Color Image Encryption Using 2D Sine-Cosine Coupling Map

In this paper, a new two-dimensional sine-cosine coupling chaos map(2D-SCCM) is proposed. The performance of the proposed two-dimensional chaotic system is analysed by using trajectory distribution maps, Lyapunov exponents, sample entropy and sequence sensitivity, etc. The results show that the 2D-SCCM has better randomness and ergodicity, as well as a wider hyperchaotic range than the existing partial two-dimensional chaos maps. To verify its application in practice, a 2D-SCCM-based color image encryption algorithm is proposed. First, the plain image is used to combine with hash function to generate the key. Then, we construct S-Boxes using the random sequence generated by 2D-SCCM combined with Arnold map. Finally, a color image encryption algorithm is proposed by using the constructed S-Boxes and the chaotic map combined with hash function. Experimental simulations and security tests show that the proposed encryption algorithm has high encryption efficiency and strong security, and can effectively protect images from various attacks.


I. INTRODUCTION
With the rapid development of modern network technology, the use of images for information transmission has become more and more frequent. The security of images has received extensive attention from researchers because of the widespread use of images in the Internet. In order to protect the information of images, many schemes have been proposed by researchers, including data hiding [1], watermarking [2], and image encryption [3]- [5]. Among all the schemes, image encryption is one of the most straightforward and effective schemes [4].
Due to the large redundancy and strong correlation between image pixels, image encryption is different from text encryption [5]. Traditional data encryption schemes mainly include Advanced Encryption Standard (AES) and Data Encryption Standard (DES). When they are applied to image encryption, it takes a lot of time resulting in low encryption efficiency [4]. Therefore, researchers have proposed many different image encryption techniques, which The associate editor coordinating the review of this manuscript and approving it for publication was Jun Wang . contain image filtering [3], DNA coding [5], frequency domain transform [6], elliptic curves [7], compressive sensing [8], S-Box [9], fractional Fourier transform [10], neural networks [11], chaotic systems [12]- [14], etc. However, among these technologies, because of the special nature of chaotic systems, cryptography based on chaotic systems has become one of the most popular cryptographic schemes.
There are many unique characteristics of chaotic systems, including high sensitivity to initial conditions and parameters, internal randomness and ergodicity, which make them suitable for image encryption. Since J. Fridrich first applied chaotic system to image encryption [14], the image encryption schemes based on chaotic systems have been developed rapidly, and various chaos-based image encryption schemes have been proposed. For example, Wu et al. proposed an image encryption algorithm based on 2D-HSM and DNA coding techniques, where they performed a diffusion operation on the image by DNA and scrambled the image by chaotic sequences [5]. Mansouri et al. proposed an image encryption algorithm based on a one-dimensional chaotic system, where they added chaotic sequences to the original image during the diffusion process to improve the security of VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the algorithm [15]. J. Arif et al. proposed an image encryption algorithm based on Logistic map and S-Box [9]. They used S-Box to perform pixel substitution in the pixel diffusion stage to enhance the security of the algorithm, but this algorithm can only be used for grayscale images. Similarly, there are many other image encryption schemes based on chaotic systems that have been proposed by researchers [16]- [19]. Although many image encryption algorithms based on chaotic systems have been proposed, some of them have been proven to be insecure [5]. Because the one-dimensional chaotic map has the characteristics of small key space and simple chaotic orbit, the attacker can easily deduce the initial conditions and chaotic orbit of the system [12]. While high-dimensional chaotic systems have relatively complex chaotic behavior and larger key space, the computational complexity and space complexity of high-dimensional chaos are much larger than those of one-dimensional chaotic systems. Thus, it is relatively difficult to implement in practical hardware, and it takes a long time. Therefore, it becomes particularly important to propose a chaotic system with complex chaotic behavior and low computational complexity [19].
In this paper, a new two-dimensional discrete chaotic map is proposed, called 2D-SCCM. On the basis of studying the classical one-dimensional discrete chaotic map, a new twodimensional discrete chaotic map is constructed by coupling the output variables of multiple one-dimensional discrete systems. At the same time, because the cosine function is introduced in the system, the output of the entire chaotic system is in the range of [−1,1]. To verify the chaotic properties of the proposed system, first the system is analyzed by using trajectory distribution diagrams, Lyapunov exponents(LE), sample entropy, and sequence sensitivity. Then the chaotic properties are compared with those of other 2D chaotic maps. Finally, the randomness of the system-generated sequences were tested by using the National Institute of Standards and Technology (NIST) statistical test tool. The results show that the proposed two-dimensional chaotic map has more complex chaotic behavior and wider hyperchaotic interval. Based on the proposed chaotic map, we further propose a color image encryption algorithm. The algorithm can be divided into two major parts: scrambling and diffusion. In the image scrambling stage, the chaotic sequences are used to perform overall scrambling. At the same time, pixels are replaced one by one by introducing S-Boxes. In the diffusion stage, XOR and the modular diffusion are used for pixel diffusion, so that the three components of the color image interact with each other to achieve better encryption performance. The experimental simulation results and security analysis show that the proposed algorithm can encrypt color images of arbitrary size and has strong resistance to attacks. This paper is organized as follows. Section II introduces the mathematical model of 2D-SCCM. Section III analyzes dynamic behavior of the 2D-SCCM. Section IV details the proposed color image encryption algorithm. Section V introduces the simulation and safety analysis. And section VI is a brief summary about the paper.

II. 2D SINE-COSINE COUPLING MAP
This section first reviews two traditional one-dimensional discrete chaotic maps, and then introduces the mathematical model of the two-dimensional sine-cosine coupling chaotic map proposed in this paper.

A. LOGISTIC MAP
Logistic map is a one-dimensional discrete chaotic map with the mathematical definition of Eq(1).

B. SINE MAP
Sine map is also a one-dimensional discrete chaotic map with the mathematical definition of Eq(2), which is derived from the sine function.

C. 2D-SCCM
The traditional one-dimensional discrete chaotic map has many disadvantages such as narrow chaotic interval and simple system structure. To solve these problems, a new twodimensional sine-cosine coupling chaotic system is proposed with the mathematical definition of Eq(3).
where α, β are both system control parameters with α = 0, β = 0. It can be seen from the mathematical definition of Eq(3) that the 2D-SCCM is mainly derived from Logistic and Sine maps. Its output signals are finally modulated with the nonlinear cosine function, which further improves the randomness of the output signals. Meanwhile, because of the boundedness of the cosine function, the final output signals x and y of the system are in the range of [−1,1].

III. BEHAVIOR ANALYSIS
In this section, the dynamical behavior of the proposed chaotic system is analyzed by chaos trajectory, Lyapunov exponent, sample entropy, and sequence sensitivity. Meanwhile, the randomness of the system output sequences are tested by using the NIST statistical test tool. In addition, we compare the relevant properties with three existing 2D discrete chaotic systems, including 2D Sine Logistic modulation map (2D-SLMM) [19], 2D Logistic-Sine-coupling map (2D-LSCM) [12], 2D hyperchaotic map (2D-HM) [17]. The 2D-SLMM has two parameters γ and η. When the control parameter η = 3, it has a better chaotic performance. Thus, it is simply defined as follow [19]: The 2D-LSCM only has one parameter θ and the definition is written as follow [12]: The 2D-HM has two parameters κ and λ. its mathematical expression is defined as follow [17]: A. CHAOS TRAJECTORY The trajectory distribution diagram of a chaotic system describes the variation of the output of the system over time. Systems with complex chaotic behavior usually have complex trajectory distribution maps. And it also occupies a larger phase space [17]. Fig.1 shows the trajectory distributions of four different 2D chaotic systems, including (a) 2D-SLMM, (b) 2D-LSCM, (c) 2D-HM, (d) 2D-SCCM. When plotting these trajectory distributions, the parameters are set that can make all corresponding chaotic maps attain a completely chaotic state [12]. At the same time, the first 5000 terms of the system output are discarded in order to make the system reach a stable chaotic state. It can be seen from the Fig.1 that the trajectory distribution diagram of 2D-SLMM only occupies a small part of the phase plane, and the trajectory distribution diagram of 2D-LSCM does not completely occupy the entire phase plane, while the trajectory distribution diagrams of 2D-HM and 2D-SCCM completely occupies the entire phase plane. In addition, the trajectory distribution of 2D-SCCM is more uniform than that of 2D-HM, indicating that the sequences generated by 2D-SCCM are more random than the other three chaotic systems.

B. LYAPUNOV EXPONENT
The initial state sensitivity is one of the main features to determine whether a dynamic system exists chaotic behavior. A good chaotic system should gradually output two distinct sequences of iterations after inputting two initial iterations with small differences. We usually use the Lyapunov Exponent (LE) [12] to characterize the sensitivity of a chaotic system. Assuming that a one-dimensional discrete chaotic system is defined by x i+1 = F(x i ), where F(x) is piecewise differentiable [12], the LE of the system is defined as LE is a numerical feature that describes the average separation rate and divergence degree of adjacent phase space trajectories [12]. A positive LE means that the system exist chaotic behavior. And the larger the value of LE is, the more complex behavior the chaotic system has. For high-dimensional chaotic systems, there are generally several different LE values in the system. When the system has two or more positive LE values, we call this system a hyperchaotic system. Compared with the general chaotic system, the hyperchaotic system has more complex chaotic behavior.
A two-dimensional chaotic system usually has two LE values, and Fig.2 shows the LE of different two-dimensional chaotic systems under their respective control parameters. When the parameter range is set to [0,1], the LE values of each system behave as follows: the 2D-SLMM shows hyperchaotic behavior when γ ∈ [0.85, 1], the 2D-LSCM shows hyperchaotic behavior when θ ∈ [0.57, 1], the 2D-HM shows hyperchaotic behavior when λ ∈ [0.2, 0.34] or λ ∈ [0.36, 0.58], and the 2D-SCCM shows hyperchaotic behavior when α ∈ [0. 28,1] or β ∈ [0. 15,1]. The result shows that the 2D-SCCM has a wider range of hyperchaotic intervals compared with the other three chaotic maps. In order to better compare the magnitude of LE, we compare the two LE values with the four chaotic systems under certain parameters in Fig.3. The results show that the LE values of 2D-SCCM are larger than the other three existing chaotic systems in a large interval, which further indicates that the proposed system have more complex chaotic behavior.

C. SAMPLE ENTROPY
Sample Entropy (SE) is a new evaluation method for testing the complexity of time series based on approximate entropy [22], which is used to describe the output similarity of dynamic systems. The SE of a given time series where m is the reconstruction dimension and r is the calculated threshold, which we usually set m = 2, r = 0.2 × std (the std is the standard deviation of the tested time series).
The K 1 represents the number of vectors whose Chebyshev distance is less than or equal to r between Z mC1 (i) and Z mC1 (j), and the K 2 represents the number of vectors whose Chebyshev distance is less than or equal to r between Z m (i) and Z m (j), where Z m (i) = {z i , z i+1 , · · · , z i+m−1 }. The larger the calculated SE value, the lower the level of regularity of the system and the more complex the chaotic behavior of the system [22]. Fig.4 shows the comparison of SE tests for Sine, Logistic and four 2D chaotic systems under different control parameter conditions. It can be seen that the SE values of the proposed 2D-SCCM system are larger than those of the other five chaotic systems, indicating that the proposed chaotic system has more complex chaotic characteristics.

D. SEQUENCE SENSITIVITY ANALYSIS
A good chaotic system should be extremely sensitive to changes in the initial values. In other words, if the system's inputs are slightly different, its output sequence should be significantly different. In order to test the sensitivity of the system, we input initial values with a difference of 10 −15 , and the changes of the output sequences are shown in Fig.5. They can be seen that the system's outputs are significantly different after 14 iterations, indicating that our proposed system is extremely sensitive to the initial values.

E. NIST
To further investigate the randomness of the system output sequences, we used the National Institute of Standards and Technology (NIST) SP800-22 test tool [24]. The NIST statistical test tool includes 15 statistical tests. It is used for evaluating the randomness of the sequences generated by 2D-SCCM [34]. After the NIST statistical test tool finishes testing, each test item will generate a corresponding test value P ∈ [0, 1]. And when the corresponding generated P-value is greater than 0.01, it indicates passing the test item. When all test items pass, it indicates that the sequence has strong randomness, while a larger P-value indicates stronger randomness [12]. The computational statistics of the NIST test are shown in Table 1. From the results, the sequences generated by the system can pass all tests of 15 items, indicating that they have strong pseudo-randomness.

IV. ENCRYPTION ALGORITHM
This section introduces a color image encryption algorithm based on 2D-SCCM combined with S-Box scrambling, and the overall encryption structure is shown in Fig.6. The hash value is generated by the hash function and plain image, and the key is updated by the hash value to generate the finally key. First, the chaotic sequences generated by 2D-SCCM are used to diffuse the three color components of the original image to change the pixel values. Then, the sequences are used to generate S-Boxes combined with Arnold mapping to confuse the diffused image, and the confusing method can change the position and value of the image pixels at the same time. Finally, combined with the hash value and chaotic sequences, the scrambled image is diffused again to improve  the security performance of image encryption. The specific encryption steps are described as follows.

A. KEY GENERATION
A separate system key increases the possibility of the system being attacked to a chosen plaintext attack. The hash value obtained from the plain image is used to update the initial key, which reduces the possibility of the system being attacked by the chosen plaintext. Here we choose the SHA-256 function, the generated hash value is divide into 64 blocks, and converted each block to a decimal number, denoted as k 1 , k 2 ,· · · ,k 64 . Then, the initial key is set to  and the set initial key is updated by using Eq (9).
where {x 1 , y 1 } is the updated initial condition and {α, β} is the updated control parameters of the chaotic system. The iter is the number of iterations of the Arnold map. The SQ is the number of iteration terms discarded in the encryption process to bring the chaotic system to full chaos. The CZ is the preset value used in taking the modal diffusion. The floor(Q) means taking the largest integer not larger than Q. The mean represents taking the average, and the mod means the modulo operation.

B. XOR-DIFFUSION
Step 1: Input a color plain image P with a size of M ×N , and decompose it into three components of R, G and B with a size of M × N , which are denoted as Pr, Pg and Pb respectively.
Step 2: Substitute {x 1 , y 1 , α, β} into 2D-SCCM, and iterate the system n = SQ+M ×N times. Then the first SQ items are VOLUME 10, 2022 discard. Finally we can get the sequence X1 and Y1 according to Eq (10).
Step 3: Let y 1 = x SQ+MN , x 1 = y SQ+MN , then substitute the updated x 1 , y 1 , α, β into the chaotic system. Iterate the system n = SQ + M × N times with discarding the first SQ items. Finally we can get the sequence X2 and Y2 according to Eq (11).
Step 4: Convert the obtained sequences X1 into matrices A1 with a size of M × N according to Eq (12). Similarly, we can get B1, A2. At this time, the element values of A1, B1, and A2 are all in the range of [0, 255], where the abs represents taking the absolute value.
Step 5: The generated A1, A2 and B1 are used to perform bitwise XOR with the three image components of the color image. The image components PR, PG and PB are obtained respectively after the first diffusion encryption.

C. CONFUSION
Generally, each pixel of natural image has strong correlation, and a good encryption algorithm is to eliminate the correlation between elements [15]. Since we only change the value of each pixel in the diffusion stage without modifying the position of the pixel, the correlation between the original pixels cannot be changed. In this section, we use a combination of chaotic sequences and S-Boxes to perform the scrambling operation on the diffused image, and the specific scrambling steps are as follows.
Step 4: Perform bit plane decomposition on the encrypted image components obtained by the diffusion in the subsection B. Then recombine the upper four bits and the lower four bits respectively. Finally obtain the decomposed image components PR1, PR2, PG1, PG2, PB1, PB2. Step 5: Use the chaotic sequence Y2 to generate the parameter H for controlling the scrambling combination according to Eq (20), in which Fig.7 shows the six ways of the set scrambling combination. The final scrambling image components WR, WG, WB are obtained according to the selected scrambling combination using Eq (21). In this paper, Eq(21) represents scrambling combination 1 , and other combinations are similarly represented.

D. MODULAR DIFFUSION
To further improve the security of the cryptosystem, we introduce the sequential mod-taking diffusion of elements after the XOR diffusion and the S-Box-based permutation. In the modtaking diffusion, each newly generated element is related to the previous element, and the elements of each color component are also related to the other components. Therefore, the mod-taking diffusion can effectively improve the security of the encryption algorithm, and the detailed steps are as follows.
Step 3: The obtained encrypted components CR, CG and CB are recombined to form the final color encrypted image C.
The decryption algorithm is the inverse of the encryption algorithm, which is not described in detail here.

V. EXPERIMENTAL SIMULATION AND SAFETY ANALYSIS
In this section, the simulation test of the proposed algorithm is carried out, and the encryption performance of the algorithm is analyzed at the same time. Our selected test images include 512*512*3 Lena, all black image, all white image and some color images from USC-SIPI [4]. Simulations are carried out on Windows10 environment using MATLAB 2020b [34]. The processor of the computer used is Intel(R) Core(TM) i5-7200 CPU @ 2.50 GHz, and the running memory of the computer is 12 GB.

A. EXPERIMENTAL SIMULATION
The essence of the image encryption algorithm is to transform the natural image to be transmitted into an unrecognizable noise-like image. Only the recipient can recover information of the original image by applying the correct key and the cipher image. Others cannot recover any usable information of the original image without the correct key even if they get the cipher image.
The simulation test results are shown Fig.9. It can be seen from Fig.9(a) that the original image has rich information. And from Fig.9(c), it can be seen that the encrypted image has no valid information. When the key is used, the decrypted image consistent with the original image can be obtained as shown in Fig.9(e). The simulation results show that our proposed algorithm can convert a rich color image into an unrecognizable ciphertext image.

B. KEY ANALYSIS 1) KEY SPACE ANALYSIS
An ideal image encryption algorithm should have as large a key space as possible so that it can resist possible brute-force exhaustive attacks. Reference [25] gives a standard for an VOLUME 10, 2022 ideal algorithm key space, which should be larger than 2 100 . The key of the encryption algorithm proposed in this paper is divided into two parts: (1) initial conditions {x 0 , y 0 } and control parameters {a 0 , b 0 }; (2) the number of chaotic sequences discards SQ. In this paper, we assume that the computational precision of the computer is 10 −15 . Thus, the key space of this paper is 10 60 × 1921 ≈ 2 209 , which is much larger than the key space standard mentioned in the reference [25].

2) KEY SENSITIVITY ANALYSIS
For a superior encryption algorithm, the key should not only have a large key space but also be extremely sensitive. In order to test the key sensitivity of our proposed algorithm, we encrypt the image with a key to obtain the ciphertext image Fig.10(b), and then decrypt to obtain the decrypted image Fig.10(e). Because the Key is divided into two parts, where the SQ is determined by the hash value. Therefore, we first change Hash to Hash+1 to test the sensitivity of the SQ in the key. Then, for another part, we add a tiny perturbation of 10 −15 to each parameter of the key respectively to form four new keys, named Key2, }.
In order to test the encryption sensitivity of the key, first we encrypt the original image with Key2 to get the cipher image Fig.10(c), then we subtract it with the cipher image obtained with Key to get Fig.10(d). The result shows that the two cipher images have a significant difference. And the mathematical analysis of the two cipher images shows that there is 99.6105% difference between the two cipher images, indicating that the key is extremely sensitive in the encryption algorithm. Meanwhile, in order to test the decryption sensitivity of the key, we decrypt the ciphertext image   It shows that the key has extremely strong decryption sensitivity. In summary, the proposed algorithm key not only has a large key space, but also has an extremely strong sensitivity.

C. STATISTICAL ANALYSIS 1) HISTOGRAM ANALYSIS
The histogram can well show the distribution of pixels in an image. Generally, the histogram of natural images has significant pixel distribution characteristics. An hacker can be able to get related information of the original image by analyzing the histogram information. Therefore, the pixels of the encrypted ciphertext image should be as evenly distributed as possible. The histogram of the original image is shown in Fig.9(b). It can be seen that the histogram distribution of the original image has obvious variability. The histogram of the ciphertext image is shown in Fig.9(d), and the results show that the histogram distribution of the encrypted image is uniform without obvious features. At the same time, in order to quantitatively analyze the distribution of image pixels, we use the chi-square test (χ 2 ) mentioned in the literature [26] to analyze the images, which is defined as where f k is the frequency of occurrence of pixel value k, f e = MN /256, and M, N are the width and height of the image pixels respectively. When the test degree is 0.05, the standard value χ 2 0.05 = 293.2478 can be obtained. When the calculated χ 2 result is less than the standard value, it means that the chi-square test passes [26]. The calculation results of χ 2 for each component of different color images are shown in Table 2. The results show that the χ 2 of all encrypted image components is smaller than the standard value, which means that the pixel value distribution of the encrypted image is uniform.

2) CORRELATION ANALYSIS
A good image encryption algorithm must be able to break the correlation about the original images, as there is generally a strong correlation between pixels in natural images. The correlation between image pixels is mainly manifested in three directions including horizontal, vertical and diagonal. In this paper, we use the method of literature [21] to measure the magnitude of correlation by calculating the autocorrelation coefficient (AC), which is defined as where X t and X t+1 are two adjacent pixel sequences, and E[•] denotes the mathematical expectation. The calculated value of the autocorrelation coefficient will fall into the range VOLUME 10, 2022  of [−1,1]. When the absolute value of AC is smaller, it indicates that the correlation between pixels is smaller. The statistical results of the correlation coefficients of each image component are shown in Table 3. The results show that the correlation coefficients about the encrypted images tend to be close to 0, which indicates that the proposed encryption algorithm can effectively reduce the correlation between pixels. Meanwhile, the comparison data of correlation under different encryption algorithms are shown in Table 4. The results show that the proposed algorithm is more destructive to the original image correlation compared to other encryption algorithms. In order to further visualize the change of correlation before and after encryption, the correlation statistics of Lena as an example are shown in Fig.11. The figures show that there are an obvious linear correlation between adjacent pixels of the original image, while the pixels of the encrypted image are uniformly distributed and almost no correlation. In summary, the proposed encryption algorithm can break the correlation between adjacent pixels of the original image.

D. INFORMATION ENTROPY
The information entropy is used to describe the uncertainty of the signal, and it can also be used to reflect average uncertainty of all pixel value [34]. For an image whose pixel value is in the range of [0, 255], the calculation expression of the information entropy(IE) is as where f (x i ) denotes the frequency of occurrence of pixel value x i . From Eq (26), it can be concluded that IE can obtain the theoretical maximum value of 8 when each pixel value has the same probability of occurrence. And the larger the calculated value is, the more uniform the pixel distribution is. The calculation results of IE values for different sizes of images are shown in Table 5. The results show that all plain images have relatively small information entropy. However, the IE values of encrypted images are all close to 8, indicating that the encrypted image pixels are distributed uniformly [18]. In addition, Table 6 shows the comparison results of the IE value for different encryption algorithms. The results show that the information entropy of the proposed algorithm is closer to the theoretical value 8, which indicates that the proposed algorithm has stronger security.

E. DIFFERENTIAL ATTACK ANALYSIS
Differential attack is a common cryptanalytic attack. Firstly, two plain images with minor differences are encrypted separately by an attacker. Then the attacker establishes an intrinsic connection between the ciphertext and the plain image by comparing the differences between the encrypted images.
Finally, the purpose of decryption is achieved in this way [4]. Therefore, a good encryption algorithm should have excellent resistance to differential attacks. In order to better analyze the ability of encryption algorithms to resist differential attacks, the Number of Pixels Change Rate (NPCR) and the Unified Average Changing  Intensity (UACI) are often used as evaluation indicators. They are defined as where C 1 and C 2 denote two different encrypted images. And they are generated by encrypting two original images whose pixels differ by only one bit. MN is the total number of image pixels. The literature [32] gives the test passing criteria in Table 7, where the ideal expectation values of NPCR and UACI are 99.6094% and 33.4635% respectively [32]. When the calculated values of both NPCR and UACI pass the test standard, the closer the calculation result is to the theoretical value, the better the encryption effect and the stronger the ability to resist differential attacks. In order to calculate without loss of generality, in this paper, we randomly perform 100 times NPCR and UACI analyses on the test images, and then we have the average value of the 100 times data. The final calculation results are shown in Table 8, which shows that the tested images can pass the test and are close to the theoretical value. In addition, the comparison of NPCR and UACI for different encryption algorithms using Lena images as an example is shown in Table 9. And the results show that the proposed encryption algorithm is closer to the theoretical value, so our algorithm is more resistant to differential attacks.

F. NOISE ATTACK ANALYSIS
Because all transmission channels exist noise, images are susceptible to noise contamination during transmission. A good encryption algorithm should be able to recover as much of the plain image information as possible in a noise-contaminated ciphertext image. In order to test the ability of our proposed algorithm to resist noise attack, we first add different degrees of noise to the ciphertext image, then use the correct key to decrypt the ciphertext image, finally observe the recovery degree of the image under different noise pollution. The salt and pepper noise with density of 1%, 5%, 10%, 20% is added to the cipher image respectively, and the decryption results are shown in Fig.12. The results show that even if the added noise density reaches 20%, the proposed algorithm can still recover a large amount of image information, indicating that our algorithm has a strong ability to resist noise attack.

G. DATA LOSS ATTACK ANALYSIS
In the process of information transmission, there is a high possibility of data loss. At the same time, there are also attackers who want to make the correct image information unavailable to the receiver by maliciously interfering with the normal transmission of the information. It is desirable for the receiver to recover the original image as much as possible in case of   partial data loss. Therefore, in order to test the ability of our proposed algorithm to cope with information loss, we first perform the following four processes on the cipher image: (a) 1/16 of the data is lost in all components of the cipher image; (b) 1/8 of the data is lost in all components of the cipher image; (c) 1/4 of the data is lost in all components of the cipher image; (d) 1/2 of the data is lost in all components of the cipher image. Then, the cipher images with data loss are decrypted using the correct key. Finally the results are shown in Fig.13. The results show that most of the information of the image can still be recovered when 1/2 of the cipher image is lost. Thus, the proposed algorithm has the ability to resist information loss.

H. SPEED ANALYSIS
In order to better analyze the encryption and decryption speed of the proposed algorithm, the speed of different encryption algorithms is shown in Table 10. The results show that the encryption and decryption speed of the proposed algorithm is better than these algorithms. Therefore, the proposed algorithm has high encryption efficiency.

VI. CONCLUSION
In this paper, we propose a new two-dimensional discrete chaotic system, called 2D-SCCM, which is designed based on the study of the Logistic map and the cosine function. In order to better evaluate the chaotic behavior of the proposed system, we use various testing methods, including trajectory distribution map, Lyapunov exponent, sample entropy, sequence sensitivity. And the randomness of the generated time series was examined by using the NIST test tool. Experimental results and comparative analyses show that the 2D-SCCM has a wider hyperchaotic interval, more complex chaotic behavior, and better ergodicity compared to some existing chaotic systems. Based on the 2D-SCCM, we further propose a new color image encryption algorithm by combining the hash function. The algorithm consists of four main parts: key update, XOR-diffusion, pixel scrambling and substitution combined with S-Box, and Modular diffusion. Firstly, the updated key associates with the plain image, which makes the whole algorithm effective against the selective plaintext attack. Then, the XOR-diffusion and the Modular diffusion can effectively change the value of the pixels. Finally, in the pixel scrambling and substitution combined with S-Box stage, not only the value of the pixels can be changed, but also the position of the pixels, thereby breaking the correlation between the plain image pixels. Experimental simulations show that the proposed encryption algorithm can effectively convert natural images into unrecognizable noiselike images. The security analyses show that the algorithm is able to resist various common cryptanalysis attacks. JINYUAN CHEN is currently pursuing the M.S. degree with the College of Engineering, Huaqiao University, Quanzhou, China. His research interests include information security, chaotic synchronization and control, and the application of chaotic cryptography in image processing.
ZHONGMING HUANG is currently pursuing the M.S. degree with the College of Engineering, Huaqiao University, Quanzhou, China. His research interests include information security and image privacy protection. VOLUME 10, 2022