Chaotic Image Encryption Algorithm Based on Fractional Order Scrambling Wavelet Transform and 3D Cyclic Displacement Operation

In order to deal with the security of image information, this paper proposes a chaotic image encryption method based on fractional wavelet transform. First, fractional wavelet transform is applied to preprocess the image as a signal. After three-level fractional wavelet decomposition, high-frequency components are obtained with low frequency components. Second, the hash function is used to generate the initial value of the chaotic system for the plaintext operation, and the initial value is brought into the Lorenz system to obtain the chaotic sequence, the chaotic sequence is scrambled for the high frequency components, and the 3D cyclic displacement scrambling is performed for the low frequency components. The high-frequency components and low-frequency components are reconstructed by fractional wavelet transform and restored to the final scrambled image. Last, in order to further deepen the encryption effect, cyclic displacement diffusion is performed on the scrambled image to obtain the final encrypted image. Experimental and simulation results show that the algorithm proposed in this paper has excellent encryption effect and can resist common attack methods.


I. INTRODUCTION
Image encryption is widely used in secure communication, information hiding and digital watermarking. The research on image encryption technology has high theoretical and practical significance [1]- [3]. With the development of the digital age, the amount of information that needs to be stored, transmitted and processed increases exponentially, and the mode of information transmission is also changing constantly. As an important information carrier, digital images are increasingly widely used in the field of information transmission [4]. Current research on digital image encryption method basic is divided into two categories: space transform encryption scheme and transform domain encryption method [5], more popular scheme mainly concentrated in the frequency domain or airspace simple encryption, the The associate editor coordinating the review of this manuscript and approving it for publication was Sun Junwei . method has some deficiencies, the characteristics of image information is a huge amount of information, and need the information representation and transmission, image the difficulty lies in the transmission process of information security assurance image data and the security of [6], [7]. How to store and transmit image data safely and effectively has become an urgent need in modern information society. The original image data has strong correlation and a lot of redundancy [8]. Eliminating this redundancy can save storage space. In most images, there is a large correlation between adjacent pixels, namely, spatial redundancy. There is a great correlation between the adjacent frames of sequential images, namely, time redundancy. The method to eliminate the redundancy is called image compression, which improves the compression ratio of image encryption, thus improving the image encryption efficiency and facilitating the transmission and storage of ciphertext information. Reference [9] proposes to add confusion and diffusion of chaos to existing alternatives to completely eliminate correlation. First, TD-ERCS chaotic map is used to displace pixels, which breaks the autocorrelation of data. The final encrypted image is generated by using the logistic map to carry out bite-or to the diffusion. At present, almost all image encryption algorithms are single image encryption system, image encryption efficiency is low, and image compression encryption algorithms based on high-resolution wavelet analysis are few. Wavelet transform coding provides multi-scale and multi-resolution image transformation, which can effectively eliminate statistical redundancy and visual redundancy. It plays an important role in the field of image compression and coding. Image compression and encryption using wavelet transform can achieve the compression ratio of any compressed image theoretically, and it is also a relatively simple practice to achieve such a goal. Of course, any method is different from other methods in terms of its advantages, but also has some defects. Therefore, in order to achieve good image compression and encryption, a variety of technologies are used. Such image compression and encryption based on wavelet analysis is no exception. In most cases, wavelet analysis needs to be dynamically combined with other related technologies, such as dynamic S-box technology, neural network technology [10], Fourier transform and chaotic system [11], etc., Reference [12] proposes a new encryption scheme, which can transform the plaintext image pixels into visually meaningful encrypted images. In this scheme, confusion and diffusion of plaintext image are introduced by using entangled logic mapping, and finally the final ciphertext image is realized by using grayscale replacement Box (S-box). Reference [13] proposes to divide the plaintext image into several blocks, and then calculate the correlation coefficient of each block. Blocks with maximum relational values are processed pixel by pixel using random Numbers generated by skew maps based on predefined thresholds. Finally, two random sequences generated by TD-ERCS chaotic map are used to sort the whole image, so as to achieve more perfect results [14]. In view of this, in view of the large amount of image data and the characteristics of the high redundancy, proves that the traditional encryption algorithms are not suitable for image encryption, we put forward the improvement scheme based on wavelet and chaos theory, we select [15], [16] lorzen chaotic system combined with the wavelet analysis to image compression encryption research, image encryption based on chaos and wavelet technology not only has important theoretical value, but also has a certain application value.
Image coding technology began in the late 1940s, and early classical coding theories such as entropy coding, predictive coding and transform coding originated from Shannon's information theory. The starting point of these coding theories is to eliminate statistical redundant information in images or visual redundancy, structural redundancy and knowledge redundancy in image data [17]. Since the 1990s, wavelet transform coding has provided a multiscale and multi-resolution image transformation method, and can effectively eliminate statistical redundancy and visual redundancy, so wavelet change analysis method has begun to take an important position in the field of image compression coding. Since the first information hiding workshop was held in Cambridge University in 1996, the research on digital image encryption technology has made great progress. The second and third International Conferences on Information Hiding held in 1998 and 1999 in the United States and Germany led to a growing number of scholars working on image encryption. The fourth International Conference on Information Hiding was held in Pittsburgh in April 2001, and the fifth and sixth International Conferences on Information Hiding were held in The Netherlands and Canada in October 2002 and May 2004, respectively [18]. Research on image encryption, wavelet theory and chaos theory has become a hot research topic. At the same time, image encryption method is implemented in the frequency domain, which combines chaos theory and wavelet transform. Compressed and encrypted images can also be used as preprocessing before embedding digital watermarks. All of these have become research hotspots.
In recent years, the research on chaotic system is more and more in-depth. The research on stability, controllability and synchronous control of chaotic systems [19] has become extremely hot. In the field of image encryption, chaotic encryption is a typical representative of spatial transformation and release methods [20]- [23]. Due to the initial value of chaotic system Sensitivity, pseudo-randomness and other good properties have been widely used in the field of encryption [24]- [27], Diaconu [28] proposed a chaotic image encryption method of pixel circular arrangement, Hua et al. [29] proposed a new two Dimensional sine logic modulation mapping is used for encryption. Sharma et al. [30] proposed a double random phase image coding encryption technology based on Lorenz system.
The typical representative method of the transform domain is the wavelet transform method [31]- [34]. The wavelet transform is very widely used in signal processing because of its good positioning characteristics and detailed processing functions [29], [35]- [37]. The wavelet transform is used in image processing [38]- [42]. The application in is also extremely popular. Tedmori [43] uses haar wavelet as the basis function to introduce a chaotic encryption method based on wavelet transform. Similarly, Ajish [44] discussed a new image encryption algorithm based on wavelet transform and dynamic S-box AES algorithm. Liu et al. [45] proposed an image encryption algorithm based on fractional Fourier transform domains. The combined use of wavelet transform and chaotic system can greatly enhance the encryption effect, and the encryption result has higher stability and resistance to attack [46].
In this paper, the image is preprocessed by the fractional wavelet transform method. The fractional wavelet has better image processing effects and can reflect the image signal information in the time domain and the frequency domain. In this paper, combined with chaotic system, cyclic displacement scrambling and diffusion are designed. Scrambling is VOLUME 8, 2020 based on a 3D cyclic displacement mechanism. Diffusion is based on sequential cyclic displacement. The image after fractional wavelet transform is scrambled by cyclic displacement. After diffusion, the final encrypted image is obtained. The structure of this paper is as follows: The second part introduces the pre-theoretical method of the encryption algorithm, the third part introduces the specific model of the circular displacement operation we proposed, the fourth part introduces the encryption and decryption process, and the fifth part is the simulation result, the sixth part is the security analysis, the seventh part is the conclusion.

II. PRELIMINARY WORKS A. KEY GENERATION
Step 1: Generate Hash array {H} with SHA-512 Hash algorithm and plaintext image [47], [48], extract 12 elements from {H} as the initial value and control parameters of Lorenz chaotic map, Every three keys form a team of key groups, and the parameter calculation process is shown in Eq. (1): (1) Step 2: The initial value composed of the key group generates 12 columns of chaotic sequences through the Lorenz system, and every three sequences are a group, the calculation process is shown in Eq. (2).
Hash algorithms can compress messages of any length to a fixed length. Often used in cryptographic systems to defend against violent attacks. As Hash functions such as sha-1 are broken, some encryption algorithms are no longer safe. At present, the encryption algorithm based on sha-2 design has not been cracked. The second part of the key selects the SHA-512 hash algorithm in SHA-2 and combines the algorithm with the plaintext image P to obtain a hash array of length of 128-bit. Even if you only change the 1-bit value of P pixel, the resulting hash array is completely different, and the complexity is 2 512 , which is sufficient to resist violent attacks and achieve the purpose of one density at a time. For the convenience of the following description, here is given a 128-bit hexadecimal Hash array, denoted as H :

B. CHAOTIC SYSTEM
The Lorenz system is a strange attractor obtained by American meteorologists during the study of convection experiments, and then a dynamic system evolved [49]. During the rise of the system energy, the fluid orbit enters from the right outer edge, from inside to outside, it jumps to the left in a random state. During the entire change process, all behaviors are acyclic and unpredictable, and the orbits never intersect themselves, thus forming a chaotic behavior. As a three-dimensional chaotic system, Lorenz has a chaotic sequence that is more complex and unpredictable than a lower-dimensional chaotic system, and has a better pseudo-random effect. The three-dimensional system has more parameters and initial values, and has a greater secret Key space. The dynamic equation of Lorenz chaotic system is: where a, b, and c are the parameters of the system, and the range of values is any real number. If a = 10, b = 8/3, and c = 28, the system is in the best state of chaos, and the difference equation of the system can be expressed as: According to the difference equation, given initial values, x 0 , y 0 , z 0 , by solving the equation, three different chaotic sequences can be obtained simultaneously.
The phase diagram of the strange attractor of the Lorenz system is shown in the Fig. 1: The sequence generated by the Lorenz system is a random sequence of chaos. During use, we need to standardize the chaotic sequence in order to be more suitable for the encryption system. The standardization steps are divided into two steps: Step 1: In order to make the irregularity of the chaotic sequence more fully, we shift the sequence worth decimal point backward by n bits to obtain the sequence value containing integer bits.
Step 2: Since the pixel value range of the image is between (0,255), we map the sequence value into this range, and take the integer closest to the original value as the new sequence value.
The mathematical formula of the standardized model of chaotic sequence is:

C. FRACTIONAL WAVELET TRANSFORM
Mendlovic and Zalevsky first proposed the definition of fractional wavelet transform in 1997 [23]. The combination of fractional Fourier transform and traditional wavelet transform into a fractional discrete wavelet transform, FRWT definition formula is: where B P (t, t ) is: The fractional order change domain is: (9) in Eq. (9), h * a mn b (x , y ) represents the scale and translation function in the process of wavelet transform, a mn = (a m , a n ) represents the discrete scale,b = (b x , b y ) represents the displacement scale, and p = [p 1 , p 2 ] represents the order of wavelet transform.
Fractional wavelet reconstruction formula is: The steps of performing a two-dimensional fractional wavelet transform on the initial image are: First performing a two-dimensional FRFT change on the matrix, the order is p 1 , p 2 , and then performing a two-dimensional wavelet decomposition to obtain the transformed image.

III. CYCLIC DISPLACEMENT OPERATION A. 3D CYCLIC DISPLACEMENT SCRAMBLING
Treat the image as a 3D stereo model. Each pixel of the image matrix contains a pair of position information and a pixel value information. The three information form a spatial position. The position information of the matrix is scrambled by rotating the rows and columns of the matrix. Specific operations Proceed as follows: Step 1: To obtain the chaotic sequence Z , first normalize the chaotic sequence according to Eq. (11): Step 2: The step size following the back displacement is given by Eq. (12): H is a mapping matrix, and the position information and pixel values of the original image matrix are regarded as three-dimensional position information, namely: where p is the image matrix, i is the row coordinate, j is the column coordinate, k is the pixel information, and the scrambling formula is: VOLUME 8, 2020  Taking the 3 × 3 matrix as an example, set the cyclic displacement variable to [2,1,2], the scrambling process is shown in the Fig. 2 and Fig. 3: The steps of cyclic displacement diffusion are: Step 1: For the chaotic sequence x 4 , y 4 , standardize the sequence: Step 2: For M ×N images, convert the intermediate ciphertext matrix into a list of vectors as shown in the Eq. (16): Step 3: To diffuse the intermediate ciphertext vector, the horizontal circular diffusion formula is: (17) convert the diffused vector P into a matrix of M × N , and convert it vertically into a column of vectors E, and perform vertical circular diffusion on the vector. The diffusion formula is:

IV. IMAGE ENCRYPTION STRATEGY
The algorithm in this paper consists of two parts: Scramble and diffusion. Next, the algorithm is described in detail.
Image scrambling is used to change the position of image pixels. However, scrambling only completes the change of the pixel position of the image. In order to further deepen the effect of encryption, it is worth to modify the pixel using the diffusion method to enhance the encryption effect of the image. In this paper, M × M images are taken as an example. The encryption process is as follows: Step 1. The plaintext image is converted into a pixel matrix P, and the matrix P is subjected to a fractional wavelet transform, and the transformed components are retained. The transform result is: Step 2. In Eq. (19), CA is the low-frequency component, and CH , CV , and CD are the high-frequency components. This is the first-order wavelet transform. The low-frequency component is used as the new matrix to continue the wavelet transform, and the matrix is subjected to three-level wavelet decomposition. The result is: 11 , CH 11 , CV 11 , CD 11 ] = FRWT (CA 1 ). (20) Finally, one low-frequency component CA 11 and nine high-frequency components CH , CV , CD, CH 1 CV 1 , CD 1 , CH 11 , CV 11 , CD 11 are obtained. The wavelet decomposition results are shown in the Fig. 4.
Step 3. Chaotic scrambling of all high-frequency components during the three-level wavelet transform of matrix P, selecting the three pairs of key groups in Eq. (1), and iterating the Lorenz system to obtain three sets of chaotic sequences: x 1 , y 1 , z 1 ; x 2 , y 2 , z 2 ; x 3 , y 3 , z 3 , converting all high-frequency components into one row Vector, use chaotic sequence to chaotically scramble the high frequency components of matrix P, first sort the chaotic sequence:      C 1 = sort(x 1 ); C 2 = sort(y 1 ); C 3 = sort(z 1 ) C 11 = sort(x 2 ); C 21 = sort(y 2 ); C 31 = sort(z 2 ) 208722 VOLUME 8, 2020 use the sorted chaotic sequence for chaotic scrambling of high frequency components: 11 = CH (C 12 (i)); CD 11 = CD 11 (C 22 (i)); CV 11 = CV 11 (C 32 (i)) (22) the scrambled high-frequency component sequence is converted into a matrix, which is the scrambled high-frequency component matrix: Step 4. Perform a 3D cyclic displacement scrambling operation on the low-frequency component CA 11 of the third-order fractional wavelet decomposition of the matrix P. Use the key group: x 4 (0) = Key 10 , y 4 (0) = Key 11 , z 4 (0) = Key 12 to iterate the Lorenz system to obtain the chaotic sequence x 4 , y 4 , z 4 . Use the sequence z 4 to set CA 11 according to Eqs. (9)-(11) scrambling operation: Step 5. The low-frequency component and the highfrequency component of the scrambled wavelet are reconstructed. The reconstruction process is: FF 3 is the final scrambled image.
Step 6. Using the chaotic sequence x 4 , y 4 , the scrambled image FF 3 is subjected to cyclic displacement diffusion according to Eqs. (12) - (15), and the final encrypted image ee is output.
The encryption flowchart as shown in Fig. 5. The decryption algorithm is the inverse operation of the encryption algorithm.

V. EXPERIMENTAL RESULTS
In order to verify the effectiveness of the encryption strategy proposed in section 4, this paper simulates the algorithm through multiple images, and the image data used is the image in USC-SIPI 'Miscellaneous' image dataset. The simulation software is MATLAB R2017a, the computer is configured with 8GB RAM, Intel (R) Core (TM)i5 Duo CPU e6550@2. 33 GHz, and the operating system is Windows 10.

VI. SECURITY ANALYSIS
A good encryption algorithm should be able to withstand a variety of attacks, such as violent attacks, statistical attacks, differential attacks, selective plaintext attacks, and so on. In order to verify the security of this algorithm, this section will conduct theoretical analysis and numerical simulation of this algorithm from these aspects, and make a comparison with. VOLUME 8, 2020

A. KEY SPACE ANALYSIS
The key space contains the keys used by the algorithm during encryption. The size of the key space directly determines the ability of an encryption algorithm to resist violent attacks. The valid keys of this algorithm are as follows: the number of iterations TNI ; Random key sequence Key with size 12, and key space 2 12 ; The Hash array H is generated by SHA-512 Hash algorithm from the plaintext image, and the key space is 2 512 ; If the precision of the computer is 10 −14 , then the key space of this algorithm is 2 12 × 2 512 × 10 14×6 , which is much larger than 2 100 [50] required by the cryptographic system, so the algorithm in this paper can resist violent attacks.

B. KEY SENSITIVITY ANALYSIS
If a cryptographic system is highly sensitive to keys, it is a good key system. We take key Key 1 as an example to test the sensitivity of image ''Lena''. The experimental results are shown in Fig. 6. When other keys remain unchanged, Fig. 7 (a) is the ciphertext image generated according to key x 1 (0); If the key x 1 (0) is changed from 0.3801 to 0.38010000000001, the generated ciphertext image is shown in Fig. 7(b). Fig. 7(c) is the difference between Fig. 7(a) and Fig. 7(b). Fig. 7(d) is the decrypted image obtained by decrypting Fig. 7(a) with the modified key x 1 (0); When the key changes slightly, the encryption image generated by the algorithm in this paper is completely different from the encryption image generated based on the original key in Fig. 7(b). It can be seen that the algorithm in this paper is extremely sensitive to the key.

C. HISTOGRAM ANALYSIS
An important measure of a cryptographic algorithm's resistance to statistical analysis is to calculate the histogram of ciphertext images. Histogram is used to describe the pixel   value distribution of the image. Statistical analysis attacks are easy to obtain a certain amount of information from images with uneven pixel distribution. If the histogram of the image is flat, that is, the pixel distribution is even, it is difficult to get information from statistical analysis attacks. In other words, the flatter the histogram of ciphertext images generated by a good encryption strategy, the better.   It can be seen that the histogram pixel distribution of the plaintext image is not uniform. On the contrary, histogram pixel distribution of ciphertext image is more uniform. Therefore, the ciphertext image obtained by the encryption algorithm in this paper can resist the attack of statistical analysis.

D. χ 2 TEST
In order to analyze the distribution situation of ciphertext images more accurately, we exploit the chi-square test to obtain the quantitative numerical results, avoiding visual spoofing. In order to describe the experimental results more intuitively, we use the formula given in [50] for comparative analysis as Eq. (26).
where, q i represents the number of times the pixel value i appears in the image, and q is defined as: Table 1 gives the chi-square test results of all images in the experimental set which shows that the chi-square values are all less than 293.2483 which is the limit value of 5% significance. Therefore, the proposed cryptosystem exhibits a good performance to withstand statistical attacks.

E. CORRELATION ANALYSIS
The high correlation of adjacent pixels indicates that the plaintext image is vulnerable to statistical analysis. Therefore, it is necessary to reduce the correlation between adjacent pixels. A good encryption algorithm can reduce the correlation between adjacent pixels. We randomly selected 10000 pixels from the plaintext and ciphertext images of the grayscale image ''Lena'', and calculated the horizontal, vertical and diagonal correlations of adjacent pixels by using Eq. (28) and Eq. (29) [51], respectively. The calculated results are shown in Fig. 9. here: In addition, the correlation coefficients of experimental images (plaintext and ciphertext) during the experiment and the comparison results with the correlation coefficients of ciphertext images in algorithm [29], [35]- [37] and algorithm [52], [53] are also given in Fig. 10, Table 2 and Table 3. The correlation between plaintext and ciphertext images is strong. However, the correlation between pixels of ciphertext images encrypted by the algorithm in this paper is much lower than the correlation coefficients of plaintext images. It can be seen from the comparison results between the algorithm in this paper and the other algorithm, that all the correlation coefficients of the algorithm in this paper are basically smaller than other algorithms. Therefore, this algorithm has a good ability to resist statistical analysis attacks.

F. INFORMATION ENTROPY ANALYSIS 1) GLOBAL INFORMATION ENTROPY
Information entropy is a commonly used index to reflect the randomness of information, and its calculation is 208730 VOLUME 8, 2020 Eq. (30). Theoretically, the possibility of information leakage decreases gradually as the information entropy approaches 8. Table 4 lists the information entropy of experimental images during the experiment. Table 5 shows The gray image entropy comparison results of this algorithm and algorithm [32]- [35]. The color image and binary image are compared with algorithm [49], [50] respectively.
here p(s i ) is the probability of s i happening.
The information entropy of the encrypted ciphertext image should be close to 8. As can be seen from Table 4, in the test images, the information entropy of ciphertext images obtained by the algorithm in this paper is close to 8. As can be seen from Table 5, the information entropy of both the algorithm in this paper and the algorithm [29], [35]- [37] is close to 8. In other words, the ciphertext images obtained in this paper are less likely to leak information and have better ability to resist statistical analysis attacks [45]- [47].

2) LOCAL SHANNON ENTROPY
For better analysis, we give experimental comparisons of local information entropy in this part. The calculation of local information entropy is Eq. (31).
where, H (S T B , L) is the information entropy of nonoverlapping image blocks S T B , k and T B denote the number here W and H represent the width and height of the image respectively; c 1 represents the original plaintext image, and c 2 represents the plaintext image that only changes the 1-bit pixel value. If c 1 (i, j) = c 2 (i, j), then D(i, j) = 1, otherwise D(i, j) = 0. The critical value of NPCR with the significance level α is marked as N * α , which can be calculated according to the following formula.
where, α is the significance level which is set as 0.05 in our experiment, −1 (α) is the inverse cumulative density function of normal distribution N (0, 1). The lower bound of UACI with the significance level α is marked as U * − α and the upper bound is marked as U * + α , which can be calculated according to the following formulas.
where, and L is grey level, M and N represent the sizes of the image.
In theory, if NPCR and UACI values are close to 99.6093% and 33.4635%, respectively, the results are good. Under the condition that the key remains unchanged, c 1 and c 2 are encrypted by the encryption algorithm in this paper. Then the NPCR and UACI of the two ciphertext images were calculated. Table 7 and Table 8 respectively show the NPCR value and UACI value obtained by this algorithm and the comparison results with other algorithms, and the gray image is compared with the algorithm [51]- [56].Color image and binary image are compared with algorithms [49], [50]. It can be seen that, compared with other algorithms, most of the NPCR and UACI values obtained by this algorithm are close to 99.6093% and 33.4635%.

H. ROBUSTNESS ANALYSIS
Robustness is an important index to measure the antiinterference ability of a cryptographic system. We test the robustness of this algorithm by noise attack and block attack respectively.
Noise interference is an important problem in practical communication. The common noise types include gaussian noise, pepper and salt noise, etc. Compared with other noises, salt and pepper noise has more influence on ciphertext image. Therefore, this paper mainly considers the effect of adding salt and pepper noise to the plaintext image. Under the condition of keeping the key unchanged, we added the pepper and salt noise to the plaintext image with different intensity, and encrypted and decrypted the image after adding the noise respectively with the algorithm in this paper. Fig. 11 shows the encrypted and decrypted images of the plaintext image ''Lena'' with pepper and salt noise intensity of 0. 01, 0. 05 and 0. 1, respectively. It can be seen that, even if the noise intensity reaches 0. 1, the plaintext image can still be distinguished from the decrypted image.
Therefore, the algorithm in this paper has a good ability to resist noise attack.
Cropping attacks is another test of robustness. The experimental method is to first block the information of different sizes of an encrypted ciphertext image, and then try to decrypt it with the algorithm. Obviously, with the increasing of blocking information, the quality of decrypted images gradually decreases. Fig. 12 shows the decrypted images obtained by blocking 1.5625%, 6.25%, 25% and color image 25% of the ciphertext image ''Lena'' respectively. It can be seen that the main information of the plaintext image ''Lena'' can still be distinguished from the decrypted image. Therefore, the algorithm in this paper has a good ability to resist blocking attacks. Therefore, the proposed algorithm is robust.

I. TIME COMPLEXITY ANALYSIS
As is known to all, speed is an important index of encryption algorithm, which not only verifies its security, but also verifies its feasibility. Encryption speed can reflect the utility of the algorithm. In this algorithm, O (MN ) is the time complexity, and N is the width and height of the plane image. In the scrambling phase, we need order O (1/4 MN ) time complexity. In the diffusion stage, the time complexity is O (3/4 MN ), so the total time complexity of the proposed encryption algorithm is O (MN ). In general, the encryption speed depends on CPU, GPU, compiler, etc. MATLAB is a more appropriate simulation tool, but its simulation efficiency is low. We propose a more reliable image encryption algorithm rather than a concrete implementation, which does not violate the original intention of this paper. In this paper, all image encryption algorithms are implemented using MAT-LAB R2017a and run on a computer with 8.0GB memory and a CPU of Intel(R) Core (TM) i5-2.30hz. The comparison of time complexity between gray image, color image and binary image is shown in Table 9.

VII. CONCLUSION
In this paper, a chaotic image encryption algorithm is proposed based on the fractional wavelet transform. First, the image is preprocessed based on the fractional wavelet transform, and the image after three-level wavelet decomposition is divided into two types of components, highfrequency components and low-frequency components. To scramble two types of components, this paper proposes a new 3D cyclic displacement scrambling method, which is used in conjunction with chaos scrambling to scramble the two types of components. After scrambling is completed, in order to deepen the effect of encryption, pixel diffusion of the image matrix is performed based on the cyclic shift operation, and finally an encrypted image is obtained. In this paper, the Lorenz system is used to ensure the chaotic characteristics of the chaotic sequence, and the initial value of the chaotic system is calculated according to the SHA-512 hash algorithm and the plain text image. Compared with other algorithms, it can be found that the algorithm in this paper can achieve very good encryption effects and can resist common attacks. The innovation of this paper lies in applying the fractional wavelet to the preprocessing of the image, treating the image as a type of signal, and proposing a new scrambling and diffusion method. Scrambling and diffusion based on cyclic displacement can complete the encryption process well. The encrypted result has good security.