Image encryption algorithm for crowd data based on a new hyperchaotic system and Bernstein polynomial

A new two-dimensional chaotic system in the form of a cascade structure is designed, which is derived from the Chebyshev system and the inﬁnite collapse system. Performance analysis including trajectory, Lyapunov exponent and approximate entropy indicate that it has a larger chaotic range, better ergodicity and more complex chaotic behaviour than those of advanced two-dimensional chaotic system recently proposed. Moreover, to protect the security of the crowd image data, the newly designed two-dimensional chaotic system is utilized to propose a visually meaningful image cryptosystem combined with singular value decomposition and Bernstein polynomial. First, the plain image is compressed by singular value decomposition, and then encrypted to the noise-like cipher image by scrambling and diffusion algorithm. Later, the steganographic image is obtained by randomly embedding the cipher image into a carrier image in spatial domain through the Bernstein polynomial-based embedding method,


INTRODUCTION
According to the dimension of the system state space, chaotic systems can be divided into one-dimensional chaotic systems (ODCS) and multi-dimensional chaotic systems (MDSC). With regard to ODCS, due to their simple structure, the resulting chaotic sequences have poor ergodicity, small chaotic range and weak chaotic behaviour, which are easy to be attacked by signal estimation algorithms [28][29][30][31][32]. To address these issues, many improved chaotic system schemes have been proposed. For example, in ref. [2], Cao et al. used an infinite collapse system to modulate the amplitude of another infinite collapse chaotic system with different control parameter to design a novel two-dimensional chaotic system (TDCS). Then the newly designed TDCS was applied to encrypt the plain image for multiple rounds adopted the classic scrambling and diffusion architecture. Hua et al. [5] also proposed an improved scheme to integrate the Logistic map and the Sine map using coupling coefficient, and then extend the integrated ODCS to two dimensions by means of cross iteration. By coupling the two ODCS sufficiently, a MDSC with more complex chaotic characteristics can be obtained. Further, the authors used the pseudorandom matrix generated by the newly designed TDCS to scramble and diffuse the plain image for two rounds to obtain the final noise-like cipher image. Besides, Liu et al. designed a new TDCS without modulus operation based on Chebyshev system and Sine system to eliminate the defect of the ODCS in ref. [14], and applied it into the color image encryption. Meanwhile, the initial values of the new chaotic system in ref. [14] are first extracted from the coloured non-Gaussian noise, and then the color cipher image with good avalanche effect can be obtained by using the diffusion matrix to perform exclusive or operation on the components of the plain image. Moreover, in 2020, Liu et al. [3] mentioned a new TDCS with modulo operation, which was obtained by cascading 2D Hénon map and Chebyshev map. During the encryption process, the newly designed TDCS is applied to generate three sets of pseudorandom matrices controlled by the default parameters and the hash value of the plain image, which are used to perform genetic mutation, scrambling and diffusion on the plain image after genetic recombination. Eventually, a cipher image with high security can be obtained. Last but not least, Peng et al. [21] designed a discrete memristor via the difference theory, and applied it to the 2D discrete Hénon map. As far as a chaotic system is concerned, if it has more than two (including two) positive Lyapunov exponents, it indicates that this chaotic system has a more complicated hyperchaotic behaviour. Although the improved chaotic systems mentioned above have very good ergodicity and complex chaotic behaviour, and their chaotic range has also been expanded. The two Lyapunov exponents of the aforementioned 2D chaotic systems are not all greater than zero in the entire parameter interval, so there is still room for improvement in the chaotic performance of these TDCS.
Image encryption technology has been widely used to transmit digital images containing confidential information. However, the cipher images generated by traditional encryption algorithms are texture-or noise-like, which are easily recognized by the human visual system. Therefore, Bao et al. [23] proposed a novel solution of embedding the noise-like cipher image into a visually meaningful carrier image after encryption process, thereby reducing the probability of information leakage and being attacked, and achieving the double security of image information and image appearance. Unfortunately, in Bao's scheme, the final steganography image is four times the size of the plain image due to the lack of compression, so additional storage space and transmission bandwidth are needed to ensure the security of the plain image information. Besides, ref. [33] also has the same deficiency. In his scheme, the plain image is subjected to pre-processing such as sparse and scrambling, and then the pre-processing image is compressed and quantized using the measurement matrix generated by the logistic-tent chaotic system. In order to improve security and anti-noise performance, random numbers are added to the quantized image before the second scrambling. Finally, the cipher image is hidden in the carrier image by singular value decomposition (SVD) in wavelet domain. To alleviate transmission pressure and save storage space, Chai et al. introduced a set of efficient visually meaningful image encryption scheme in ref. [34]. In Chai's scheme, before zigzag scrambling the plain image needs to be compressed to a quarter of its original size through compressive sensing technology, and then the cipher image is split and randomly embedded into the carrier image through the least significant bit (LSB) embedding method. Wang et al. [35] also proposed a set of improved visual image encryption algorithm using parallel compressive sensing and embedding technology. Both parallel compressive sensing and zigzag scrambling are applied to generate cipher images in ref. [35]. Additionally, during the embedding process, the embedding order will be disrupted by the index sequence generated by the 3D Cat system to improve the security. Similarly, Wen et al. [36] proposed an improved scheme that uses semi-tensor product compressive sensing to compress the plain image, and then embeds the compressed image after Arnold scrambling into a carrier image through embedding technology, which can not only simultaneously reduce the storage space of the cipher image and the secret keys, but also the quality of the reconstructed image is improved. So far, many excellent visually meaningful image encryption algorithms have been proposed [37][38][39][40][41][42][43][44][45], but most of them have a common deficiency. That is to say, the identical noise-like cipher image is embedded into different carrier images, and the visual quality of the resulting steganographic images is uneven. If an improper carrier images is selected, the final generated steganographic image will has low visual quality. Meanwhile, the imperceptibility of the cipher image will also be reduced.
For the sake of obtaining a chaotic system with more complex chaotic behaviour, in this paper, a new TDCS with hyperchaotic behaviour based on the Chebyshev system and the infinite collapse system is designed. Moreover, we apply the newly designed TDCS to protect the crowd image data, and propose a meaningful image cryptosystem combined with the SVD and Bernstein polynomial (BP). In proposed scheme, the plain image is compressed to half of its original size by SVD in wavelet domain. Then using the stochastic matrix generated by the newly designed TDCS to scramble and diffuse the compressed image. Finally, the embedding algorithm deduced from the BP is adopted to embed the cipher image into the carrier image randomly. Additionally, the embedding effect is controlled by the adjustment factor, which means that the visual quality of the steganographic image can be adjusted through AF in the light of different carrier images. The contributions of this paper are summarized as follows. a. A new TDCS with hyperchaotic behaviour over the entire parameter range is designed. Compared with other advanced TDCS proposed in refs. [2][3][4], performance analysis including system trajectory, Lyapunov exponent and approximate entropy (ApEn) indicate that it has more complex chaotic behaviour and better ergodicity (see Section 3.2). b. Combining the newly designed TDCS with SVD and BP, a visually meaningful image cryptosystem (VMIC) is proposed. And it does not require additional storage space and transmission bandwidth like refs. [23,33] to improve the visual quality of the steganographic image. The remainder of this paper mainly includes following five sections. The second section introduces some indispensable prerequisites. The newly designed 2D chaotic system and performance analysis are mentioned in the third section. Besides, the encryption and decryption process of VMIC are detailly recorded in the fourth section. And the fifth section is to analyze the security performance of proposed image cryptosystem. The last section will sum up all of our research work.

PRELIMINARY KNOWLEDGE
Because of its simple structure and low computational complexity, the ODCS have been widely used in many fields. In this section, two classical ODCS in detail will be introduced, which are Chebyshev system and infinite collapse system. At the same time, the chaotic performance of these two systems is also analyzed.

Chebyshev system
As one of the classic ODCS, the Chebyshev system with a control parameter k and an initial value x 0 is widely used in digital communication [46][47][48], neural network [49], security issue [50,51] and radio frequency identification [52]. Its mathematical expression is shown in Equation (1). In addition, Equation (2) is expressed as the probability density function of the chaotic sequence generated by Equation (1).
where x n represents the discrete state of the output of Chebyshev system, which is located in the interval [−1, 1]. The bifurcation chart and Lyapunov exponent chart of the Chebyshev system are shown in Figure 1. It can be seen from Figure 1(a) that under different control parameter conditions, the discrete state of the output of the Chebyshev system presents an uneven distribution. At the same time, when the control parameter k is greater than 1, the Chebyshev system slowly begins to appear bifurcation and enters a chaotic state, which can be verified by the Lyapunov exponent chart in Figure 1(b). Besides, when k is greater than 2, the discrete state of the output is evenly distributed over the entire interval. At this time, the Chebyshev system has very good ergodicity and unpredictability.

Infinite collapse system
The infinite collapse system, a classical ODSC, was firstly proposed by He et al. in 2001 [53], which has the characteristic of infinite collapse times in the interval [−1, 1]. Equation (3) represents the mathematical expression of this ODSC.
where a is the control parameter of the infinite collapse system, which belongs to the interval (0, 4]. It can be seen from Equation (3) that this system has odd symmetry for the parameter a. The bifurcation chart and Lyapunov exponent chart of infinite collapse system are plotted in Figure 2. We can clearly see from Figure 2 that this chaotic system has non-uniform ergodicity and short chaotic range, which will bring negative effects to its application.

Mathematical definition
As can be seen from the previous section, the chaotic performance and ergodicity of Chebyshev system and infinite collapse system are unevenly distributed, which are easily attacked by the signal estimation algorithm, thereby obtaining the entire trajectory of the chaotic system. For the sake of solving these problems, in this section, we will introduce a new TDCS cascading Chebyshev system and infinite collapse system, which is defined in Equation (4).
In Equation (4), is the control parameter of the newly designed TDCS in the interval [0,4]. Both x n and y n represent the discrete states outputted by this chaotic system, which all belong to the interval [−1, 1]. It can be seen from the mathematical expression of the newly designed chaotic system that we firstly couple Chebyshev system and infinite collapse system with the parameter . Then the coupled ODCS is extended to two dimensions by means of cross iteration. Finally, map the 2D system to the interval [−1, 1] by Sine function. By combining Chebyshev system with infinite collapse system, not only the key space and chaotic range of the obtained system can be enlarged, but also the chaotic performance can be improved.

Performance evaluation
In this subsection, the performance of the newly designed TDCS will be analyzed through system trajectory, Lyapunov exponent, ApEn and National Institute of Standards and Technology (NIST) test. At the same time, in order to demonstrate the superiority, the newly designed TDCS is compared with three advanced TDCS recently proposed by other researchers, The contrasting chaotic systems are the 2D infinite collapse system (referred to as 2D ICS) [2], which is defined in Equation (5), the 2D Hénon-Chebyshev modulation system (referred to as 2D HCMS) [3], which is defined in Equation (6), and the 2D Hénon-Sine system (referred to as 2D HSS) [4], which is defined in Equation (7). Among them, the symbol "mod" denotes the modulo operation.

Trajectory analysis
The trajectory chart of a chaotic system is used to analyze the distribution of its attractor in the phase space. As far as a chaotic system is concerned, if its attractor has a complicated geometric distribution and occupies a large area in the phase space, which indicates that this chaotic system has good ergodicity [5]. To visually observe the attractor distribution of the four chaotic systems mentioned above, under the condition that the initial states are all set as (0.06, 0.03), we set the control parameters as 0.62, 1.99 and 2.79 respectively and iterate these chaotic systems 30,000 times. The resulting trajectories are plotted in Figure 3. It can be seen from the figure that (a) when the control parameters are set as 0.62, the attractor distribution of the newly designed The trajectory charts of different 2D chaotic systems 2D chaotic system in this paper is evener than that in refs. [2][3][4].
(b) Compared with the other three TDCS, the attractors of the proposed 2D chaotic system are also uniformly distributed and occupy almost all the places in the phase space when the control parameters are changed. These indicate that the newly designed chaotic system has good and stable ergodicity.

Lyapunov exponent analysis
The Lyapunov exponent (LE) is a significant indicator for quantitatively measuring the characteristics of a dynamical system. It represents the average exponential rate of convergence or divergence between two infinitely close orbitals in the phase space over time. Whether the dynamical system has chaotic behaviour or not can be directly judged by its LE. If its LE is greater than 0, it means that no matter how close the two orbits are in the phase space, the difference would increase exponentially over time to such an extent that it would be unpredictable. The larger the value of its LE is, the more complicated the chaotic behaviour is. In this section, we use the algorithm introduced in ref. [54] to calculate the LE of different 2D chaotic systems. The experimental results are demonstrated in Figure 4. First, compared Figure 4 with Figures 1b and 2b, it can be clearly seen that the Lyapunov exponent of the TDCS designed in this paper is greater than 0 in the entire parameter interval. That is to say, this newly designed chaotic system has a wider range of chaos than its seed system. Besides, by analyzing Figure 4, we can know that the newly designed chaotic system has a larger Lyapunov exponent compared with other advanced TDCS, such as that in refs. [2][3][4], and has hyperchaotic characteristics over the entire parameter interval. It further indicates that the newly designed TDCS has superior chaotic behaviour.

Approximate entropy analysis
ApEn, a nonlinear dynamical parameter, is frequently used to qualitatively describe the fluctuation regularity and unpredictability of a time sequence, which was firstly proposed by Pincus [55]. It uses a non-negative number to indicate the complexity and irregularity of a time sequence, reflecting the possibility of new information occurring in the time sequence. The more complex the time sequence is, the higher the ApEn corresponding to it will be. Due to its strong anti-noise performance, easy implementation and low computational demand, we will evaluate the complexity of the above-mentioned 2D chaotic systems through ApEn in this section. Table 1 demonstrates the pseudo-code used to quantitatively calculate the ApEn of a chaotic system. As can be seen from the 12-th row in Table 1, the ApEn actually measures the probability of generating a new pattern in the time sequence when the dimension changes. The greater probability of generating a new pattern, the more complicated the sequence will be, and the greater the corresponding ApEn will be. Therefore, the ApEn can be used to represent the unpredictability and complexity of a time sequence. Through this method, we obtain the approximate entropy curves of four TDCS, as shown in Figure 5. Through the analysis of Figure 5, the time sequence generated by the newly designed TDCS has a higher complexity and unpredictability in the interval [−1,1], compared with the remaining three ApEn curves.

National Institute of Standards and Technology analysis
Pseudo-randomness is one of the inherent characteristics of the chaotic systems. To test the pseudo-randomness of the newly designed TDCS, we will adopt the SP800-22 randomness test suite released by the NIST for experiments. It is widely used to test the randomness of binary sequences generated by secret random or pseudo-random number generators. This test suite uses probability statistics to determine whether the   (1) , sy (1) , ), Iteration times len.
Output: Binary sequence text document.

1.
for n← from 1 to len−1 2. sx (n+1) ← sin( (cos((4 − )cos −1 (sx (n) )) + sin( ∕sy (n) ))) 3. sy (n+1) ← sin( (cos((4 − )cos −1 (sy (n) )) + sin( ∕sx (n) ))) detected sequence passes each randomness sub-test item or not [56]. Assuming that the significance level is 0.01, if the P value generated by the sub-test item is greater than the preset significance level , then the detected sequence passes this randomness sub-test item. Table 2 below shows the pseudo-code for generating the binary sequence file for testing. We set the initial state as (0.721, 0.591, 3.47) and iterate the newly designed TDCS 1,000,000 times. The final test results are displayed in Table 3 below. It can be seen from the table that the two sets of sequences generated by the newly designed TDCS have passed all sub-item tests, indicating that the chaotic system designed in this paper has good pseudo -randomness.

Encryption algorithm
The flow chart of the proposed visually meaningful encryption scheme is shown in Figure 6. It can be seen that the encryption scheme mainly consists of compression, encryption and embedding processes. In the compression process, first, perform integer wavelet transform on the plain image to obtain four sub-images LL, HL, LH and HH. Then the combined image is compressed by SVD, which is obtained by combining the three sub-images HL, LH and HH. At the same time, a stochastic matrix is also added for filling. Before the encryption process, we need to map the compressed image between [0, 255], and then the pseudo-random matrixes generated by the newly designed chaotic system are used to scramble and diffuse the new combined image of the compressed image and the subimage LL. In the embedding process, the newly designed TDCS is firstly controlled by secret keys to generate a position index matrix, which is used to determine the embedded position of the cipher image in the carrier image. Secondly, the AF needs to be preset in advance, which is used to control the visual effect of steganographic image. Next, the meaningful image encryption algorithm is described in detail below.

Compression process
Singular value is an important feature of matrix [57]. In numerical analysis, SVD is an algorithm that diagonalizes the matrix, which has many applications in image compression, feature recognition, digital watermarking and other aspects. In this paper, the SVD is utilized to compress the plain image in wavelet domain. Assume that the plain image to be encrypted is I sized of h × w. And the image generated in this process is JL ∈ ℕ h∕2×w .
Step 1: First, perform 2D integer wavelet transform on the plain image I to obtain four sub-images, namely LL ∈ ℕ (h∕2×w∕2) , HL ∈ mathbbN (h∕2×w∕2) , LH ∈ ℕ (h∕2×w∕2) and HH ∈ ℕ h∕2×w∕2 . Step 2: The combined image CL ∈ ℕ h∕2×3w∕2 can be obtained by combining the sub-images HL, LH and HH in light of Equation (8). Then perform SVD on the combined image CL by Equation (9), in which svd(⋅) means to perform SVD on the matrix in parentheses, and return the left singular value matrix IS, the right singular value matrix ID, and the singular value diagonal matrix IV.
Step 3: Quantify partial data in matrix IS, IV and ID. The quantization equation is shown in  (12). where the max (⋅) and min (⋅) represent the maximum and minimum value of the matrix in parentheses, respectively. ⌊⋅⌋ indicates rounding to negative infinity.
Step 4: Extract the diagonal data in the matrix QIV and denote it as DQIV ∈ ℕ k×1 . The compressed image CPI is then generated on the basis of the combination rule in Figure 7. The purpose of adding a stochastic matrix in the combination process is to make the size of the compressed image exactly h∕2 × w∕2. Additionally, because of the compressed image containing stochastic numbers, in this case the cipher images generated by each encryption are different. This plays a certain ability to resist chosen-plaintext attacks.
Step 5: Splice the compressed image CPI and the subimage LL. The joint equation is shown in Equation (14).

Encryption process
Since the spliced image JL obtained in the first process has rich contour information, which needs to be encrypted to cover up. The specific encryption operation is shown as follow.
Step 2: Sort the chaotic sequences generated in the previous step by Equation (15) to obtain the index sequence. where sort(⋅) means sort the matrix in parentheses in ascending order and return the corresponding index sequence.
Step 3: The scrambled image SCI can be obtained by using index sequence to scramble the spliced image in the column-wise manner. The scrambling equation is shown in Equation (16). Where i = 1, 2, 3, … , wh∕2. Step 4: The final noise-like cipher image CI can be obtained by using Equation (18) to diffuse the scrambled image.

Embedding process
In order for the cipher image to be attacked as little as possible, it is necessary to embed the noise-like cipher image in a visually meaningful carrier image. Therefore, in this sub-section, a novel embedding method based on BP is proposed. BP is widely used in the differential equation and approximate theory [58,59]. Its mathematical model is shown in Equation (19). Since only two images (cipher image and carrier image) are needed in embedding process, N is set to 2. Then the image embedding equation is obtained by simplifying Equation (19), as shown in Equation  (20). while Equation (21) is the corresponding image extraction equation, where f (0) and f (1) represent the meaningless cipher image and the carrier image, respectively. Besides, AF represents the adjustment factor which is used to adjust the hidden visual effect. The schematic diagram of image embedding is demonstrated in Figure 8.
Step 1: Input the second set of secret keys B: (cu 0 , cv 0 , 1 ) and iterate the newly designed TDSC (Equation (4)) for hw + T 0 times to generate a set of chaotic sequence u(u T 0 +1 , u T 0 +2 , u T 0 +3 , …, u hw+T 0 ). Step 2: Sort the chaotic sequence u in light of Equation (22) to obtain the corresponding position index sequence.
Step 3: Create a blank image SI and assign all the pixel values of the carrier image ZI to it. Then the final steganographic image containing cipher information can be obtained through Equation (23), where i = 1, 2, 3, … , hw∕2.

Decryption algorithm
Similarly, the decryption algorithm also includes three processes, namely extraction, decryption and reconstruction. The detailed decryption steps are as follows.
Step 1: Input the second set of secret keys B: (cu 0 , cv 0 , 1 ) and iterate the newly designed TDSC (Equation (4)) to generate the chaotic sequence u. And sort it to obtain the position index matrix Tcu.
Step 2: Extract the cipher image CI which is hidden in the steganographic image by Equation (24).
Step 3: Utilize the first set of secret keys A:(cx 0 , cy 0 , 0 ) to iterate the newly designed TDCS (Equation (4)) to generate the chaotic sequences x and y. And then the Tcx and Tdy are obtained by Equation (15) and Equation (17). Later, the inverse scrambling and diffusion operates are executed to obtain the spliced image.
Step 4: Extract the compressed image CPI from the spliced image. Then perform decomposition, inverse quantization and inverse SVD operations on the compressed image to obtain three sub-images HL, LH and HH.
Step 5: The sub-image LL extracted from the spliced image is stitched with three other sub-images, and then the inverse integer wavelet transform is performed to obtain the recovered plain image I.

Simulation results
In this section, simulation experiments will be performed on the proposed VMIC, which are all carried out on a laptop (1.8 GHz i7-8550U CPU and 16G RAM), which is equipped with MTA-LAB 2018B simulation platform. Six plain images (512 × 512) and six carrier images (512 × 512) are randomly selected for experiments. The secret keys used in the encryption process are set to A: (0.5829, 0.7936, 2.481) and B: (0.9328, −0.5832, 0.9723). In addition, the adjustment factor is set to 0.08. The simulation results demonstrated in Figure 9. It can be clearly seen from the figure that (a) The steganographic images (third column) have a good visualization quality, which are similar to the corresponding carrier images (second column). That is to say the imperceptibility of the cipher image is high. (b) The reconstructed decrypted images (fourth column) obtained by the decryption algorithm are of high quality. Using the BP to embed the cipher image into the carrier image can reduce the probability of illegal acquisition and malicious attacks, which can improve the security of the plain image.

Key space analysis
The key space is a set consisting of all legal initial secret keys during the encryption process. If the key space of the encryption algorithm is too small, the attacker can use the exhaustive attack method to brute force the encryption algorithm. The secret keys of the VMIC proposed in this paper include two parts. The first part of the secret keys is A:(cx 0 , cy 0 , 0 ), which is used to generate the scrambled sequence and diffused sequence. Another part of the secret keys is B:(cu 0 , cv 0 , 1 ), which is used to generate the positional index matrix for hiding the noise-like cipher image. Assuming the precision of the floating-point data is 10 −14 , the total key space is about 2 744 ((10 14 ) 2 × (10 14 ) 2 × (10 14 ) 4 × (10 14 ) 2 × (10 14 ) 2 × (10 14 ) 4 ), which is larger than that of in refs. [33,34,37,38,45]. Since its key space is much larger than 2 100 [60,61], the image cryptosystem proposed in this paper has enough ability to resist exhaustive attack.

5.2.2
Key sensitivity analysis An excellent encryption system not only needs to have a large enough key space to resist exhaustive attacks, but also needs to be sensitive enough to the secret keys. In this subsection, we investigate the sensitivity of the decryption algorithm to the secret keys. Given an infinitesimal change in one of the secret keys and the rest of secret keys remaining unchanged, the corresponding decryption images are shown in Figure 10. It can be seen from the figure that the decrypted images visually do not contain any texture information of the corresponding plain image under the condition of using the wrong secret keys for decryption, even if the secret keys have infinitesimal changes. Next, we use the mean square error (MSE) [62] for further analysis, which is defined in Equation (25).
where P and D represent the plain image and the decrypted image respectively. The MSE curves of two sets of secret keys are plotted in Figure 11. The abscissa in Figure 11 represents the added disturbance, and the ordinate represents the MSE between the decrypted image and the plain image. We can clearly see from Figure 11 that whether a small positive or negative disturbance is added to one of the secret keys or not, an errordecrypted image is obtained, significantly different from the plain image. It shows that the VMIC is extremely sensitive to the secret keys.

Histogram analysis
The histogram of natural image depicts the intensity distribution of pixel values. However, as far as the visually meaningful cipher image is concerned, the more similar its histogram is to the corresponding carrier image, the more difficult it is to make statistical analysis on it. Then, the histogram intersection [35] is adopted to measure the similarity between the meaningful cipher image and the carrier image. Its mathematical model can be calculated from Equation (26).
Where J and V represent the histograms of the two images, respectively. In the tests, the images Brain and Cameraman, Girl and Barbara, Jet and Baboon are subjected to the proposed meaningful encryption scheme, and the numerical results are listed in Table 4. It can be clearly seen from the table that the distance of histogram intersection of our scheme is higher than that of in refs. [35,43], indicating that it has a good ability to withstand the statistical attacks.

Visualization quality analysis
In this subsection, we evaluate the visual quality of the steganographic image and the imperceptibility of the cipher image, which mainly includes pixel-based evaluation methods and structure-based evaluation methods. Since the peak signal-tonoise ratio (PSNR), a pixel-based evaluation method, can quantitatively measure the distortion introduced by information hiding, which is easy to be calculate. Its mathematical expression is shown in Equation (27) [63], where D and I denote the carrier image and the steganographic image, respectively. Additionally, Wang et al. [64] proposed a structure-based distortion evaluation standard, namely the structure similarity in the spatial domain. It expresses the distortion as the three components of structure, brightness and contrast, and then utilizes the total effect formed by these three components to evaluate the quality of an image. And Equation (28) is the mathematical expression for calculating the mean structural similarity (MSSIM). Where i D and i D represent the mean and standard deviation of the i-th non-overlapping sub-block in the image D, respectively.
We select two groups of images (Lena and Woman, Boat and Peppers) for the experiments under different adjustment factors. And the experimental results are displayed in Table 5. It can be clearly seen from the table that as the AF decreases, the visual quality of the steganographic images and the imperceptibility of the cipher images get better and better. Especially when the adjustment factor AF is 0.04, the PSNR and MSSIM of the steganographic images are about 39 dB and 98%, respectively, indicating that the steganographic images have a high similarity with the corresponding carrier images. Beside, when AF = 0.08, the comparison results of the visual quality for multiple meaningful image encryption algorithms are listed in Table 6. As described in the table, the cipher image provided by the proposed meaningful image encryption has better imperceptibility compared with the refs. [33,37,45]. However, ref. [34] embeds the meaningless cipher image into a carrier image via the LSB embedding method, which can reduce the damage intensity to the pixel of the carrier image. Therefore, the visual quality of the proposed scheme is inferior to that of ref. [34].

Reconstruction quality analysis
The quality of the decrypted image is also an important indicator to judge whether a set of image encryption algorithm is excellent or not. Moreover, the visually meaningful image encryption scheme proposed in this paper has data loss in using SVD to compress the plain image and using BP to hide the corresponding meaningless cipher image. In this subsection, we also adopt the PSNR and MSSIM to measure the difference between the decrypted image and the plain image under different adjustment factors. And the experiment results are shown in Table 7. The table indicates that as the AF increases, the quality of the decrypted image also increases, and the difference from the plain image also gradually decreases. Comparing Table 7 with Table 5, we can find that when the adjustment factor is 0.08, the visual quality of the steganographic images generated during the encryption process is the best, and the decrypted images generated during the decryption process also has a high quality. The biggest advantage of this VMIC is that users can weigh the relationship between the visual quality of steganographic image and the quality of reconstructed image, and set appropriate AF to apply to different carrier images. Next, we will compare the quality of the decrypted images in different meaningful image encryption algorithms. In the comparative experiment, the compression rates in refs. [33,34,37,45] are set to 0.25, and the AF in this paper is set to 0.08. The experimental results are demonstrated in Table 8. As can be seen from the table, the quality of the decrypted images produced by the proposed VMIC is better than that of other algorithms in refs. [33,34,37,45]. Besides, if a higher quality decrypted image is required, the AF can be increased appropriately.

5.2.6
Noise and cropping attack analysis  Tables 9 and 10. It can be seen from the two tables that, with the increase of attack intensity, the texture information of the decrypted image becomes more and more blurred, but the contour information contained in the decrypted image can still be clearly seen, indicating that the algorithm presented in this paper has good anti-noise and anti-cropping properties. In addition, it has the same ability to resist noise and cropping attacks under different adjustment factors.

Running efficiency analysis
Operational efficiency is also an indispensable indicator for measuring the overall performance of an encryption algorithm. Figure 12 shows the time required to encrypt a Lena image with the size of 512 × 512 and the percentage consumed by each part of the encryption algorithm. It can be seen from the Figure 12 that the hiding process takes the longest time, accounting for about 60% of the total time. Besides, the compression process also occupies a considerable percentage in the encryption process. Next, the efficiency is compared with other different encryption schemes, and the experimental results are demonstrated in Table 11. As can be clearly seen from the table, the operating efficiency of the VMIC is significantly better than that of in the refs. [33,34,37,38]. The proposed encryption algorithm mainly in this paper involves the following three operations: floating-point arithmetic, floatingpoint sorting and integer exchange. As far as refs. [33,34,37] are concerned, the compressive sensing technology that is used Ref. [45] 32.62 Ref. [33] 32.98 Ref. [34] 33.06 Ref. [37] 28.54 to compress the plain images in the encryption process requires a series of processes such as plain image sparsification, measurement matrix construction and sparse image compression, which are time-consuming compared with the three operations mentioned above.

CONCLUSION
In this paper, a new TDCS with hyperchaotic behaviour over the entire parameter range is designed by cascading the  Chebyshev system and the infinite collapse system to solve the shortcomings of ODCS applied in secure communication.
The performance analyses indicate that compared with other current existing advanced TDCS, the newly designed TDCS has more complex chaotic behaviour and better ergodicity. Besides, the key space and chaos range are all expanded. Then the newly designed chaotic system is utilized to protect the crowd data, and a novel visually meaningful image encryption algorithm is proposed, which consists of compression, encryption and embedding. Finally, simulation results and security performance analyses demonstrate that the proposed visually meaningful image crypto-system has good visual security, decryption quality and robustness, indicating that it has a wide range of application scenarios. In the following work, we will aim to further improve the efficiency of information transfer by hiding multiple cipher images into a carrier image using the fractional Fourier transform-based embedding method.