A Novel Five-Dimensional Three-Leaf Chaotic Attractor and Its Application in Image Encryption

This paper presents a novel five-dimensional three-leaf chaotic attractor and its application in image encryption. First, a new five-dimensional three-leaf chaotic system is proposed. Some basic dynamics of the chaotic system were analyzed theoretically and numerically, such as the equilibrium point, dissipative, bifurcation diagram, plane phase diagram, and three-dimensional phase diagram. Simultaneously, an analog circuit was designed to implement the chaotic attractor. The circuit simulation experiment results were consistent with the numerical simulation experiment results. Second, a convolution kernel was used to process the five chaotic sequences, respectively, and the plaintext image matrix was divided according to the row and column proportions. Lastly, each of the divided plaintext images was scrambled with five chaotic sequences that were convolved to obtain the final encrypted image. The theoretical analysis and simulation results demonstrated that the key space of the algorithm was larger than 10150 that had strong key sensitivity. It effectively resisted the attacks of statistical analysis and gray value analysis, and had a good encryption effect on the encryption of digital images.


Introduction
In recent years, chaotic and hyperchaotic systems that can produce various types and are suitable for secure communication have become topics of great interest in the fields of physics, biomathematics, and information security [1]. Compared with the traditional encryption method, the complex structure and dynamic behavior of chaotic attractors have a better encryption effect for digital image encryption [2]. Therefore, it has become more important to construct chaotic attractors with multiple scrolls. In the process of encrypting digital images, the core purpose is to change the position of pixels and the size of pixel values. Therefore, experts and scholars have proposed many encryption algorithms such as using chaotic sequences to perform bit disturb of images [3,4], using the chaotic sequence and image pixel value for the XOR operation [5,6], and scrambling the pixel [7][8][9]. A nonlinear state feedback controller was proposed in Reference [10] based on the original three-dimensional (3D) autonomous chaotic system to construct a new four-dimensional hyperchaotic system. An image encryption algorithm based on five-dimensional hyper-chaos and bit-level disturbance was proposed in Reference [11]. Chaotic image encryption algorithms based on bit-level scrambling and dynamic DNA coding were proposed in Reference [12]. Liu et al. [13] proposed an image encryption algorithm for bit position chaos on the upper four bits. On the basis of arranging the diffusion structure, to improve safety and sensitivity, a chaotic image encryption algorithm based on a breadth-first search and dynamic diffusion was proposed in Reference [14]. The cryptosystem in Reference [15] uses a diffusion layer and then positions the image in layers instead of byte permutations to disturb the position of the image pixels. A new chaotic system with hidden attractors and a chaotic-based image encryption algorithm with a random number generator was proposed in Reference [16]. To reduce the processing time, Enayatifar et al. [17] performed simultaneous replacement and diffusion steps for any pixel.
Many chaotic-based image encryption algorithms have been inspired by Fridrich's method. Therefore, this architecture has become most well-known [18]. However, security of an efficient image encryption method is a fundamental issue in an image encryption algorithm. Recent cryptanalytical studies have proven that some chaos-based algorithms are not adequately secure to resist against a common attack such as Reference [19].
In the proposed scheme, the new five-dimensional chaotic system can generate three-leaf chaotic attractors in multiple directions. Simultaneously, its dynamic characteristics are analyzed. In the process of encryption, to increase the key space, we perform convolution operations on five chaotic sequences. Second, we scale the image matrix proportionally and scramble the image matrix for each segment separately. Through these processes, the difficulty of the exhaustive attack is increased. All except for the exhaustive method based on the explicit plaintext ciphertext mapping, the method will be invalid and have high security.
The rest of the paper is organized as follows. In Section 2, the chaotic system model is given. In Section 3, circuit design and experimental results are described. In Section 4, related knowledge is introduced. The encryption scheme is described in Section 5. Simulation results and performance analyses are reported in Section 6. Lastly, the conclusions are drawn in Section 7.

New Five-Dimensional Chaotic System
In 1994, Sprott [20] summarized many 3D chaotic systems. After analyzing these 3D chaotic systems, we propose a new 3D chaotic system. The system equations are as follows.
Based on system (1), we introduce two controllers, w, v, and feedback w into the original controller y, feedback v into the original controller w, original controller x feedback into the new controller v, original controller y feedback into the new controller w, v, original controller z feedback to the new controller w, and feedback v to w. These six operations make the five controllers of this system interact with each other, which makes the relationship more complicated. The newly constructed five-dimensional chaotic system is as follows.

Dissipative Analysis
Because of Entropy 2020, 22, 243 3 of 27 Therefore, system (2) is dissipative, and convergence is in an exponential form of V 0 e −(a+c+e)t . Clearly, the volume element V 0 shrinks to the volume V 0 e −(a+c+e)t at moment t. Now consider when t → ∞ . Each volume element that contains the system trajectory shrinks to 0 at an exponential rate a + c + e = −9.

Balance Point Analysis
Let x = y = z = w = v = 0, that is, where the real part of 4,5  is positive, and the balance point 2 S is an unstable saddle focus.

Phase Diagram Analysis
For system parameters the 3D phase diagram generated by system (2) is shown in Figure 2. The resulting planar phase diagram is shown in Figure  3. It can be clearly seen from Figures 2 and 3 that the chaotic system can generate three-leaf chaotic attractors in multiple directions.

Phase Diagram Analysis
For system parameters a = −1, b = −3, c = −3, d = 1.8, e = −5, the 3D phase diagram generated by system (2) is shown in Figure 2. The resulting planar phase diagram is shown in Figure 3. It can be clearly seen from Figures 2 and 3 that the chaotic system can generate three-leaf chaotic attractors in multiple directions.

Phase Diagram Analysis
For system parameters the 3D phase diagram generated by system (2) is shown in Figure 2. The resulting planar phase diagram is shown in Figure  3. It can be clearly seen from Figures 2 and 3 that the chaotic system can generate three-leaf chaotic attractors in multiple directions.

Bifurcation Diagram
For the equation parameters the bifurcation diagram of the change in parameter a is shown in Figure 4. Figure 4 shows that, when the parameter ( 1.1, 1) a  − − , system (2) is in a chaotic state. (1, 2) , system (2) is in a chaotic state.

Bifurcation Diagram
For the equation parameters b = −3, c = −3, d = 1.8, e = −5, the bifurcation diagram of the change in parameter a is shown in Figure 4. Figure 4 shows that, when the parameter a ∈ (−1.1, −1), system (2) is in a chaotic state.

Bifurcation Diagram
For the equation parameters the bifurcation diagram of the change in parameter a is shown in Figure 4. Figure 4 shows that, when the parameter ( 1.1, 1) a  − − , system (2) is in a chaotic state. (1, 2) , system (2) is in a chaotic state. For equation parameters a = −1, b = −3, c = −3, e = −5, the bifurcation diagram of the change in parameter d is shown in Figure 5. As can be seen from Figure 5, when the parameter d ∈ (1, 2), system (2) is in a chaotic state.

Power Spectrum Analysis
The power spectrum of the chaotic sequence is a continuum, and the calculation results of the sequence ,, x y w , and v power spectrums of system (2) are shown in Figure 6a-d, respectively. It can be seen from Figure 6 that system (2) is in a chaotic state.

Power Spectrum Analysis
The power spectrum of the chaotic sequence is a continuum, and the calculation results of the sequence x, y, w, and v power spectrums of system (2) are shown in Figure 6a-d, respectively. It can be seen from Figure 6 that system (2) is in a chaotic state.

Power Spectrum Analysis
The power spectrum of the chaotic sequence is a continuum, and the calculation results of the sequence ,, x y w , and v power spectrums of system (2) are shown in Figure 6a-d, respectively. It can be seen from Figure 6 that system (2) is in a chaotic state.

Circuit Design and Experimental Results
In this section, we design an analog circuit, as shown in Figure 7, which is mainly composed of an inverting adder, integrator, and inverter composed of an operational amplifier TL082CD. The power supply voltage of the operational amplifier TL082CD is E = ±15V. This circuit has a simple structure and is easy to implement. The experimental results for this circuit are shown in Figure 8. Figure 8a shows the y − v plane, Figure 8b shows the x − w plane, and Figure 8c shows the w − v plane. It can be seen from the oscilloscope that the experimental results for the circuit for the three-leaf chaotic attractor in each plane were consistent with the results of the numerical simulation experiments.

Circuit Design and Experimental Results
In this section, we design an analog circuit, as shown in Figure 7, which is mainly composed of an inverting adder, integrator, and inverter composed of an operational amplifier TL082CD. The power supply voltage of the operational amplifier TL082CD is

EV =
. This circuit has a simple structure and is easy to implement. The experimental results for this circuit are shown in Figure 8. Figure 8a shows the yv − plane, Figure 8b shows the xw − plane, and Figure 8c shows the wv − plane. It can be seen from the oscilloscope that the experimental results for the circuit for the threeleaf chaotic attractor in each plane were consistent with the results of the numerical simulation experiments.   The convolution operation steps are as follows.

Convolution Operation
is an n × n matrix, where m < n. Then a n × n matrix C is obtained by a convolution operation between matrix A and convolution kernel h.
The convolution operation steps are as follows.
Step2: Obtain matrix C = (c ij ) nn using a convolution operation between matrix A and convolution kernel h, where

"Same OR" Operation
The "same OR" operation and the "exclusive OR" operation have the same effect. The "same OR" operation is defined as follows: when the input variables are the same, the output is 1, and when the input variables are different, the output is 0. The calculation results are presented in Table 1.

Encryption Algorithm Description
The flow chart of the encryption is shown in Figure 9.

"Same OR" Operation
The "same OR" operation and the "exclusive OR" operation have the same effect. The "same OR" operation is defined as follows: when the input variables are the same, the output is 1, and when the input variables are different, the output is 0. The calculation results are presented in Table 1.

Encryption Algorithm Description
The flow chart of the encryption is shown in Figure 9.
Enter initial chaos value   Given an M × N grayscale image A, the encryption steps are as follows.
Step 4: The five chaotic sequences are treated separately as follows. Step 5: Given a convolution kernel of 3 × 3.
Step 7: Divide grayscale image A into five regions starting from the center area in the proportion l = M : N. Input the value of m 1 using n 1 = m 1 ÷ l, and obtain n 1 . The formulas for m i and n i are as follows.
Step 2: Find the total iteration time Step 3: Call the ode45 function, iterate system (2), and generate five chaotic sequences.
Step 4: The five chaotic sequences are treated separately as follows.
There are Figure 10. Step 8: Divide the region 1 H in matrix (:,:,1) Y according to the starting point and size of 1 T in grayscale image A , as shown in Figure 11.   Step 8: Divide the region H1 in matrix Y(:, :, 1) according to the starting point and size of T1 in grayscale image A, as shown in Figure 11.
Step 2: Find the total iteration time Step 3: Call the ode45 function, iterate system (2), and generate five chaotic sequences.
Step 4: The five chaotic sequences are treated separately as follows.
There are Figure 10.    Step 9: Convert matrix T1 and matrix H1 sums of one row and (m 1 × n 1 ) column matrix T11 and H11, respectively. H11 is treated as follows.
Step 10: Combine T11 and HH, and perform a bitwise "XOR" operation to obtain matrix B1. Step 11: Process H11 as follows to obtain M1 and M2.
Step 13: Disturb each line of binary numbers in each row of F1, and then obtain matrix C1.
The disturbance formula is as follows.
where F1(i1, :) denotes all the columns of row i1 of matrix A and circshi f t(A, k, 2) moves all elements of row vector A clockwise by k units.
Step 14: Disturb all column elements of C1, and then obtain C2. The disturbance formula is as follows.
where C1(:, i2) denotes all the rows of column i2 of matrix C1 and circshift(A,k,1) moves all the elements of column vector A clockwise by k units.
Step 15: Convert binary number matrix C2 to decimal number matrix D1.
Step 16: D1 is used to replace the area T1. The results are shown in Figure 12.
Step 9: Convert matrix 1 T and matrix 1 H sums of one row and ( ) 11 mn  column matrix 11 T and 11 H , respectively. 11 H is treated as follows.
Step 12: Convert matrix 1 B into binary matrix 1 F .
Step 13: Disturb each line of binary numbers in each row of 1 F , and then obtain matrix 1 C .
The disturbance formula is as follows.
1( 1,: where 1( 1,:) Fi denotes all the columns of row 1 i of matrix A and ( , , 2) circshift A k moves all elements of row vector A clockwise by k units.
Step 14: Disturb all column elements of 1 C , and then obtain 2 C . The disturbance formula is as follows.
Step 15: Convert binary number matrix 2 C to decimal number matrix 1 D .
Step 16: 1 D is used to replace the area 1 T . The results are shown in Figure 12.   H is treated as follows. Step 17: The area H2 is taken out according to T2 in the starting point and size in A from Y(:, :, 2), as shown in Figure 13.
Step 9: Convert matrix 1 T and matrix 1 H sums of one row and ( ) 11 mn  column matrix 11 T and 11 H , respectively. 11 H is treated as follows.
Step 12: Convert matrix 1 B into binary matrix 1 F .
Step 13: Disturb each line of binary numbers in each row of 1 F , and then obtain matrix 1 C .
The disturbance formula is as follows.
1( 1,: where 1( 1,:) Fi denotes all the columns of row 1 i of matrix A and ( , , 2) circshift A k moves all elements of row vector A clockwise by k units.
Step 14: Disturb all column elements of 1 C , and then obtain 2 C . The disturbance formula is as follows.
Step 15: Convert binary number matrix 2 C to decimal number matrix 1 D .
Step 16: 1 D is used to replace the area 1 T . The results are shown in Figure 12. Step 17: The area 2 H is taken out according to 2 T in the starting point and size in A from (:,:,2) Y , as shown in Figure 13.  H is treated as follows. Step 18: Convert matrix T2 and matrix H2 sums of one row and (m 2 × n 2 ) column matrix T22 and H22, respectively. H22 is treated as follows.
Step 19: Combine T22 and HH1, and perform a bitwise "same OR" operation to obtain matrix B2 Step 20: Process H22 to obtain M3 and M4 as follows.
Step 22: Scramble each row of binary numbers in F2 to obtain matrix C3. The scrambling formula is as follows.
Step 25: Replace area T2 with D2. The result is shown in Figure 14.

B
Step 20: Process Step 21: Convert matrix 2 B into binary matrix 2 F .
Step 22: Scramble each row of binary numbers in 2 F to obtain matrix 3 C . The scrambling formula is as follows.
Step 24: Convert binary number matrix 4 C to decimal number matrix 2 D .
Step 25: Replace area 2 T with 2 D . The result is shown in Figure 14.
Step 28: Combine 33 T and 2 HH , and perform a bitwise "same OR" operation to obtain matrix 3 B .
Step 29: Process 33 H as follows to obtain 5 M and 6 M .
Step 30: Convert matrix 3 B into binary matrix 3 F . Step26: According to T3 from the starting point and size in the A, take the area H3 from Y(:, :, 3), as shown in Figure 15.

B
Step 20: Process Step 21: Convert matrix 2 B into binary matrix 2 F .
Step 22: Scramble each row of binary numbers in 2 F to obtain matrix 3 C . The scrambling formula is as follows. Step 23: Disturb all the column elements of 3 C to obtain 4 C . The scrambling formula is as follows.
Step 24: Convert binary number matrix 4 C to decimal number matrix 2 D .
Step 25: Replace area 2 T with 2 D . The result is shown in Figure 14.
Step 28: Combine 33 T and 2 HH , and perform a bitwise "same OR" operation to obtain matrix 3 B .
Step 29: Process 33 H as follows to obtain 5 M and 6 M .
Step 30: Convert matrix 3 B into binary matrix 3 F . Step 27: Convert the matrices T3 and H3 into matrices T33 and H33 with one row and (m 3 × n 3 ) columns, respectively, and process H33 as follows.
Step 28: Combine T33 and HH2, and perform a bitwise "same OR" operation to obtain matrix B3.
Step 29: Process H33 as follows to obtain M5 and M6.
Step 31: Scramble the binary numbers in each row of F3 to obtain matrix C5. The scrambling formula is as follows.
Step 34: Replace area T3 with D3. The result is shown in Figure 16.
formula is as follows.
Step 34: Replace area 3 T with 3 D . The result is shown in Figure 16. Step 35: Take region Step 37: Combine 44 T and 3 HH , and perform a bitwise "XOR" operation to obtain matrix 4 B . Step Step 39: Convert matrix 4 B into binary matrix 4 F .
Step 40: Scramble the binary numbers in each row in 4 F to obtain matrix 7 C . The scrambling formula is as follows. 7( 7,:) ( 4( 7,:), 7( 7), 2), C i circshift F i M i = 44 7 1, 2,3, , i m n = (12) Step 41: Disturb all the column elements of 7 C to obtain 8 C . The scrambling formula is as follows.  Step 35: Take region H4 from Y(:, :, 4) according to the starting point and size of T4 in A, as shown in Figure 17.
formula is as follows. Step 32: Disturb all the column elements of 5 C to obtain 6 C . The scrambling formula is as follows.
Step 34: Replace area 3 T with 3 D . The result is shown in Figure 16. Step 35: Take region Step 37: Combine 44 T and 3 HH , and perform a bitwise "XOR" operation to obtain matrix 4 B .
Step  Step 39: Convert matrix 4 B into binary matrix 4 F .
Step 40: Scramble the binary numbers in each row in 4 F to obtain matrix 7 C . The scrambling formula is as follows. 7( 7,:) ( 4( 7,:), 7( 7), 2), Step 41: Disturb all the column elements of 7 C to obtain 8 C . The scrambling formula is as follows.  Step 36: Convert matrices T4 and H4 into matrices T44 and H44 with one row and (m 4 × n 4 ) columns, respectively, and process H44 as follows.
Step 37: Combine T44 and HH3, and perform a bitwise "XOR" operation to obtain matrix B4.
Step 38: Process H44 to obtain M7 and M8 as follows.
Step 40: Scramble the binary numbers in each row in F4 to obtain matrix C7. The scrambling formula is as follows.
Step 41: Disturb all the column elements of C7 to obtain C8. The scrambling formula is as follows.
Step43: Replace area T4 with D4. The result is shown in Figure 18. Step42: Convert binary number matrix 8 C to decimal number matrix 4 D . Step43: Replace area 4 T with 4 D . The result is shown in Figure 18.
Step 45: Combine 55 T and 4 HH , and perform a bitwise "same OR" operation to obtain matrix 5 B .
Step 46: Process 55 H as follows to obtain 9 M and 10 M .
Step 47: Convert matrix 5 B into binary matrix 5 F .
Step 48: Scramble the binary numbers in each row of 5 F to obtain matrix 9 C . The scrambling formula is as follows. Step 49: Disturb all the column elements of 9 C . The scrambling formula is as follows.
Step 50: Convert binary number matrix 10 C to decimal number matrix 5 D , and 5 D is the final encrypted image. The result is shown in Figure 19.

Decryption Algorithm Description
Step 1: Input the initial value of the chaotic system 0 [0.6, 0.1, 0.2, 0.5, 0.4] y = and step size L = 0.02 , and find the total iteration time Step 2: Call the ode45 function, iterate system (2), and generate five chaotic sequences.
Step 3: The five chaotic sequences are treated as follows.  Step 44: Convert matrix T5 and Y(:, :, 5) to matrix T55 and H55 with one row and (M × N) columns, respectively, and H55 is treated as follows.
Step 45: Combine T55 and HH4, and perform a bitwise "same OR" operation to obtain matrix B5.
Step 46: Process H55 as follows to obtain M9 and M10.
Step 48: Scramble the binary numbers in each row of F5 to obtain matrix C9. The scrambling formula is as follows.
Step 49: Disturb all the column elements of C9. The scrambling formula is as follows. Step 50: Convert binary number matrix C10 to decimal number matrix D5, and D5 is the final encrypted image. The result is shown in Figure 19. Step42: Convert binary number matrix 8 C to decimal number matrix 4 D . Step43: Replace area 4 T with 4 D . The result is shown in Figure 18.  Step 45: Combine 55 T and 4 HH , and perform a bitwise "same OR" operation to obtain matrix 5 B .
Step 46: Process 55 H as follows to obtain 9 M and 10 M .
Step 47: Convert matrix 5 B into binary matrix 5 F .
Step 48: Scramble the binary numbers in each row of 5 F to obtain matrix 9 C . The scrambling formula is as follows.
Step 3: The five chaotic sequences are treated as follows.

Experiment Platform
The PC configuration was as follows: Intel(R) Core (TM) i5-6500 CPU @ 3.70 GHz 3.70 GHz, memory 8 GB, and Windows 7 64-bit operating system. The above encryption algorithm was implemented in a program in MATLAB R2014a.

Experimental Result
For the experiment, six types of grayscale images of classic images were selected: Lena, boat, baboon, peppers, couple, and leaf, which were all 256 256  . This algorithm is also applicable to grayscale images of any sizes. The plaintext image, encrypted image, and decrypted image are shown in Figure 21. Step6: Restore E5, E4, E3, E2, E1 in turn to obtain decrypted image A.

Experiment Platform
The PC configuration was as follows: Intel(R) Core (TM) i5-6500 CPU @ 3.70 GHz 3.70 GHz, memory 8 GB, and Windows 7 64-bit operating system. The above encryption algorithm was implemented in a program in MATLAB R2014a.

Experimental Result
For the experiment, six types of grayscale images of classic images were selected: Lena, boat, baboon, peppers, couple, and leaf, which were all 256 × 256. This algorithm is also applicable to grayscale images of any sizes. The plaintext image, encrypted image, and decrypted image are shown in Figure 21.

Key Space Analysis
The size of the key space is one of the most important factors that determines the strength of the image encryption algorithm. The larger the key space, the stronger the ability to resist brute force attacks. The secret keys of this proposed encryption algorithm include five initial values y 0 = [0.6, 0.1, 0.2, 0.5, 0.4] and five system parameters a, b, c, d, e. Since the precision is 10 −15 by computer with accuracy, the key space is (10 15 ) 10 = 10 150 . In addition, when the convolution kernel size is 3 × 3, the secret key also needs to consider nine convolution kernel parameters. Therefore, the total key space of the algorithm is much larger than 10 150 > 2 100 . For a security encryption algorithm, its key space should be larger than 2 100 [21]. Therefore, this algorithm was sufficiently secure.

Experiment Platform
The PC configuration was as follows: Intel(R) Core (TM) i5-6500 CPU @ 3.70 GHz 3.70 GHz, memory 8 GB, and Windows 7 64-bit operating system. The above encryption algorithm was implemented in a program in MATLAB R2014a.

Experimental Result
For the experiment, six types of grayscale images of classic images were selected: Lena, boat, baboon, peppers, couple, and leaf, which were all 256 256  . This algorithm is also applicable to grayscale images of any sizes. The plaintext image, encrypted image, and decrypted image are shown in Figure 21.

Key Space Analysis
The size of the key space is one of the most important factors that determines the strength of the image encryption algorithm. The larger the key space, the stronger the ability to resist brute force attacks. The secret keys of this proposed encryption algorithm include five initial values . In addition, when the convolution kernel size is 33  , the secret key also needs to consider nine convolution kernel parameters. Therefore, the total key space of the algorithm is much larger than 150 100 10 2  . For a security encryption algorithm, its key space should be larger than 100 2 [21]. Therefore, this algorithm was sufficiently secure.

Convolution Nuclear Sensitivity Analysis
In the encryption process, we multiplied each element of the five chaotic sequences produced by 10 11 and performed convolution operations with the convolution kernel of 33  . The convolution kernel of 33  in this algorithm was [1, 2,3; 4,5, 6;7, 8,9] c = . In the decryption process, when any of the parameters in the convolution kernel were slightly changed, the original image could not be successfully decrypted. When any parameter in the convolution kernel changed slightly with 15 10 − , the decrypted image was blurred, but the outline could be seen, as shown in Figure 22c. When any of the parameters in the convolution kernel was slightly changed with 14 10 − , the decrypted image could not substantially display the plaintext image information, as shown in Figure 22d. When any parameter in the convolution kernel changed slightly with 13 10 − , the plaintext image information could not be solved at all, as shown in Figure 22e. Taking the Lena image as an example, we made a slight change to the parameters of the second row and the second column of the convolution kernel: 1 =[1, 2,3;4,5.000000000000001, 6;7,8,9] c , 2 =[1, 2,3;4,5.00000000000001, 6;7,8,9] c , 3 =[1, 2,3;4, c 5.0000000000001, 6;7,8,9] . The plaintext images, ciphertext images, and the corresponding decrypted images of 1 c , 2 c , and 3 c are shown in Figure 22.

Convolution Nuclear Sensitivity Analysis
In the encryption process, we multiplied each element of the five chaotic sequences produced by 10 11 and performed convolution operations with the convolution kernel of 3 × 3. The convolution kernel of 3 × 3 in this algorithm was c = [1, 2, 3; 4, 5, 6; 7, 8, 9]. In the decryption process, when any of the parameters in the convolution kernel were slightly changed, the original image could not be successfully decrypted. When any parameter in the convolution kernel changed slightly with 10 −15 , the decrypted image was blurred, but the outline could be seen, as shown in Figure 22c. When any of the parameters in the convolution kernel was slightly changed with 10 −14 , the decrypted image could not substantially display the plaintext image information, as shown in Figure 22d. When any parameter in the convolution kernel changed slightly with 10 −13 , the plaintext image information could not be solved at all, as shown in Figure 22e. Taking the Lena image as an example, we made a slight change to the parameters of the second row and the second column of the convolution kernel: 2, 3; 4, 5.000000000000001, 6; 7, 8, 9], c 2 = [1, 2, 3; 4, 5.00000000000001, 6; 7, 8, 9], c 3 = [1, 2, 3; 4, 5.0000000000001, 6; 7, 8, 9]. The plaintext images, ciphertext images, and the corresponding decrypted images of c 1 , c 2 , and c 3 are shown in Figure 22.
A small change in the convolution kernel can lead to a great change in the ciphertext. In this study, only the Lena image was used as an example. Figure 23 shows the sensitivity of this algorithm to convolution kernels. Figure 23a is a plaintext image. Figure 23b,c are convolution kernels, c 0 = [1, 2, 3; 4, 5, 6; 7, 8, 9] and c 1 = [1, 2, 3; 4, 5.00000000000001, 6; 7, 8, 9] for encrypted ciphertext image C 0 and C 1 , respectively. Figure 23d is the result of the correct decryption for C 0 using c 0 . Figure 23e,f show the results of C 0 and C 1 using the wrong convolution kernel c 1 and c 0 decryption, respectively. Figure 23 illustrates that, despite only minor changes between the convolution kernels c 0 and c 1 , the ciphertext images C 0 and C 1 could not be properly decrypted with the convolution kernels. A small change in the convolution kernel can lead to a great change in the ciphertext. In this study, only the Lena image was used as an example. Figure 23 shows the sensitivity of this algorithm to convolution kernels. Figure 23a Figure 23 illustrates that, despite only minor changes between the convolution kernels 0 c and 1 c , the ciphertext images 0 C and 1 C could not be properly decrypted with the convolution kernels. The difference between the two images can also be measured by the pixel change rate (NPCR) and the normalized mean change intensity (UACI), which are described as Equations (29)  The difference between the two images can also be measured by the pixel change rate (NPCR) and the normalized mean change intensity (UACI), which are described as Equations (29) and (30).

Key Sensitivity Analysis
A small change in the key results in a great change in the ciphertext, which is the key sensitivity. In the experiment, the Lena image was considered as an example. Figure 24 shows the sensitivity of the algorithm to the initial key. Figure 24a Figure 24d shows the result of the correct decryption of Y 0 using y 0 . Figure 24e,f show the results of Y 0 and Y 1 using the wrong decryption keys y 1 and y 0 , respectively. Figure 24 illustrates that, despite only minor changes between the keys y 0 and y 1 , the ciphertext images Y 0 and Y 1 could be decrypted correctly using the corresponding keys y 1 and y 0 .
To better evaluate the key sensitivity of the algorithm, we tested the NPCR and UACI values between the keys y 0 = [0.6, 0.1, 0.2, 0.5, 0.4] and y = [0.600000000000001, 0.1, 0.2, 0.5, 0.4] for the encrypted image using Equations (29) and (30). The test values in this study and those in the literature [4,14,17] are shown in Table 3. Table 3 shows that the algorithm had good test results, so the encryption algorithm proposed in this paper has good key sensitivity.  Figure 24 illustrates that, despite only minor changes between the keys 0 y and 1 y , the ciphertext images 0 Y and 1 Y could be decrypted correctly using the corresponding keys 1 y and 0 y .   [4,14,17] are shown in Table 3. Table 3 shows that the algorithm had good test results, so the encryption algorithm proposed in this paper has good key sensitivity.

Information Entropy Analysis
Information entropy is an important indicator of randomness, which reflects the distribution of gray values of images. The more uniform the gray value distribution, the larger the information entropy of the image. The information entropy calculation formula of an image is shown below.
where p(x i ) is the probability of C and L is the total number of x i . For grayscale images, the theoretical maximum value of information entropy is 8. The closer the image information entropy is to the theoretical maximum, the more random the image pixel gray value distribution is. The information entropy before and after the encryption of Lena, baboon, boat, peppers, couple, and leaf is shown in Table 4. The simulation results show that the pixel value distribution of the encrypted image was very uniform, and the algorithm had a good encryption effect.

Correlation Analysis of Adjacent Pixels
A feature of digital images is the strong correlation of adjacent pixels. To calculate the correlation of adjacent pixels before and after encryption, 5000 sets of adjacent pixels were randomly selected in the horizontal, vertical, and diagonal directions of the plaintext and ciphertext images. The horizontal, vertical, and diagonal correlation coefficients were calculated using Equations (35).
The test results are shown in Table 8. The pixel correlation of the Lena plaintext image and ciphertext image in the horizontal direction, vertical direction, and diagonal direction are shown in Figure 26.   Quality evaluation of digital images can use the Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR) for measurement [23,24]. The MSE is a parameter to measure the difference between two images, which is described as Equation (36).
where H × W is the size of original image, X(i, j) is the original image, and (i, j) is the encrypted image. The higher value of MSE represents better encryption quality. This MSE analysis is a useful test for a plain image and encrypted image with pixel values in the range of [0-255]. The PSNR (expressed in logarithmic scale and decibels) determines the ratio between the maximum possible power of a signal and the power of distorting noise that affects the quality of its representation. It is calculated by Equation (37).
The smaller the MSE value is, the larger the PSNR value is, which means that there is a high degree of similarity between the tested images. By calculation, the MSE between the original image and the decrypted image is 0, and the value of PSNR is Inf. In this algorithm, the MSE between the original Lena image and the decrypted image is 77,012, and PSNR is 9.265. The results show that the quality metrics of the tested images is good.

Occlusion Attack Analysis
In an occlusion attack, we choose 12.5%, 25%, and 50% of occlusion in an encrypted image. In Figure 27, the attack results are shown. For 12.5% of occlusion, the MSE value is 7853 and the PSNR value is 9.1802. For 25% of occlusion, the MSE value is 10,148 and the PSNR value is 8.0672. For 50% of occlusion, the MSE value is 13,376 and the PSNR value is 6.8676. The results show that the proposed cryptographic algorithm can effectively resist occlusion attack. decrypted image is 0, and the value of PSNR is Inf. In this algorithm, the MSE between the original Lena image and the decrypted image is 77,012, and PSNR is 9.265. The results show that the quality metrics of the tested images is good.

Occlusion Attack Analysis
In an occlusion attack, we choose 12.5%, 25%, and 50% of occlusion in an encrypted image. In Figure 27, the attack results are shown. For 12.5% of occlusion, the MSE value is 7853 and the PSNR value is 9.1802. For 25% of occlusion, the MSE value is 10,148 and the PSNR value is 8.0672. For 50% of occlusion, the MSE value is 13,376 and the PSNR value is 6.8676. The results show that the proposed cryptographic algorithm can effectively resist occlusion attack.

Noise Attack Analysis
In order to verify the anti-noise performance of the proposed algorithm, Salt and pepper noise with different intensities was added to the encrypted image. The intensities were 10, 15, and 20, respectively, and they were then decrypted. The results are shown in Figure 28. For 10 of intensity, the MSE value is 8758.9 and the PSNR value is 8.706. For 15 of intensity, the MSE value is 9302.9 and the PSNR value is 8.446. For 20 of intensity, the MSE value is 9866.8 and the PSNR value is 8.189. It can be seen that the original image can be basically recovered after the noise image is decrypted. Therefore, the proposed algorithm has a certain anti-noise attack capability. In order to verify the anti-noise performance of the proposed algorithm, Salt and pepper noise with different intensities was added to the encrypted image. The intensities were 10, 15, and 20, respectively, and they were then decrypted. The results are shown in Figure 28. For 10 of intensity, the MSE value is 8758.9 and the PSNR value is 8.706. For 15 of intensity, the MSE value is 9302.9 and the PSNR value is 8.446. For 20 of intensity, the MSE value is 9866.8 and the PSNR value is 8.189. It can be seen that the original image can be basically recovered after the noise image is decrypted. Therefore, the proposed algorithm has a certain anti-noise attack capability.

Conclusions
In this paper, a five-dimensional chaotic system was proposed, which had a simple structure and was easy to implement. Basic dynamic analysis of the system was conducted, including the equilibrium point, phase diagram, bifurcation diagram, and power spectrum. Based on the theoretical analysis, a chaotic circuit was designed using the analog device amplifier TL082CD. The consistency of the numerical simulation results confirmed the feasibility of the method. Simultaneously, five chaotic sequences generated by the system were applied to the hybrid image encryption algorithm for physical chaotic encryption and advanced encryption standard encryption.

Conclusions
In this paper, a five-dimensional chaotic system was proposed, which had a simple structure and was easy to implement. Basic dynamic analysis of the system was conducted, including the equilibrium point, phase diagram, bifurcation diagram, and power spectrum. Based on the theoretical analysis, a chaotic circuit was designed using the analog device amplifier TL082CD. The consistency of the numerical simulation results confirmed the feasibility of the method. Simultaneously, five chaotic sequences generated by the system were applied to the hybrid image encryption algorithm for physical chaotic encryption and advanced encryption standard encryption. In the algebraic encryption process, we performed convolution operations on five chaotic sequences, which was followed by convolution operations. The latter sequence was applied to the image scaled block encryption, and a numerical simulation experiment was conducted on the hybrid encryption system. The simulation results verified the correctness of the encryption algorithm. Therefore, the encryption algorithm proposed in this paper has a good application prospect in secure communication, particularly digital image encryption.