Remote sensing image and multi-type image joint encryption based on NCCS

In this paper, an encryption algorithm for remote sensing image based on a new type of Novel Chebyshev chaotic system (NCCS) and a combined encryption algorithm for remote sensing image, gray image and color image are proposed. Aiming at the problem of large amount of remote sensing image data, this paper proposes NCCS algorithm, which effectively reduces the time complexity of the algorithm, and the generated pseudo-random sequence is more uniform, and the performance is better. On this basis, the remote sensing image encryption, first of all, each band of remote sensing image in a different channel, to obtain a three-dimensional matrix, using three-dimensional spiral curve to read each section of the three-dimensional matrix, a two-dimensional matrix composed of several one-dimensional sequences is obtained. This method makes each channel produce some coupling and reduces the dimension of the matrix, thus effectively improving the scrambling effect. Chaotic maps scramble one-dimensional sequences, then scramble one-dimensional sequences, and diffuse them by cyclic left shift based on additive modules. Because this method is suitable for multi-channel image encryption, it can be used not only for remote sensing image encryption, but also for remote sensing image, gray image, and color image encryption. Simulation results and performance analysis show that the method has good security. Compared with some existing encryption schemes, this method has a wider application range.


Introduction
In recent years, remote sensing images have developed very rapidly and play an important role in many fields. As an important spatio-temporal data, remote sensing images have the characteristics of multi-precision, multi-tense, multi-semantic, and multi-band [1][2][3][4][5]. In terms of information acquisition, it also shows its unique charm [6]. At the same time, the security of remote sensing images has gradually attracted the attention of some experts and scholars [7][8][9][10][11]. However, the encryption schemes of remote sensing images are very few and the scope of application is narrow, after a large number of literature searches, only the following work was found, such as the mixed domain remote sensing image encryption technology proposed by Zhang [12]; a new remote sensing image fragmentation chaos encryption scheme proposed by Guo et al. is based on fragmentation of remote sensing images and then scrambling the blocks and combining them with the Lorenz chaotic system [13]; however, these schemes are only for grayscale remote sensing images and color remote sensing images and cannot be applied to remote sensing images with more than three bands, and the scheme proposed in this paper solves this problem very well.
At present, the proposal of ordinary image encryption algorithms can be described as endless, and a series of image encryption algorithms based on chaos theory proposed by Wang et al. have been highly recognized by the industry [14]. It also includes the application of fractal ordering theory proposed by Xian et al. in image encryption [15][16][17], image encryption schemes based on DNA coding theory [18][19][20], and encryption schemes based on matrix half-tensor products and Boolean networks [21]. Common image and multi-image encryption schemes are growing, but joint encryption schemes for multitype images are currently absent [22][23][24].
In this regard, the encryption scheme based on chaos theory proposed in this paper can effectively solve the above problems. When encrypting remote sensing images, there is no limit on the size and number of bands of remote sensing images, and when multi-type images are jointly encrypted, there is no limit on the number of grayscale images, color images, and remote sensing images. The entire encryption process consists of two parts: scrambling and diffusion. The scrambling process is divided into three stages and each stage is accompanied by the reduction of the matrix dimension, the first stage uses the spiral curve to pre-scramble the three-dimensional matrix and convert the three-dimensional matrix into a twodimensional matrix; the second stage uses the chaotic sequence generated by NCCS to index the twodimensional matrix and convert the two-dimensional matrix into a one-dimensional matrix; the third stage is to randomize the one-dimensional matrix in onedimensional Arnold to obtain the scrambled image. The diffusion phase uses the classic method of adding and taking the mold to the left of the cycle to obtain the final ciphertext image. This paper mainly includes seven parts, the first part is a brief introduction to the whole text; the second part is the introduction of the relevant knowledge used in the article; the third part is the newly proposed chaotic system and the performance analysis tests of the system, the results show that the proposed chaotic system is better; the fourth part is the encryption process, which gives the detailed process and steps of chaos and diffusion; the fifth part is the decryption process, gives the decryption algorithm of the detailed process and steps; the sixth part is the simulation results and performance analysis part of the remote sensing image, indicating the accuracy of the algorithm proposed in this article; the seventh part is the simulation results, and performance analysis of the joint encryption of grayscale images and color images show that the proposed algorithm is also suitable for the joint encryption of multiple types of images and has better results.

Traditional Chebyshev chaotic system
The Chebyshev map is a map whose order is the parameter, and the Cosine form is defined as Eq. (1): where x n 2 ð À 1; 1Þ, l j j 2 ð2; þ1Þ, when l 2 ðÀ2; 2Þ, chaotic systems do not produce chaotic behavior.

Storage format
Remote sensing images with multiple bands are one of its more significant features, Tif format is characterized by having multiple channels, so remote sensing images mostly exist in Tif format, each band occupies a channel. One band of a remote sensing image corresponds to a two-dimensional matrix, and n bands correspond to n two-dimensional matrices, so a remote sensing image corresponds to a three-dimensional matrix. This article uses remote sensing images in Tif format for encryption.

Display mode
There are three ways to display remote sensing images, namely grayscale image display, pseudocolor image display, and true color image display.
Grayscale image display is to save one band of the remote sensing image as a grayscale image; true color image is to put the red, green, and blue bands of the remote sensing image into the R, G, B channels, respectively; the false color image display is selected from the remote sensing image to be placed in the R, G, B channel [25][26][27]. The encryption of remote sensing images in this paper is for all bands, but the display of remote sensing images can only select up to three bands, and this article uses false color images to display the simulation results of remote sensing images.

One-dimensional Arnold mapping
Arnold mapping is also known as cat mapping. It is a chaotic mapping method that repeatedly folds and stretches transformations in a finite area, which is widely used in the scrambling process of image encryption, and for NN matrices, two-dimensional Arnold matrix transformations such as Eq. (2): where x n ; y n is the coordinates before the transformation of the two-dimensional matrix,x nþ1 ; y nþ1 is the coordinates after the transformation of the twodimensional matrix; a, b is the parameter, and n is the number of transformations. This method is only suitable for cases where the length and width of the matrix are equal and have certain limitations [28][29][30][31][32]. This leads to the one-dimensional Arnold map. The one-dimensional Arnold map is suitable for matrix transformations of unequal length and width, and the time performance of the algorithm is better, for the matrix of MN; it is converted to a one-dimensional matrix, the size is 1 9 MN, then the coordinates of any point are (i = 1, 2, 3,…, MN), and the Arnold transformation obtains new coordinates.
x n y n " # x nþ1 ¼ 1 þ by n y nþ1 ¼ a þ ðab þ 1Þy n ð4Þ By Eq. (3) available Eq. (4), where a = 1 does not consider the transformation of the horizontal axis, and ab ? 1 is a new pseudo-random number, then Eq. (4) becomes Eq. (5). Thus, the formula for a one-dimensional Arnold matrix transformation can be expressed as Eq. (6): where y n is the ordinate coordinate before the onedimensional vector transformation, and y nþ1 is the ordinate coordinate after the transformation; a, b is the parameter.

The new Chebyshev chaotic system
This paper proposes an NCCS based on the Cosine form of the Chebyshev map, which has a wider range of parameters, a more uniform distribution of pseudorandom sequences generated, and better chaotic behavior. NCCS is a mapping of order l, defined as a result of the rest of the chordal forms such as Eq. (7): x nþ1 ¼ cos(larccos(x n ÞÞ Â 10 6 À floor(cos(larccos(x n ÞÞ Â 10 6 Þ ð 7Þ where x n 2 ð0;1Þ;l 6 ¼ 0; AE 2; AE 4; AE 6, when the initial value of the system x 0 and the value of the parameter l meet the above range, the chaotic behavior of NCCS is better.

Comparative analysis of Bifurcation diagram
According to the knowledge of dynamics, the uniformity of pseudo-random sequences generated by chaotic systems iteratively within the constraint range is an important criterion for evaluating the quality of a system [33][34][35][36]. The forked diagram of the Chebyshev chaotic system and the forked diagram of NCCS are given below. It can be seen that when the initial values of x 0 and the values of parameter l of the system meet the custom ranges, the distribution of the pseudorandom sequences generated by NCCS is much more uniform than that of the pseudo-random sequences generated by the Chebyshev chaotic system. Therefore, it is determined that the chaotic performance of NCCS is very good Fig. (1).

Comparative analysis of the Lyapunov Index
The Lyapunov Index (LE) is an important indicator of the dynamic stability of chaotic systems, accurately determining whether the system is in a chaotic state [37][38][39][40][41]. LE is calculated by formulas such as Eq. (8): where f ðx i Þ is the formula for chaos mapping, when LE is negative, it indicates that the system is in a contraction state; when LE is positive, it indicates that the system is in a chaotic state. It can be seen from Fig. 2 that the LE value of NCCS is larger than that of the traditional Chebyshev chaotic system, Logistic map and Sine map, that is, the dynamic stability of NCCS is better.

Comparative analysis of Shannon entropy
Shannon entropy (SE) reflects the degree of chaos of pseudo-random sequences, the larger the SE, the higher the degree of chaos, the better the chaotic performance [42][43][44]. The comparison of Shannon entropy between NCCS and Chebyshev system, Logistic system and Sine system is shown in Fig. 3, which shows that the chaotic performance of NCCS is better than that of other chaotic systems.

NIST test
The National Institute of Standards and technology (NIST) is a method for evaluating the performance of chaotic systems [45]. The randomness of sequences generated by chaotic systems is described by means of probability theory and statistics. The NIST test consists of 15 sub-tests, each of which generates a P value. Only when the P value is within the range of the interval [0.01,1] can we consider the test passed. The NIST test results are shown in Table 1, and all P values can be found to fall into the interval. Therefore, it is concluded that the chaotic behavior of the pseudorandom sequence generated by the proposed chaotic system is better.

0-1 Test
In addition, we also use 0-1 test to evaluate the performance of chaotic systems [46], which is also a more popular method in recent years. As shown in Fig. 4, NCCS has better chaotic performance than the existing Logistic map, Sine map, and Chebyshev map.

Encryption algorithm
This algorithm is based on NCCS, one-dimensional Arnold scrambling and addition modulus cyclic left shift diffusion method of remote sensing image encryption and remote sensing image combined encryption with color image and grayscale image. You can use remote sensing images or a union of multiple types of images as input to the algorithm. There are four parts, the first step is to generate the key through the SHA-512 algorithm; the second part is to generate two pseudo-random sequences by substituting the processed key into NCCS, which are used for chaos and diffusion; the third part is to first use a spiral curve to reduce the dimensionality of the threedimensional matrix, and then to perform index disorder and one-dimensional Arnold chaos; the fourth part is to use the classic cyclic left shift method of adding modulus for diffusion. The result is a ciphertext image. The encryption flowchart is shown in Fig. 5.

Key processing
Step 1: This article uses the SHA-512 algorithm to generate the key and brings the image P with a size of MNK into the SHA-512 algorithm to obtain a set of hexadecimal key with a length of 128 bits.
Step 2: Converts the hexadecimal key key to the binary string key 1 , and one hexadecimal number is equal to a four-digit binary number, so the string length becomes 512.  Step 3: Each adjacent bit of the string key 1 is xor or different, resulting in a string key 2 with a length of 256 bits.
Step 4: Place the key key 2 as described in Eq. (9). Convert into four parts of equal length to obtain four values, namely: K 1 ;K 2 ;K 3 ;K 4 .
Step 5: According to Eq. (10) and Eq. (11), generate the parameters and initial values required for the two sets of NCCS:

Chaotic sequence generation
Step 1: The parameters l ¼ l 1 and initial values x 0 ¼ x 1 required for NCCS generated by the above formula are substituted into NCCS and grow into a set of sequence A of M ? 3MNK, which is divided into four parts A 1 ; which is used for interline index scrambling, A 2 long is MNK, which is used for inline index scrambling, and A 3 and A 4 long are MNK, which is used to generate one-dimensional Arnold scrambled parameters.
Step 2: The parameters l ¼ l 2 and initial value x 0 ¼ x 2 required for NCCS generated by the above formula are substituted into NCCS, and the growth is generated into a set of sequence B of 2MNK, and the sequence is divided into two parts B 1 ;B 2 . The length is MN and is used in diffusion.

Scramble algorithm
This method is suitable for MNK multi-channel image P, where there is no limit to the range of M, N, K.
Considering the problem of time complexity during the operation of the algorithm, the three-dimensional matrix is processed twice. Each dimensionality reduction is accompanied by the occurrence of chaos, which is divided into three steps: the first step is to use a spiral curve to scan the three-dimensional side-slice surfaces one by one, each tangent as a row of the twodimensional matrix, as shown in the figure; the second step is to scramble the two-dimensional matrix and convert the two-dimensional matrix into a onedimensional matrix; the third step is to mess up the one-dimensional matrix.
Step 1: The preset mess process is shown in Fig. 6, for the three-dimensional matrix P, from the far left side of the spiral curve to obtain pixel values one by one, as a row of the two-dimensional matrix P 1 , that is the three-dimensional matrix is reduced to a twodimensional matrix, the size is M 9 (NK).
Step 2: The index scramble process, the twodimensional matrix P 1 is scrambled once according to the pseudo-random sequence A 1 , and then converted to a one-dimensional matrix P 2 with a size of 1 9 (MNK). An index scramble is performed according to sequence A 2 to obtain a one-dimensional matrix P 3 .
Step 3: The one-dimensional Arnold is confused, and the one-dimensional Arnold transformation is performed on the one-dimensional matrix P 3 to obtain C 0 .

Diffusion algorithm
Step 1: The pseudo-random sequences B 1 and B 2 are decimal places of (0,1) and the length is MNK, which are mapped to (0,255) by the following formula to obtain the pseudo-random sequences S 1 and S 2 .
Step 2: Convert the original matrix P to onedimensional matrix P 0 with a size of 1 9 MNK; The size of the scrambled matrix C 0 is 1 9 MNK, and the pseudo-random sequences S 1 and S 2 are all 1 9 MNK in size.
Step 3: According to the original matrix P 0 , the pseudo-random sequence S 1 , the matrix C 0 is forward diffused according to Eq. (12) to obtain matrix C 1 .
Step 4: According to the original matrix P 0 , pseudorandom sequence S 2 , according to Eq. (13) reverse diffusion of matrix C 1 to obtain matrix C.

Decryption algorithm
The decryption algorithm is the inverse operation of the encryption algorithm. The flowchart is shown in Fig. 7. It consists of two stages. It spreads the ciphertext image in reverse and then inverts it to get the original image after decryption. The decryption process needs several parameter values, as the initial value of NCCS, the specific introduction and the calculation process can be seen Chapter 4.1 and 4.2.
The specific decryption steps are as follows.

The reverse process of diffusion
Step 1: According to the diffusion formula, the inverse diffusion process of ciphertext image C is carried out, and the pre-diffusion image C 1 is obtained.
Step 2: The image C 0 is transformed from onedimensional matrix to two-dimensional matrix, and C 1 is obtained.

The reverse process of scrambling
Step 1: One-dimensional Arnold scrambling is used to invert the C 1 image, and the chaotic sequence generated by NCCS system is needed in the scrambling process.
Step 2: Finally, the original image is obtained by the inverse process of image prescrambling.

Simulation results and performance analysis of remote sensing images
This section selects the size of 512 9 512 9 6 Landsat4-5 remote sensing image fragment for encryption, the remote sensing image used in this chapter and the resulting ciphertext image are containing six channels of Tif format, cannot be viewed directly, so here the remote sensing image of the false color image displays method to reflect the experimental results that is, optional three bands as the R, G, B channel of the color image.

Simulation results of remote sensing images
The results of the original image of this simulation experiment are shown in false color as shown in Fig. 9a, the image of each band of the original image is shown in Fig. 8, the encrypted ciphertext image is in Tif format, and the results displayed in false color are shown in Fig. 9b; the decrypted image is still in Tif format, and the display method of the false color image here is shown in Fig. 9c.

Keyspace analysis
The keyspace refers to the total number of different keys that can be used in a cryptographic system, and it is an important measure of the cryptography's resistance to brute force attacks. Theoretically, the larger the key space, the stronger the algorithm's ability to  resist various attacks. The medium key of the proposed algorithm in this paper is converted from 512-bit binary, and its key space size is 2 512 . The key space is large enough to effectively resist brute force attacks and enhance the security of encryption.

Key sensitivity analysis
The level of key sensitivity is also an important indicator of a cryptographic algorithm. In the process of image encryption, for small changes in the key, the image cannot be decrypted correctly to ensure the security of image encryption. As follows, decryption using the correct key key 3 yields an image as shown in Fig. 8a, changing bit 18 in key key 3 to change ''9'' to ''e'' to obtain key key 4 , and decrypting with key 4 to obtain a decrypted image as shown in Fig. 10b. Obviously, the 128-bit hexadecimal key key 3 obtains a far difference from the original image after changing one character, which can show that the algorithm has good key sensitivity and can ensure the security of the encrypted image. In addition, the correlation of two ciphertext images generated by two different encryption keys is tested [47]. As shown in Fig. 10c, d, e and f, the diagonal correlation coefficients of the two ciphertext images are 0.001922 and 0.000902. This numerical result is a good proof of the sensitivity of the key.

Histogram analysis
The histogram of the image reflects the distribution characteristics of pixel values. The histogram of the original image is mostly uneven, and attackers often use statistical analysis methods to select the pixel values of important information in the image as a Lansat4-5 remote sensing image histogram as shown in Fig. 12a, the histogram of each band as shown in Fig. 11, the histogram of the ciphertext  image as shown in Fig. 12b, it can be seen that the histogram of the original image and its various bands is uneven, and the histogram of the ciphertext image is very uniform, and it will be difficult for the attacker to use statistical analysis methods to obtain important information in the image, which ensures the security of the image to a certain extent.

Correlation analysis between adjacent pixels
In plaintext images, the correlation of adjacent pixels tends to be stronger, and one of the purposes of encryption is to break the correlation between adjacent pixels. The size of the correlation coefficient is an important indicator of the algorithm's ability to resist attacks. The closer the correlation coefficient is to the ideal value of 0, the better the effect of the encryption algorithm. Therefore, the encryption algorithm should ensure that the correlation coefficient between adjacent pixels is as close as possible to the ideal value. In this paper, the parties are randomly selected from the various bands of the plaintext image, the ciphertext image, and the plaintext image to perform correlation analysis on adjacent pixels.
b Fig. 11 Histogram of each band of Landsat4-5 (a) Histogram of Landsat4-5 plaintext (b) Histogram of Landsat4-5 ciphertext  The test results of the correlation of adjacent pixels in the diagonal direction of each band of the Landsat4-5 remote sensing image are shown in Fig. 13. The test results for the correlation of adjacent pixels in the horizontal, vertical, and diagonal directions of Land-sat4-5 remote sensing images are shown in Fig. 14a, b and c, respectively. The test results for the correlation of adjacent pixels in horizontal, vertical, and diagonal directions of redaction images are shown in Fig. 14d, e and f, respectively. It can be observed from the plot that the distribution of adjacent pixels in the plaintext image and its various bands is relatively concentrated, while the distribution of adjacent pixels in the redaction image is relatively uniform. In order to more accurately show the correlation between pixels in different directions, the correlation coefficient is calculated using Eqs. (14)- (17), which is shown in Table 2 and compared with other schemes as Table 3. The value of the correlation coefficient in the ciphertext image is close to the ideal value of 0, indicating that the correlation between adjacent pixels is greatly reduced, which ensures the security of the image to a certain extent. DðxÞ EðxÞ 6.6 v 2 test v 2 test is used to describe whether the distribution of image pixels is uniform, the more uniform the pixel distribution of the image, the better the performance of the encryption algorithm, the less valid information contained in the secret map, so the value of a v 2 should be as small as possible, and the security of the algorithm will be high. v 2 test formula such as Eqs. (18) and (19) are shown.
where v i represents the frequency at which pixel values i appear in the image, M represents the length of the three-dimensional matrix, N represents the width of the three-dimensional matrix, and K represents the height of the three-dimensional matrix. v is the average frequency. Table 4 shows the v 2 value of the Landsat4-5 remote sensing image, the v 2 value of each band and the v 2 value of the redaction image, which can be obtained by comparison, and the v 2 value of the secret map encrypted by this algorithm is much lower than the v 2 value of the plaintext image. Therefore, it can be shown that the pixel distribution of the ciphertext image obtained by the modification scheme encryption is relatively uniform, the performance of the encryption algorithm is better, and the security of the image can be better guaranteed.

Information entropy analysis
Information entropy can be used to describe the degree of confusion of pixel information in an image, and the closer the information entropy is to the ideal value8, the higher the randomness of the image pixel value. It is an important indicator of the quality of image encryption algorithms. Formulas for information entropy such as Eq. (20) are shown: where L is the length of a pixel in binary and pðs i Þ represents the probability of pixel s i appearing. When the entropy of information is close to 8, the randomness of the secret map is better, that is, the safety is better. The second column in Table 5 lists the information entropy of the original remote sensing image compared to the ciphertext image; columns 3-8 show the information entropy of the original bands of the remote sensing image compared to the encrypted bands. The test results show that the encryption result obtained by the algorithm is very close to the ideal value of 8, so it can be concluded that the algorithm has better randomness, that is, better security.

Robustness analysis
Robustness is used to reflect whether the encryption algorithm is robust enough to resist interference that images may encounter during transmission. When the image encounters the interference of uncertainty such as attack during transmission, the pixel value of the image will be damaged and some information is missing, and a good algorithm can still decrypt the useful information in the plaintext image through the partially destroyed ciphertext image, and this section performs noise attack and cropping attack on the picture to test the robustness of the algorithm. The cropping attack uses three different degrees of attacks to crop the various channels of the ciphertext image, which are 1/49 degree, 1/25 degree and 1/16 degree of cropping, and the cropped ciphertext image is displayed in a false color image, as shown in Fig. 15a, b and c. The corresponding Landsat4-5 remote sensing image under this attack decrypted image is shown in Fig. 15d, e and f. It can be seen that the algorithm can still restore the original image in the case of different degrees of pixel value loss in the secret map, and the security is better. The noise attack uses the classic salt and pepper noise attack and Gaussian noise attack, Fig. 16a, b and c is a ciphertext image after adding 0.01, 0.03 and 0.05, respectively, and the corresponding Landsat4-5 remote sensing image under this attack is shown in Figs. 16d, e and f; Figs. 17a, b and c shows the ciphertext image after Gaussian attack with mean 0, 0.05, 0.05 and variance 0.05, 0.05, 0.1 are added successively. The decrypted image is shown in Figs. 17d, e and f. Obviously, this algorithm can effectively resist noise attacks, and the original plaintext image can still be displayed under different levels of noise attacks.

Differential attack
A differential attack is the comparison and analysis of the results of different plaintext encryptions to attack cryptographic algorithms. This indicates that small changes in the pixels of a plaintext image produce two different ciphertext images. The two metrics that measure whether an image can resist differential attacks are the pixel number conversion rate (NPCR) and the uniform average change intensity (UACI), which are often used to qualitatively analyze encrypted images. NPCR and UACI can be used with Eqs. (21)(22)(23) to calculate: In Eq. (22), c 1 and c 2 are encrypted images in which the plaintext image changes a pixel value before and after the change. The test results for NPCR and UACI are shown in Table 6. From the results, the averages of NPCR and UACI were 99.6089% and 33.4374%, respectively, very close to the ideal values of 99.609% and 33.464%. This suggests that improved cryptographic algorithms are highly sensitive to small changes in plaintext. Compared with other algorithms, the algorithm has a good ability to resist differential attacks.  It is known that plaintext and select plaintext attacks are common attacks in the field of image encryption, which refers to testing the performance of encryption algorithms against special pixel values by encrypting two special images, pure white and pure black images. Pure white images and pure black images have pixel values of 255 and 0, respectively, so these two special images are selected to test. Figure 18 is the experimental result of pure white image encryption; Fig. 19 is the experimental result of pure white image encryption. Table 7 lists some common test data, including v 2 tests, information entropy tests, and correlation coefficient analysis. The results are more in line with the ideal value, and the algorithm is sufficient to resist known plaintext and selective plaintext attacks.

Operational efficiency analysis
The operational efficiency of an algorithm is one of the important criteria for measuring the quality of an algorithm, and the algorithm runtime tests are performed on MATLAB 2020b, Intel Core i5-7300 CPU, 8 GB RAM, and Window 10 operating systems. For the remote sensing image encryption scheme mentioned in this article, the encryption time for images with dimensions 512 9 512 9 6 is 7.32 s, and the decryption time is 6.91 s. The running efficiency of the algorithm is still relatively ideal.
7 Simulation results and performance analysis of multi-type image joint encryption According to the characteristics of multi-channel encryption, this section performs joint encryption of  remote sensing images, color images, and grayscale images. In the theory of joint encryption, there is no limit to the number and size of images, and the maximum value of the length of all pictures will be selected as the length of the three-dimensional matrix, the maximum value of the width as the width of the three-dimensional matrix, the sum of the channel number of all pictures as the height of the threedimensional matrix, and the pixel value where there are no pixels is filled with 1. Although there is no limit on the size of this scheme, if the size difference between the pictures is too large, it will cause a waste of space, so the author recommends that you choose images of similar size for joint encryption.   The remote sensing image selects Landsat4-5 with the same size as Chapter 5, which is 512 9 512 9 6; the color image is selected from the classic Baboon diagram, which is 512 9 512 9 3; the grayscale image is selected from the classic Lena diagram, which is 256 9 256; as shown in Fig. 20. The three images form a three-dimensional matrix with 10 channels, with sizes of 512 9 512 9 10. Where the Lena diagram is undersized and filled with 1. The ciphertext image obtained by federated encryption is 512 9 512 9 10, and the ciphertext image of each channel is shown in Fig. 21. The decrypted image is shown in Fig. 22.

Common statistical analysis
The histogram of the Lansat4-5 remote sensing image is shown in Chapter 6 Figs. 11 and 12a, and the histogram of the color image Baboon and the grayscale image Lena is shown in Fig. 23a, b shown; the histogram of the redaction image is shown in Fig. 23c. The correlation coefficient test results of the   Table 8.

Robustness analysis
This section tests the robustness of the algorithm by performing noise attacks and cropping attacks on images. Figure 24a, b shows ciphertext images of 1/49 degree cropping attack and 1% pretzel noise attack, respectively, and is displayed in the form of false color images; the corresponding decrypted images under the corresponding attacks are shown in Figs. 25 and 26. Table 9 gives some common security analysis test results, including v 2 tests, information entropy tests, and NPCR and UACI in differential attack tests. The encryption time of the joint image of 512 9 512 9 10 is 10.03 s, and the decryption time is 9.84 s. The operation efficiency is better.

Conclusion
In this paper, a remote sensing image encryption scheme based on low-dimensional chaotic system is proposed, which is also applicable to the joint encryption of remote sensing images, grayscale images and color images. Compared with the existing remote sensing image encryption scheme, this paper mainly solves two problems, one is to solve the encryption problem of remote sensing images for multiple bands, the algorithm proposed in this paper has no limit on the size and number of bands of remote sensing images, and the second is that the algorithm can be used for the joint encryption of remote sensing images, grayscale images and color images and the encryption effect is better. Based on the traditional Chebyshev chaotic system, this paper proposes a new type of Chebyshev chaotic system, and the performance test results of the chaotic system show that the performance of the system is better. The encryption process of this paper includes chaos and diffusion, of (d) Landsat4-5 (e) Baboon (f) Lena  which the scrambling method includes the pre-placement disorder dominated by spiral curve scanning, index scrambling and one-dimensional Arnold scrambling three, the diffusion method adopts the classic cyclic left shift method of adding and taking modulus, and the simulation experimental results and performance analysis show that the algorithm is sufficient to resist common attacks, with better security and good running efficiency. In addition, the encrypted data of the multi-type images proposed in this paper are large, so it is often combined with the knowledge of image compression to achieve the purpose of improving the efficiency of the algorithm, and the author will continue to study this issue in depth.