3xy.A chaotic system is defined as a system that has several causes and multiplication, addition, and differentiation, and differentiation exists in the system equations, and a realistic expression of this system is obtained utilizing a summation, an integrator, and a transformer. The relevant circuit implementation is depicted in Figure4 based on the above explanations. The state variables x, y, and z in the system (1) correspond to the state voltages of the capacitors C1, C1, and C3 in the simplified circuit, and the corresponding circuit components can be selected as follows: V1=V2=15V, C1=C2=C3=10nF, R1=R2=R4=R5=R6=R7=R8=100kΩ, R3=106kΩ; LM741CN is selected as an operational amplifier; there is a general time scale 1000 for better displaying in the oscilloscope.Figure 4
Circuit realization of the system (1).Figure5 shows the phase trajectory of the system (1) in the analog oscilloscope. The area of the phase track will change with the organization of the resistance R1, as shown in Figure 5(d).Figure 5
Chaotic attractor of system (1): (a) (x)-(y), (b) (x)-(z), (c) (y)-(z), and (d) the attractor changes due to the difference of R1 resistance.


## 5. Application in Image Encryption

The chaotic system with a linear equilibrium in this section has a higher level of unpredictability, a larger keyspace, and a higher level of complexity, all of which make the encryption stronger secure in concept.


### 5.1. Encryption Application with a Chaotic System

The control parameters and initial conditions of the system are (Table1) μ = 3.9999 and x0 = 0.6209, respectively, and a standard color image is chosen for testing, as illustrated in Figure 6. The new chaotic system parameters are a = 0.9, b = 1, and c = 1. The initial conditions are set as X0=0.7361,Y0=0.4663,Z0=0.1501, and U0  = 0.7653. Table 2 describes the selected keys that were chosen. M1 and N1 are the zeroing parameters used throughout encryption; k1 is the average gray level of G in the original image, and the original image k2 is the average gray level of B. Figure 6(b) shows the encrypted image after simulation. The image after encryption is chaotic and fundamentally different from the image before processing, as it can be observed. The properly decrypted image, which can be seen in Figure 6(c), is identical to the original image (Table 2).Table 1
Algorithm key.
Keyμx0X0Y0Z0U0M1N1k1k2Value3.99990.62090.73610.46630.15010.7653000.47610.1255.Figure 6
Encryption images: (a) original pepper image; (b) encryption pepper image; (c) decryption pepper image.Table 2
The information entropy of the three channels in the original and encrypted on the pepper image.
ImageR channel entropyG channel entropyB channel entropyOriginal picture5.68717.68564.686Encrypted image7.99977.99977.9998


### 5.2. Security Analysis

Secure analysis is the most essential and fundamental requirement for an encrypted system. In general, a chaotic image encryption algorithm requires a substantial keyspace, reversible encryption and decryption, strong antiattack characteristics, and the other performances, and then the performance of chaos-based encryption is the next to determine by using the following eligibility requirements: keyspace analysis, histogram analysis, information entropy analysis, and correlation analysis.


### 5.3. Keyspace Analysis

The keyspace can approach101610 = 10160, satisfying the security level of the keyspace (greater than 2128) using a 64-bit processor, floating-point precision up to 10−16. The modified system can be observed to be more resistant to the attacker’s comprehensive attack. The decrypted image cannot be acquired accurately when the algorithm key is changed slightly, as shown in Figure 7. As a result, the new system has a high level of key sensitivity. The images used throughout the investigations are 256 × 256 conventional images with a 23 = 8-bit grayscale.Figure 7
Initial condition perturbation ciphertext.


### 5.4. Histogram Analysis

The distribution of pixel intensity values within an image is represented by a histogram. It can be utilized to defend against statistical attacks. The frequency distribution of each gray level pixel can be visually displayed using a gray histogram, which is a statistical analytical method. The histogram of the image becomes smooth even after encryption, compared to the fluctuation before encryption, thus preventing the attacker from accessing the original image information through statistical analysis, resulting in information leakage, and ensuring information. Figures8(a)–8(c) are the histograms of the original image, and Figures 8(d)–8(f) are the histograms of the encrypted images. It is clear that the new system can withstand a more powerful onslaught.Figure 8
Histogram experiment images. (a–c) Histogram of the original image. (d–f) Histogram of the encrypted image.
(a)(b)(c)(d)(e)(f)


### 5.5. Information Analysis for Entropy

The image’s information entropy can be used to evaluate the level of uncertainty and randomness in the image distribution of its pixel gray value. Typically, the higher the image’s entropy, the more consistent the image’s gray distribution. In a grayscale image, each pixel is coded in 8 bits. As a result, an image’s maximum entropy value is 8. The information entropy of the encrypted image should be near 8, which indicates the best amount of uncertainty, owing to a decent encryption process. To compute the information entropy, many users use the method as follows:(13)entropyHm=−∑i=1Hpxilog2pxi,where pxi represents the probability of the occurrence of the gray value xi and H indicates the gray level of the image. Theoretically, for a completely random digital image with a grayscale of 256 has an evenly distributed pixel value in [0, 255], then pxi = 1/256 (i ∈0,255), and the estimated information entropy is 8 bits. If the image is encrypted, the closer the image’s information entropy is near 8, the better the encryption features.


### 5.6. Correlation Statistical Analysis

On the one hand, the correlation level of adjacent pixels is larger when the correlation coefficient degree of adjacent pixels is higher. On the other hand, the lower the correlation, the smaller the coefficient. As a result, calculating the correlation coefficient can be justified the algorithm’s security. The lower the correlation and the advanced security, the smaller the coefficient. The correlation coefficients were calculated from the three channels (R, G, and B) with three directions: horizontal, vertical, and diagonal, to measure the correlation between the original image and adjacent pixels of the ciphertext image;N pairs of adjacent pixels were selected from the image, and the correlation coefficients were calculated from the three channels (R, G, B) with three directions: horizontal, vertical, and diagonal. To equivalence the autocorrelation of an unadorned and encrypted image, we have calculated the correlation coefficient r of each pair of pixels by using the following formula:(14)rXY=covX,YDXDY,EX=1N∑i=1Nxi,DX=1N∑i=1Nxi−EX,covX,Y=1N∑i=1Nxi−EXyi−EY,where covX,Y represents the correlation and autocorrelation function, X and Y are the grayscale values of two adjacent pixels in the image, and N denotes the sample. E is the expected value operator, and DX represents the variance of the variable x. The values of rXY lie in the range [−1, 1], with 1 indicating perfect correlation, −1 indicating anticorrelation, and 0 representing no correlation.Here, we can see from Table3 that the correlation coefficients of the original image are all very close to 1, whereas the correlation coefficients of the encrypted image are all very close to 0, indicating that the encrypted image’s pixel point distribution is very highly discrete.Table 3
A correlation coefficient of two adjacent pixels in the original and encrypted on the pepper image.
ImageChannelHorizontalVerticalDiagonalOriginal imageB0.986450.987820.97851G0.995290.992820.98965R0.994280.991680.98792Encrypted imageB0.00494−0.025310.00540G−0.01262−0.016060.00444R−0.00282−0.016560.00919If we compare Tables3 and 4 with each other, we can see some different points. In Table 3, the original correlation coefficient values are average horizontal 99.2%, vertical 99.07%, and diagonal 98.54%, and in Table 4, the original correlation coefficient values are average horizontal 96.72%, vertical 98.17%, and diagonal 95.18%, that is, the original points in Table 3 are closer to 1 and the encrypted points are also closer to 0 than Table 4. So, Table 3 indicates that the encrypted points of distribution are highly discrete.Table 4
Correlation coefficient test result [36].
ImageChannelHorizontalVerticalDiagonalOriginal imageRed0.974890.986600.96227Green0.975320.987310.96377Blue0.951670.971120.92931Encrypted imageRed0.000700.011750.01539Green0.00855−0.01537−0.01660Blue0.001220.001350.01235The correlation and autocorrelation plots for each image are shown in Figures9–11. In Table 3, horizontal, vertical, and diagonal directions provide their correlation and autocorrelation coefficients. Table 3 also includes significant data from references for comparison.Figure 9
Correlation and autocorrelation of R channel adjacent pixels of the pepper image and its ciphered image: (a) horizontal direction of the pepper image; (b) vertical direction of the pepper image; (c) diagonal direction of the pepper image; (d) horizontal direction of the pepper ciphered image; (e) vertical direction of the pepper ciphered image; (f) diagonal direction of the pepper ciphered image.Figure 10
Correlation and autocorrelation of G channel adjacent pixels of the pepper image and its ciphered image: (a) horizontal direction of the pepper image; (b) vertical direction of the pepper image; (c) diagonal direction of the pepper image; (d) horizontal direction of the pepper ciphered image; (e) vertical direction of the pepper ciphered image; (f) diagonal direction of the pepper ciphered image.Figure 11
Correlation and autocorrelation of B channel adjacent pixels of the pepper image and its ciphered image: (a) horizontal direction of the pepper image; (b) vertical direction of the pepper image; (c) diagonal direction of the pepper image; (d) horizontal direction of the pepper ciphered image; (e) vertical direction of the pepper ciphered image; (f) diagonal direction of the pepper ciphered image.


## 5.1. Encryption Application with a Chaotic System

The control parameters and initial conditions of the system are (Table1) μ = 3.9999 and x0 = 0.6209, respectively, and a standard color image is chosen for testing, as illustrated in Figure 6. The new chaotic system parameters are a = 0.9, b = 1, and c = 1. The initial conditions are set as X0=0.7361,Y0=0.4663,Z0=0.1501, and U0  = 0.7653. Table 2 describes the selected keys that were chosen. M1 and N1 are the zeroing parameters used throughout encryption; k1 is the average gray level of G in the original image, and the original image k2 is the average gray level of B. Figure 6(b) shows the encrypted image after simulation. The image after encryption is chaotic and fundamentally different from the image before processing, as it can be observed. The properly decrypted image, which can be seen in Figure 6(c), is identical to the original image (Table 2).Table 1
Algorithm key.
Keyμx0X0Y0Z0U0M1N1k1k2Value3.99990.62090.73610.46630.15010.7653000.47610.1255.Figure 6
Encryption images: (a) original pepper image; (b) encryption pepper image; (c) decryption pepper image.Table 2
The information entropy of the three channels in the original and encrypted on the pepper image.
ImageR channel entropyG channel entropyB channel entropyOriginal picture5.68717.68564.686Encrypted image7.99977.99977.9998


## 5.2. Security Analysis

Secure analysis is the most essential and fundamental requirement for an encrypted system. In general, a chaotic image encryption algorithm requires a substantial keyspace, reversible encryption and decryption, strong antiattack characteristics, and the other performances, and then the performance of chaos-based encryption is the next to determine by using the following eligibility requirements: keyspace analysis, histogram analysis, information entropy analysis, and correlation analysis.


## 5.3. Keyspace Analysis

The keyspace can approach101610 = 10160, satisfying the security level of the keyspace (greater than 2128) using a 64-bit processor, floating-point precision up to 10−16. The modified system can be observed to be more resistant to the attacker’s comprehensive attack. The decrypted image cannot be acquired accurately when the algorithm key is changed slightly, as shown in Figure 7. As a result, the new system has a high level of key sensitivity. The images used throughout the investigations are 256 × 256 conventional images with a 23 = 8-bit grayscale.Figure 7
Initial condition perturbation ciphertext.


## 5.4. Histogram Analysis

The distribution of pixel intensity values within an image is represented by a histogram. It can be utilized to defend against statistical attacks. The frequency distribution of each gray level pixel can be visually displayed using a gray histogram, which is a statistical analytical method. The histogram of the image becomes smooth even after encryption, compared to the fluctuation before encryption, thus preventing the attacker from accessing the original image information through statistical analysis, resulting in information leakage, and ensuring information. Figures8(a)–8(c) are the histograms of the original image, and Figures 8(d)–8(f) are the histograms of the encrypted images. It is clear that the new system can withstand a more powerful onslaught.Figure 8
Histogram experiment images. (a–c) Histogram of the original image. (d–f) Histogram of the encrypted image.
(a)(b)(c)(d)(e)(f)


## 5.5. Information Analysis for Entropy

The image’s information entropy can be used to evaluate the level of uncertainty and randomness in the image distribution of its pixel gray value. Typically, the higher the image’s entropy, the more consistent the image’s gray distribution. In a grayscale image, each pixel is coded in 8 bits. As a result, an image’s maximum entropy value is 8. The information entropy of the encrypted image should be near 8, which indicates the best amount of uncertainty, owing to a decent encryption process. To compute the information entropy, many users use the method as follows:(13)entropyHm=−∑i=1Hpxilog2pxi,where pxi represents the probability of the occurrence of the gray value xi and H indicates the gray level of the image. Theoretically, for a completely random digital image with a grayscale of 256 has an evenly distributed pixel value in [0, 255], then pxi = 1/256 (i ∈0,255), and the estimated information entropy is 8 bits. If the image is encrypted, the closer the image’s information entropy is near 8, the better the encryption features.


## 5.6. Correlation Statistical Analysis

On the one hand, the correlation level of adjacent pixels is larger when the correlation coefficient degree of adjacent pixels is higher. On the other hand, the lower the correlation, the smaller the coefficient. As a result, calculating the correlation coefficient can be justified the algorithm’s security. The lower the correlation and the advanced security, the smaller the coefficient. The correlation coefficients were calculated from the three channels (R, G, and B) with three directions: horizontal, vertical, and diagonal, to measure the correlation between the original image and adjacent pixels of the ciphertext image;N pairs of adjacent pixels were selected from the image, and the correlation coefficients were calculated from the three channels (R, G, B) with three directions: horizontal, vertical, and diagonal. To equivalence the autocorrelation of an unadorned and encrypted image, we have calculated the correlation coefficient r of each pair of pixels by using the following formula:(14)rXY=covX,YDXDY,EX=1N∑i=1Nxi,DX=1N∑i=1Nxi−EX,covX,Y=1N∑i=1Nxi−EXyi−EY,where covX,Y represents the correlation and autocorrelation function, X and Y are the grayscale values of two adjacent pixels in the image, and N denotes the sample. E is the expected value operator, and DX represents the variance of the variable x. The values of rXY lie in the range [−1, 1], with 1 indicating perfect correlation, −1 indicating anticorrelation, and 0 representing no correlation.Here, we can see from Table3 that the correlation coefficients of the original image are all very close to 1, whereas the correlation coefficients of the encrypted image are all very close to 0, indicating that the encrypted image’s pixel point distribution is very highly discrete.Table 3
A correlation coefficient of two adjacent pixels in the original and encrypted on the pepper image.
ImageChannelHorizontalVerticalDiagonalOriginal imageB0.986450.987820.97851G0.995290.992820.98965R0.994280.991680.98792Encrypted imageB0.00494−0.025310.00540G−0.01262−0.016060.00444R−0.00282−0.016560.00919If we compare Tables3 and 4 with each other, we can see some different points. In Table 3, the original correlation coefficient values are average horizontal 99.2%, vertical 99.07%, and diagonal 98.54%, and in Table 4, the original correlation coefficient values are average horizontal 96.72%, vertical 98.17%, and diagonal 95.18%, that is, the original points in Table 3 are closer to 1 and the encrypted points are also closer to 0 than Table 4. So, Table 3 indicates that the encrypted points of distribution are highly discrete.Table 4
Correlation coefficient test result [36].
ImageChannelHorizontalVerticalDiagonalOriginal imageRed0.974890.986600.96227Green0.975320.987310.96377Blue0.951670.971120.92931Encrypted imageRed0.000700.011750.01539Green0.00855−0.01537−0.01660Blue0.001220.001350.01235The correlation and autocorrelation plots for each image are shown in Figures9–11. In Table 3, horizontal, vertical, and diagonal directions provide their correlation and autocorrelation coefficients. Table 3 also includes significant data from references for comparison.Figure 9
Correlation and autocorrelation of R channel adjacent pixels of the pepper image and its ciphered image: (a) horizontal direction of the pepper image; (b) vertical direction of the pepper image; (c) diagonal direction of the pepper image; (d) horizontal direction of the pepper ciphered image; (e) vertical direction of the pepper ciphered image; (f) diagonal direction of the pepper ciphered image.Figure 10
Correlation and autocorrelation of G channel adjacent pixels of the pepper image and its ciphered image: (a) horizontal direction of the pepper image; (b) vertical direction of the pepper image; (c) diagonal direction of the pepper image; (d) horizontal direction of the pepper ciphered image; (e) vertical direction of the pepper ciphered image; (f) diagonal direction of the pepper ciphered image.Figure 11
Correlation and autocorrelation of B channel adjacent pixels of the pepper image and its ciphered image: (a) horizontal direction of the pepper image; (b) vertical direction of the pepper image; (c) diagonal direction of the pepper image; (d) horizontal direction of the pepper ciphered image; (e) vertical direction of the pepper ciphered image; (f) diagonal direction of the pepper ciphered image.


## 6. Conclusion

The single nonbifurcation parameter of the system can effectively modify the amplitude and frequency of the demonstrated three-dimensional chaotic system. Numerical simulation and circuit experiment based on multisim agree to each other by proving the phenomenon. As a typical application, the property of image encryption is exhaustively analyzed. With the chaotic signal from the new system, a color image is well encrypted and decrypted in the keyspace. Histogram and correlation of adjacent pixels are used for showing the high encryption performance.



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*Source: 3086747-2022-03-07.xml*