Chaos-Based Synchronized Dynamic Keys and Their Application to Image Encryption with an Improved AES Algorithm

: This study aimed to design chaos-based synchronized dynamic keys and develop an improved chaos-based advanced encryption standard (AES) algorithm with the proposed synchronized random keys. First, based on sliding mode control (SMC) technology, a rippling control scheme was introduced to guarantee the synchronization between master–slave discrete chaotic systems. Under the synchronization, the same dynamic random chaos signals could be simultaneously obtained at the transmitter and receiver in communication systems. Then, a novel modiﬁed AES cryptosystem with dynamic random keys based on chaos synchronization was presented. In a traditional AES cryptosystem, a static key is used, and it must be exchanged in advance and conﬁrmed to be safely kept. However, in the proposed design, by introducing the synchronization technology of chaotic systems, the static key becomes dynamic and random, and it does not need to be kept or transmitted in open channels. Consequently, the disadvantage of key storage could be eliminated and the security of encryption could be improved. Finally, the developed chaos-based AES (CAES) algorithm has been applied to construct a novel image encryption algorithm. The statistical analysis, histogram, information entropy, and correlation indexes have been calculated and analyzed through simulation experiments in order to demonstrate the capability and improvement of this presented CAES cryptosystem.


Introduction
Since the high-speed growth and wide application in the field of the multimedia information and internet technology, the security demands for data communication and image information confidentiality have continually increased [1]. Recently, because of the good advantages of chaotic signals, such as the broadband random responses, and sensitivity to the system's initial conditions (butterfly effect), many studies have applied these properties to solve and enhance information security and image encryption and decryption issues [2][3][4][5][6][7][8][9]. In the literature [2], a new algorithm that integrated a chaotic system was proposed in order to construct a new transforming-scrambling-diffusion model for color image encryption. In [3], logistic and rectangular chaotic maps were used to implement the image encryption. However, the approaches introduced in the proposed algorithms were simple processes and could not generate uniform distribution for resisting statistical attacks [2]. In [4], the hyper-chaotic system and DNA plane were utilized to propose an image encryption approach. In [5], a new image encryption algorithm was proposed based on complex number an ultra-high computing speed would cause the existing AES encryption to be cracked. Motivated by this, this study aimed to develop a modified chaos-based AES algorithm with dynamic synchronized random keys. Firstly, a control scheme was proposed to solve the problem of synchronization for master-slave chaotic systems. Under synchronization, the same random chaos signals could be simultaneously obtained at the transmitter and receiver in communication systems. Then, the dynamic and unpredictable features of synchronized chaotic signals were extended to design a synchronized dynamic key generator for improving AES cryptosystems. These random keys were dynamically updated and hidden. Unlike the traditional AES algorithm, these keys did not need to be saved in advance, so security could be promoted. To verify the security, the developed chaos-based AES (CAES) algorithm was applied to construct a novel image encryption algorithm. The statistical analysis, histogram, information entropy, and correlation indexes were provided and examined through the simulation experiment in order to demonstrate the efficiency and security of the presented CAES algorithm.
This paper is organized as follows. Section 2 formulates chaos synchronization and the design of a dynamic chaos-based random key generator. The synchronization controller is proposed using discrete sliding mode control. Numerical simulations are given to illustrate the derived results. Section 3 introduces the structure of the improved CAES algorithm. Section 4 proposes and analyzes the performance of the image cryptosystem based on the proposed CAES. Finally, brief conclusions and future works are given in Section 5.

Chaos Synchronization Design
In this research, a novel CAES cryptosystem was considered with a synchronized chaos-based dynamic random key generator. Before generating the new dynamic chaosbased random keys, the first problem to be well solved was the chaos synchronization. Therefore, the synchronization problem and the controller design were first formulated so as to achieve synchronization for master-slave hyperchaotic Henon maps [22].
In other words, the states of master-slave hyperchaotic systems described in Equations (1) and (2) could be fully synchronized with random chaos dynamics. To obtain a good robustness, similar to that of the literature [23], the sliding mode control was introduced to design the controller ( ( ) u k ) in order to ensure the exact synchronization. Generally, there are two steps for completing the design of the sliding mode controller ( ( ) u k ). First, it is necessary to design a proper sliding surface (s(k)) and to ensure the state synchronization of the controlled system under this assumption, such that the controlled trajectory of the systems can hit and operate in the sliding mode (i.e., ( ) = 0). Then, it is necessary to design the sliding mode controller ( ( ) u k ) to guarantee that the state trajectory of the controlled states can be forced to the sliding manifold as expected. In order to complete the above design steps, the sliding surface for the sliding manifold was selected as follows: where ( ) ∈ and , are the chosen parameters. If the controlled system can be forced to enter the sliding motion under the effect of u(k) (in other words, ( ) = 0), then ( ) = − ( ) − ( ). By substituting the relation of ( ) = − ( ) − ( ) into the error Equation (3) of the master-slave system, it can be rewritten as follows: From Equation (5) After discussing the synchronization under the condition of ( ) = 0, the sliding mode controller ( ) is given as follows: where | | < 1. If the sliding mode controller is given as Equation (6), the following equation can be obtained: To discuss the synchronization controller design, the error state was defined as e i (k) = y i (k) − x i (k), i = 1, 2, 3. This resulted in the following error system: Here, the aim was to design a controller, u(k), to result in lim k→∞ e i (k) = 0, i = 1, 2, 3. In other words, the states of master-slave hyperchaotic systems described in Equations (1) and (2) could be fully synchronized with random chaos dynamics. To obtain a good robustness, similar to that of the literature [23], the sliding mode control was introduced to design the controller (u(k)) in order to ensure the exact synchronization. Generally, there are two steps for completing the design of the sliding mode controller (u(k)). First, it is necessary to design a proper sliding surface (s(k)) and to ensure the state synchronization of the controlled system under this assumption, such that the controlled trajectory of the systems can hit and operate in the sliding mode (i.e., s(k) = 0). Then, it is necessary to design the sliding mode controller (u(k)) to guarantee that the state trajectory of the controlled states can be forced to the sliding manifold as expected. In order to complete the above design steps, the sliding surface for the sliding manifold was selected as follows: where s(k) ∈ R and α, β are the chosen parameters. If the controlled system can be forced to enter the sliding motion under the effect of u(k) (in other words, s(k) = 0), then e 1 (k) = −αe 2 (k) − βe 3 (k). By substituting the relation of e 1 (k) = −αe 2 (k) − βe 3 (k) into the error Equation (3) of the master-slave system, it can be rewritten as follows: From Equation (5), obviously, if α and β are chosen to ensure |λ i (A)| < 1, i = 1, 2, then E(k) = e 2 (k) e 3 (k) T can converge to zero. Furthermore, as the controlled systems operate in the sliding mode, e 1 (k) = −αe 2 (k) − βe 3 (k) and e 1 (k) = 0 are ensured when E(k) = 0. Therefore, the state responses of the controlled slave Henon map (2) can fully track the random chaotic behavior of master Henon map (1) in the sliding manifold (s(k) = 0). After discussing the synchronization under the condition of s(k) = 0, the sliding mode controller u(k) is given as follows: where |δ| < 1. If the sliding mode controller is given as Equation (6), the following equation can be obtained: According to Equation (7), s(k + 1) = δs(k). Because |δ| < 1 was specified, the trajectory could be forced to hit and operate in the sliding manifold (s(k) = 0). In addition, the state behavior of the master-slave Henon maps would be synchronized and would have the same chaotic dynamics as in Figure 1.
Next, the simulation tool of MATLAB was used to examine the control design discussed above. In this simulation, the initial conditions were selected as x(0) = 1.5 −0.3 0.4 T and y(0) = − 0.3 −1.1 0.8 T . α = 0.1 and β = 0.3 were selected, such that the eigenvalues of A were λ 1 = 0.6, λ 2 = −0.5, and δ = 0.5. Therefore, the sliding surface could be constructed as follows: According to Equation (6), the controller was designed as follows: Figures 2-4 show the simulation results. Figure 2 shows the error state with the proposed control input (Equation (9)). Figure 3 shows the state responses of the controlled systems. The responses of the switching function of s(k) and the sliding mode controller u(k) were, respectively, given in Figure 4. By observing the simulation results, it was revealed that the trajectory of the controlled systems could be forced to the sliding manifold of s(k) = 0; the error state responses between the master-slave chaotic system converged to zero, as expected; and the controlled master-slave system reached exact synchronization.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 16 According to Equation (7), ( + 1) = ( ). Because | | < 1 was specified, the trajectory could be forced to hit and operate in the sliding manifold ( ( ) = 0). In addition, the state behavior of the master-slave Henon maps would be synchronized and would have the same chaotic dynamics as in Figure 1.
Next, the simulation tool of MATLAB was used to examine the control design discussed above. In this simulation, the initial conditions were selected as such that the eigenvalues of A were = 0.6, = −0.5, and = 0.5. Therefore, the sliding surface could be constructed as follows: According to Equation (6), the controller was designed as follows: Figures 2-4 show the simulation results. Figure 2 shows the error state with the proposed control input (Equation (9)). Figure 3 shows the state responses of the controlled systems. The responses of the switching function of s(k) and the sliding mode controller u(k) were, respectively, given in Figure 4. By observing the simulation results, it was revealed that the trajectory of the controlled systems could be forced to the sliding manifold of ( ) = 0; the error state responses between the master-slave chaotic system converged to zero, as expected; and the controlled master-slave system reached exact synchronization.  Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 16 According to Equation (7), ( + 1) = ( ). Because | | < 1 was specified, the trajectory could be forced to hit and operate in the sliding manifold ( ( ) = 0). In addition, the state behavior of the master-slave Henon maps would be synchronized and would have the same chaotic dynamics as in Figure 1.
Next, the simulation tool of MATLAB was used to examine the control design discussed above. In this simulation, the initial conditions were selected as such that the eigenvalues of A were = 0.6, = −0.5, and = 0.5. Therefore, the sliding surface could be constructed as follows: According to Equation (6), the controller was designed as follows: Figures 2-4 show the simulation results. Figure 2 shows the error state with the proposed control input (Equation (9)). Figure 3 shows the state responses of the controlled systems. The responses of the switching function of s(k) and the sliding mode controller u(k) were, respectively, given in Figure 4. By observing the simulation results, it was revealed that the trajectory of the controlled systems could be forced to the sliding manifold of ( ) = 0; the error state responses between the master-slave chaotic system converged to zero, as expected; and the controlled master-slave system reached exact synchronization.

Dynamic Chaos-Based Random Key Generator
From the above discussion, it was possible to suppress the butterfly effect between master-slave systems through the effect of the controller, and the states of the controlled slave system could be synchronized with those random states of the master system. Next, we designed a dynamic random key generator. There are three types of AES advanced encryption technologies, which use cryptographic keys of 128, 192, and 256 bits [24], respectively. Obviously, from the above chaotic systems, it was possible to generate three random states; however, if these random states were used as the AES encryption key, the lengths would be somewhat insufficient. Therefore, to solve this problem, dynamic chaotic signals were integrated with the hash function to establish the dynamic key generator. Because of the avalanche effect of Secure Hash Algorithm 256 (SHA256) [25] and the butterfly effect of the chaotic systems, each dynamic key was never repeated, which made frequency analysis impossible and thus was difficult to crack. The dynamic random key generator with SHA256 is shown in Figure 5.

Design of the Improved AES Algorithm
AES encryption is a popular symmetric-key cryptosystem. The main architecture of the AES encryption algorithm is given in Figure 6. In order to generate a uniform distribution of cipher text, a series of linked operations were included in AES cryptosystem design. Some of the linked operations were performed by substituting the inputs using some output feedback (substitutions). The left operations involved shuffling many bits around (permutations). AES runs computations using bytes. Therefore, the AES separated 128 bits of information block into 16 bytes. These 16 bytes were then formatted as a matrix form with four columns and four rows in order to process the operations. The operation rounds in the AES cryptosystem were dependent on the key length we selected. Here, 10

Dynamic Chaos-Based Random Key Generator
From the above discussion, it was possible to suppress the butterfly effect between master-slave systems through the effect of the controller, and the states of the controlled slave system could be synchronized with those random states of the master system. Next, we designed a dynamic random key generator. There are three types of AES advanced encryption technologies, which use cryptographic keys of 128, 192, and 256 bits [24], respectively. Obviously, from the above chaotic systems, it was possible to generate three random states; however, if these random states were used as the AES encryption key, the lengths would be somewhat insufficient. Therefore, to solve this problem, dynamic chaotic signals were integrated with the hash function to establish the dynamic key generator. Because of the avalanche effect of Secure Hash Algorithm 256 (SHA256) [25] and the butterfly effect of the chaotic systems, each dynamic key was never repeated, which made frequency analysis impossible and thus was difficult to crack. The dynamic random key generator with SHA256 is shown in Figure 5.

Dynamic Chaos-Based Random Key Generator
From the above discussion, it was possible to suppress the butterfly effect between master-slave systems through the effect of the controller, and the states of the controlled slave system could be synchronized with those random states of the master system. Next, we designed a dynamic random key generator. There are three types of AES advanced encryption technologies, which use cryptographic keys of 128, 192, and 256 bits [24], respectively. Obviously, from the above chaotic systems, it was possible to generate three random states; however, if these random states were used as the AES encryption key, the lengths would be somewhat insufficient. Therefore, to solve this problem, dynamic chaotic signals were integrated with the hash function to establish the dynamic key generator. Because of the avalanche effect of Secure Hash Algorithm 256 (SHA256) [25] and the butterfly effect of the chaotic systems, each dynamic key was never repeated, which made frequency analysis impossible and thus was difficult to crack. The dynamic random key generator with SHA256 is shown in Figure 5.

Design of the Improved AES Algorithm
AES encryption is a popular symmetric-key cryptosystem. The main architecture of the AES encryption algorithm is given in Figure 6. In order to generate a uniform distribution of cipher text, a series of linked operations were included in AES cryptosystem design. Some of the linked operations were performed by substituting the inputs using some output feedback (substitutions). The left operations involved shuffling many bits around (permutations). AES runs computations using bytes. Therefore, the AES separated 128 bits of information block into 16 bytes. These 16 bytes were then formatted as a matrix form with four columns and four rows in order to process the operations. The operation rounds in the AES cryptosystem were dependent on the key length we selected. Here, 10

Design of the Improved AES Algorithm
AES encryption is a popular symmetric-key cryptosystem. The main architecture of the AES encryption algorithm is given in Figure 6. In order to generate a uniform distribution of cipher text, a series of linked operations were included in AES cryptosystem design. Some of the linked operations were performed by substituting the inputs using some output feedback (substitutions). The left operations involved shuffling many bits around (permutations). AES runs computations using bytes. Therefore, the AES separated 128 bits of information block into 16 bytes. These 16 bytes were then formatted as a matrix form with four columns and four rows in order to process the operations. The operation rounds in the AES cryptosystem were dependent on the key length we selected. Here, 10 rounds, 12 rounds, and 14 rounds, were used for the key lengths of 128-bit, 192-bit, and 256-bit, respectively, as shown in Figure 6. Using different round keys, K R , derived from the original static AES key, each of these rounds was performed, where R was the round. rounds, 12 rounds, and 14 rounds, were used for the key lengths of 128-bit,192-bit, and 256-bit, respectively, as shown in Figure 6. Using different round keys, R K , derived from the original static AES key, each of these rounds was performed, where R was the round. For AES symmetric-key cryptosystems, the keys represented a shared secret in the communication systems that could guarantee private information transmission. However, compared with asymmetric key encryption, one of the main weaknesses of symmetric key cryptosystems was the requirement of the secret key [25]. Once this static secret key is stolen, all information will be exposed. Therefore, to solve this problem, in this paper, a new CAES encryption method was proposed based on synchronized chaotic dynamic keys, as shown in Figure 7. In this proposed improved chaos-based AES architecture, the dynamic random states of synchronized chaos systems were used as the secure keys. Obviously, this improved design could provide synchronized random states to a random key generator and then generate dynamic random keys with 256 bits for AES to promote its security. By introducing a designed controller to the slave chaotic system, the slave states could be synchronized with the master states. Furthermore, to promote the security of practical applica- For AES symmetric-key cryptosystems, the keys represented a shared secret in the communication systems that could guarantee private information transmission. However, compared with asymmetric key encryption, one of the main weaknesses of symmetric key cryptosystems was the requirement of the secret key [25]. Once this static secret key is stolen, all information will be exposed. Therefore, to solve this problem, in this paper, a new CAES encryption method was proposed based on synchronized chaotic dynamic keys, as shown in Figure 7.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 16 rounds, 12 rounds, and 14 rounds, were used for the key lengths of 128-bit,192-bit, and 256-bit, respectively, as shown in Figure 6. Using different round keys, R K , derived from the original static AES key, each of these rounds was performed, where R was the round. For AES symmetric-key cryptosystems, the keys represented a shared secret in the communication systems that could guarantee private information transmission. However, compared with asymmetric key encryption, one of the main weaknesses of symmetric key cryptosystems was the requirement of the secret key [25]. Once this static secret key is stolen, all information will be exposed. Therefore, to solve this problem, in this paper, a new CAES encryption method was proposed based on synchronized chaotic dynamic keys, as shown in Figure 7. In this proposed improved chaos-based AES architecture, the dynamic random states of synchronized chaos systems were used as the secure keys. Obviously, this improved design could provide synchronized random states to a random key generator and then generate dynamic random keys with 256 bits for AES to promote its security. By introducing a designed controller to the slave chaotic system, the slave states could be synchronized with the master states. Furthermore, to promote the security of practical applica- In this proposed improved chaos-based AES architecture, the dynamic random states of synchronized chaos systems were used as the secure keys. Obviously, this improved design could provide synchronized random states to a random key generator and then generate dynamic random keys with 256 bits for AES to promote its security. By introducing a designed controller to the slave chaotic system, the slave states could be synchronized with the master states. Furthermore, to promote the security of practical applications, the synchronization controller (u(k)) was decomposed into u m (k) and u s (k), calculated from the master and slave chaotic systems, respectively. The controller was then realized at the receiver side with u(k) = F(u m (k), u s (k)). Even if hackers had these parameters in the synchronization controller, they would still not be able to obtain the states of the master and slave systems separately, and they would not be able to reorganize the synchronization controller. Therefore, the security of the dynamic keys and the encryption system would be ensured. After achieving synchronization, the same random chaos signals could be simultaneously obtained, and the synchronized dynamic random keys at both sides of transmitter and receiver could be constructed. The static keys in the traditional AES were replaced by synchronized random signals, and did not need to be given in advance or be communicated through public channels. As a result, the disadvantage of key storage could be eliminated and the security of the encryption could be improved because of the synchronized dynamic random keys.

Implementation and Performance Analysis of CAES-Based Image Encryption
After designing the CAES algorithm shown in Figure 7, this proposed CAES was applied to image encryption, and its encryption performance and security were analyzed.
Here, Python was used as the developing environment. The program modules used included Pycrypto, Numpy, Numpydoc, OpenCV, and Matplotlib. In this image encryption, PNG images were used as the plain text data. The encrypted files could also be displayed normally by pixel processing. At the same time, the encrypted pixel data were stored in bytes as the test data for the National Institute of Standards and Technology (NIST) test. In this application, the master-slave hyperchaotic Henon Maps in Equations (1) and (2) were used with a synchronization controller from Equation (6) to synchronize dynamic random key generators with SHA256, as shown in Figure 7. The encryption and decryption of the CAES flow chart are shown in Figures 8 and 9, respectively. tions, the synchronization controller ( ( ) u k ) was decomposed into ( ) and ( ), calculated from the master and slave chaotic systems, respectively. The controller was then realized at the receiver side with ( ) = ( ( ), ( )). Even if hackers had these parameters in the synchronization controller, they would still not be able to obtain the states of the master and slave systems separately, and they would not be able to reorganize the synchronization controller. Therefore, the security of the dynamic keys and the encryption system would be ensured. After achieving synchronization, the same random chaos signals could be simultaneously obtained, and the synchronized dynamic random keys at both sides of transmitter and receiver could be constructed. The static keys in the traditional AES were replaced by synchronized random signals, and did not need to be given in advance or be communicated through public channels. As a result, the disadvantage of key storage could be eliminated and the security of the encryption could be improved because of the synchronized dynamic random keys.

Implementation and Performance Analysis of CAES-Based Image Encryption
After designing the CAES algorithm shown in Figure 7, this proposed CAES was applied to image encryption, and its encryption performance and security were analyzed. Here, Python was used as the developing environment. The program modules used included Pycrypto, Numpy, Numpydoc, OpenCV, and Matplotlib. In this image encryption, PNG images were used as the plain text data. The encrypted files could also be displayed normally by pixel processing. At the same time, the encrypted pixel data were stored in bytes as the test data for the National Institute of Standards and Technology (NIST) test. In this application, the master-slave hyperchaotic Henon Maps in Equations (1) and (2) were used with a synchronization controller from Equation (6) to synchronize dynamic random key generators with SHA256, as shown in Figure 7. The encryption and decryption of the CAES flow chart are shown in Figures 8 and 9 Figure 8. Flow chart of encryption using chaos-based AES (CAES).

Visual Effect of Encrypted Images
In this example, an image with a 256 × 256 resolution was used as the plaintext data. Several of the plain and cipher images are shown below.
From Figure 10, it can be observed that when the AES-ECB was used to encrypt the data, the intervals with the same color in the background would be encrypted into the same cipher text. This was a drawback of AES-ECB for image encryption. The encrypted picture looked confusing enough when AES-CBC was used to encrypt the image. When using the proposed improved CAES for pictures with a size of 256 × 256 and 49,152 dynamic keys, as the original AES already had a qualified security, it would be impossible to completely solve all of the keys in short time. Furthermore, because of the design of the dynamic key, the 49,152 keys were updated immediately in the next round of encryption and the old password had no effect. Therefore, in terms of security, CAES was greatly improved compared with the original AES-ECB and AES-CBC.

Visual Effect of Encrypted Images
In this example, an image with a 256 × 256 resolution was used as the plaintext data. Several of the plain and cipher images are shown below.
From Figure 10, it can be observed that when the AES-ECB was used to encrypt the data, the intervals with the same color in the background would be encrypted into the same cipher text. This was a drawback of AES-ECB for image encryption. The encrypted picture looked confusing enough when AES-CBC was used to encrypt the image. When using the proposed improved CAES for pictures with a size of 256 × 256 and 49,152 dynamic keys, as the original AES already had a qualified security, it would be impossible to completely solve all of the keys in short time. Furthermore, because of the design of the dynamic key, the 49,152 keys were updated immediately in the next round of encryption and the old password had no effect. Therefore, in terms of security, CAES was greatly improved compared with the original AES-ECB and AES-CBC. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 16

Statistical Analysis by NIST
In this section, the National Institute of Standards and Technology (NIST) [26] test suite was introduced to evaluate the randomness of the encrypted images. The OpenCV function library of Python was utilized to collect the three values of R, G, and B from image 1 in Figure 10; then convert the values from 0 to 255 into bytes for encryption, as shown in Figure 8; and then output the results as a NIST test file through the Numpy library. The NIST SP800-22 test consisted of 15 items. The method of detection compared the calculated outputted p-value for each item with a prescribed significance level of 0.01. If the outputted p-values were all greater than 0.01 for all of the test items, it passed the NIST test. The NIST test results shown in Table 1 were obtained with a stream length of 10,000,000 bits and a bit stream count of 30. The obtained data revealed that the encrypted data from the CAES algorithm passed the NIST SP 800-22 random number detection (pvalue > 0.001) and that the test performance of each item was better than the original AES-CBC and AES-ECB. This result indicates that the presented CAES algorithm had a good randomness.

Statistical Analysis by NIST
In this section, the National Institute of Standards and Technology (NIST) [26] test suite was introduced to evaluate the randomness of the encrypted images. The OpenCV function library of Python was utilized to collect the three values of R, G, and B from image 1 in Figure 10; then convert the values from 0 to 255 into bytes for encryption, as shown in Figure 8; and then output the results as a NIST test file through the Numpy library. The NIST SP800-22 test consisted of 15 items. The method of detection compared the calculated outputted p-value for each item with a prescribed significance level of 0.01. If the outputted p-values were all greater than 0.01 for all of the test items, it passed the NIST test. The NIST test results shown in Table 1 were obtained with a stream length of 10,000,000 bits and a bit stream count of 30. The obtained data revealed that the encrypted data from the CAES algorithm passed the NIST SP 800-22 random number detection (p-value > 0.001) and that the test performance of each item was better than the original AES-CBC and AES-ECB. This result indicates that the presented CAES algorithm had a good randomness.

Histogram Analysis and Mean Absolute Deviation
For an image, the histogram is an important statistical feature. A good cryptosystem can encrypt an image, and the histogram of the cipher image is evenly distributed, such that attackers can not analyze the statistical features of the original image from its cipher image. Figure 11a-d presents the visual effect of the plain and cipher images, respectively. Figure 11e-h shows the histograms of the plain and cipher images, respectively, demonstrating that the encryption can safely obscure the histogram information.

Histogram Analysis and Mean Absolute Deviation
For an image, the histogram is an important statistical feature. A good cryptosystem can encrypt an image, and the histogram of the cipher image is evenly distributed, such that attackers can not analyze the statistical features of the original image from its cipher image. Figure 11a-d presents the visual effect of the plain and cipher images, respectively. Figure 11e-h shows the histograms of the plain and cipher images, respectively, demonstrating that the encryption can safely obscure the histogram information. To evaluate the uniform distribution of histograms, the mean absolute deviation (MAD) [27] was calculated using the following equation: where i x is the number of pixel values (i),  Table 2, it is revealed that not all of the histograms of the encrypted images from AES-EBC had a uniform distribution. However, AES-EBC and CAES-CBC always had good mean absolute deviations. Furthermore, CAES-CBC had the smallest average MAD. This means that the uniform distribution of CAES-CBC was slightly better than AES-CBC.  To evaluate the uniform distribution of histograms, the mean absolute deviation (MAD) [27] was calculated using the following equation:  Table 2, it is revealed that not all of the histograms of the encrypted images from AES-EBC had a uniform distribution. However, AES-EBC and CAES-CBC always had good mean absolute deviations. Furthermore, CAES-CBC had the smallest average MAD. This means that the uniform distribution of CAES-CBC was slightly better than AES-CBC. Table 2. Mean absolute deviation (MAD) uniform distribution test.

Image
Histogram MAD Uniform Distribution Test  To evaluate the uniform distribution of histograms, the mean absolute deviation (MAD) [27] was calculated using the following equation:

AES-ECB AES-CBC CAES-CBC
where i x is the number of pixel values (i),  Table 2, it is revealed that not all of the histograms of the encrypted images from AES-EBC had a uniform distribution. However, AES-EBC and CAES-CBC always had good mean absolute deviations. Furthermore, CAES-CBC had the smallest average MAD. This means that the uniform distribution of CAES-CBC was slightly better than AES-CBC.  To evaluate the uniform distribution of histograms, the mean absolute deviation (MAD) [27] was calculated using the following equation: where i x is the number of pixel values (i),  Table 2, it is revealed that not all of the histograms of the encrypted images from AES-EBC had a uniform distribution. However, AES-EBC and CAES-CBC always had good mean absolute deviations. Furthermore, CAES-CBC had the smallest average MAD. This means that the uniform distribution of CAES-CBC was slightly better than AES-CBC.  To evaluate the uniform distribution of histograms, the mean absolute deviation (MAD) [27] was calculated using the following equation:  Table 2, it is revealed that not all of the histograms of the encrypted images from AES-EBC had a uniform distribution. However, AES-EBC and CAES-CBC always had good mean absolute deviations. Furthermore, CAES-CBC had the smallest average MAD. This means that the uniform distribution of CAES-CBC was slightly better than AES-CBC.

Information Entropy Analysis
Image information entropy (IE) represents the aggregation feature of the distribution of the image pixel values [28]. The image IE was calculated using the following formula: where is the frequency of each greyscale. For a grayscale image, the pixel had a data field of [0, 255], and the maximum value of IE will be 8. The IE test report with the grayscale of image 1 in Figure 10 is given in Figure 12. From Figure 12, the IE value of CAES-ECB is the closest to the ideal value of 8. Therefore, it can be concluded that the encrypted image with CAES-ECB possessed the true random signal property, and the proposed algorithm had the ability to resist entropy attacks.

Connected Component Analysis
Generally speaking, for a plain image, a high correlation between adjacent pixels is an obvious feature. Therefore, it would be better if the encryption algorithm could result in smaller correlation values between adjacent pixels. Here, 3000 pixels were randomly selected from the information of the plaint and encrypted images of image 1 in Figure 10. The correlation coefficients for the horizontal, vertical, and diagonal directions were calculated using the following equations [2]: where i x and i y denote the pixel values of the two adjacent pixels, and N is the number of pair ( , ) i i x y . Figure 13 shows the correlation in a diagonal direction for the plain and encrypted image information according to CAES-ECB, seperately. A detailed report of the connected component analysis is given in Figure 14. According to Figure 14, it can be seen that AES-

Information Entropy Analysis
Image information entropy (IE) represents the aggregation feature of the distribution of the image pixel values [28]. The image IE was calculated using the following formula: where p i is the frequency of each greyscale. For a grayscale image, the pixel had a data field of [0, 255], and the maximum value of IE will be 8. The IE test report with the grayscale of image 1 in Figure 10 is given in Figure 12. From Figure 12, the IE value of CAES-ECB is the closest to the ideal value of 8. Therefore, it can be concluded that the encrypted image with CAES-ECB possessed the true random signal property, and the proposed algorithm had the ability to resist entropy attacks.

Information Entropy Analysis
Image information entropy (IE) represents the aggregation feature of the distribution of the image pixel values [28]. The image IE was calculated using the following formula: where is the frequency of each greyscale. For a grayscale image, the pixel had a data field of [0, 255], and the maximum value of IE will be 8. The IE test report with the grayscale of image 1 in Figure 10 is given in Figure 12. From Figure 12, the IE value of CAES-ECB is the closest to the ideal value of 8. Therefore, it can be concluded that the encrypted image with CAES-ECB possessed the true random signal property, and the proposed algorithm had the ability to resist entropy attacks.

Connected Component Analysis
Generally speaking, for a plain image, a high correlation between adjacent pixels is an obvious feature. Therefore, it would be better if the encryption algorithm could result in smaller correlation values between adjacent pixels. Here, 3000 pixels were randomly selected from the information of the plaint and encrypted images of image 1 in Figure 10. The correlation coefficients for the horizontal, vertical, and diagonal directions were calculated using the following equations [2]: where i x and i y denote the pixel values of the two adjacent pixels, and N is the number of pair ( , ) i i x y . Figure 13 shows the correlation in a diagonal direction for the plain and encrypted image information according to CAES-ECB, seperately. A detailed report of the connected component analysis is given in Figure 14. According to Figure 14, it can be seen that AES-

Connected Component Analysis
Generally speaking, for a plain image, a high correlation between adjacent pixels is an obvious feature. Therefore, it would be better if the encryption algorithm could result in smaller correlation values between adjacent pixels. Here, 3000 pixels were randomly selected from the information of the plaint and encrypted images of image 1 in Figure 10. The correlation coefficients for the horizontal, vertical, and diagonal directions were calculated using the following equations [2]: where x i and y i denote the pixel values of the two adjacent pixels, and N is the number of pair (x i , y i ). Figure 13 shows the correlation in a diagonal direction for the plain and encrypted image information according to CAES-ECB, seperately. A detailed report of the connected component analysis is given in Figure 14. According to Figure 14, it can be seen that AES-ECB had the highest pixel correlation. In AES-CBC, although it was superior to AES-ECB, the correlation of the vertical and horizontal directions was higher than those of the proposed CAES-ECB. Although AES-CBC had a lower pixel correlation in the diagonal direction, the proposed CAES-ECB still had the lowest average correlation. This result indicates that the proposed improved CAES algorithm had better diffusion characteristics than AES-ECB and AES-CBC. ECB had the highest pixel correlation. In AES-CBC, although it was superior to AES-ECB, the correlation of the vertical and horizontal directions was higher than those of the proposed CAES-ECB. Although AES-CBC had a lower pixel correlation in the diagonal direction, the proposed CAES-ECB still had the lowest average correlation. This result indicates that the proposed improved CAES algorithm had better diffusion characteristics than AES-ECB and AES-CBC.
(a) plain image (b) encrypted image by CAES_ECB

Conclusions and Future Works
In this paper, by integrating the random property of chaotic signals with synchronization control, the synchronized random key generator was established. Then, a novel modified CAES cryptosystem with the proposed dynamic random keys was presented. Unlike the traditional AES algorithm, the traditional AES static key was replaced by the proposed synchronized dynamic random keys, not necessary to be saved or exchanged in advance, so that the security could be prompted. This proposed CAES was applied to image encryption, and several tests and analyses, such as visual effect, statistical analysis, histogram, information entropy analysis, and connected component analysis, were all performed to determine the capability and security of the CAES algorithm. Recently, as a ECB had the highest pixel correlation. In AES-CBC, although it was superior to AES-ECB, the correlation of the vertical and horizontal directions was higher than those of the proposed CAES-ECB. Although AES-CBC had a lower pixel correlation in the diagonal direction, the proposed CAES-ECB still had the lowest average correlation. This result indicates that the proposed improved CAES algorithm had better diffusion characteristics than AES-ECB and AES-CBC.
(a) plain image (b) encrypted image by CAES_ECB

Conclusions and Future Works
In this paper, by integrating the random property of chaotic signals with synchronization control, the synchronized random key generator was established. Then, a novel modified CAES cryptosystem with the proposed dynamic random keys was presented. Unlike the traditional AES algorithm, the traditional AES static key was replaced by the proposed synchronized dynamic random keys, not necessary to be saved or exchanged in advance, so that the security could be prompted. This proposed CAES was applied to image encryption, and several tests and analyses, such as visual effect, statistical analysis, histogram, information entropy analysis, and connected component analysis, were all performed to determine the capability and security of the CAES algorithm. Recently, as a

Conclusions and Future Works
In this paper, by integrating the random property of chaotic signals with synchronization control, the synchronized random key generator was established. Then, a novel modified CAES cryptosystem with the proposed dynamic random keys was presented. Unlike the traditional AES algorithm, the traditional AES static key was replaced by the proposed synchronized dynamic random keys, not necessary to be saved or exchanged in advance, so that the security could be prompted. This proposed CAES was applied to image encryption, and several tests and analyses, such as visual effect, statistical analysis, histogram, information entropy analysis, and connected component analysis, were all performed to determine the capability and security of the CAES algorithm. Recently, as a result of the impact of COVID-19 and to reduce the risk of infection caused by conversation contact, people have become accustomed to using wireless networks for communication. In addition, with the ultra-high computing speed of quantum computers, the existing encryption algorithms might be able to be cracked. Therefore, knowing how to ensure the security of data exchange is a very important issue. In the future, the robustness and synchronization speed of the proposed random key generators should be further discussed and promoted, such that the chaos-based cryptosystem could be applied in order to ensure the communication security of video/audio streaming in the network environment.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest:
The authors declare no conflict of interest.