
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)01942-4
10.1016/j.isci.2024.110717
110717
Article
Image encryption hiding algorithm based on digital time-varying delay chaos model and compression sensing technique
Jin Bingxue 1
Fan Liuqin 1
Zhang Bowen 1
Lei Rongqing 1
Liu Lingfeng vatanoilcy@163.com
12∗
1 School of Software, Nanchang University, Nanchang 330029, Jiangxi, China
∗ Corresponding author vatanoilcy@163.com
2 Lead contact

15 8 2024
20 9 2024
15 8 2024
27 9 11071722 3 2024
27 6 2024
7 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
Summary

Most of the existing image encryption algorithms encrypt images as meaningless cryptographic images, which can easily attract the attention of attackers during transmission. To address this problem, scholars have proposed to hide the cipher image in a meaningless carrier image. However, larger carrier images are often required, which occupy more bandwidth. In order to solve this problem, this paper realizes embedding the color secret image into the carrier image whose size is equal to or even smaller than the original image by combining the chaotic compressed sensing model. First of all, the original image is sparsely processed using discrete wavelet transform. Then the time varying delay chaotic model is used to generate pseudo random sequence and then the measurement matrix is constructed to compress and encrypt the image. In the end, using singular value decomposition to achieve image embedding, the carrier image carrying information is obtained.

Graphical abstract

Highlights

• A method that can suppress the degradation of chaotic system dynamics is proposed

• Proposing a new measurement matrix construction method

• Proposing a strong robust embedding algorithm

Applied sciences; Artificial intelligence; Computer science

Subject areas

Applied sciences
Artificial intelligence
Computer science
Published: August 15, 2024
==== Body
pmcIntroduction

With the rapid development of the information age, how to ensure the security of digital information in the transmission process is the focus of network security. As digital images are intuitive, vivid, and so on and are widely used in different fields such as military, business, etc., so how to ensure the security of the image is to become the top priority of network security.

Traditional cryptography usually treats plaintext information as a binary stream for processing, but due to the large amount of image data and the two-dimensional distribution of the data, traditional encryption methods are not applicable to image encryption. Since the 1990s, a variety of image encryption techniques have been proposed by researchers incorporating the characteristics of digital images. Image encryption algorithms can be categorized into space domain encryption and transform domain encryption depending on the encryption domain. Space-domain encryption1,2,3,4 involves treating the image as a two-dimensional matrix and performing invertible operations on the pixels of the image, which mainly involves rearranging the pixel positions and changing the pixel values. Transform domain encryption5,6,7,8 involves performing operations such as discrete wavelet transform (DWT), discrete cosine transform (DCT), and Fourier transform (FT) on the image before encrypting it in the frequency domain.

The pseudo-randomness, initial value sensitivity, and ergodicity characteristic of chaotic systems are in line with the principles of disruption and diffusion proposed by Shannon in modern cryptographic regimes.9 In 1997, Fridrich10 first applied chaotic map to image encryption. In recent years, image encryption based on chaos theory has become the research direction of many scholars1,11,12,13,14,15,16,17; chaotic encryption algorithms usually use the chaotic system as a pseudo-random sequence generator, and the generated pseudo-random sequence is used as a key to operate with the plaintext so as to obtain the ciphertext, and this process usually acts in two phases of disruption and diffusion.

In order to improve the efficiency of image transmission, traditional image encryption techniques are combined with compression-aware techniques18 so that they can protect the image security while compressing the amount of data. Although compressed perception itself has the ability to protect information, encrypting an image using only compressed perception techniques is not secure enough, and it is usually combined with other algorithms. Encryption schemes that combine chaotic systems with compressed perception techniques are proposed.19,20,21,22,23,24,25,26 In a study by Liang and Xiao,20 the original image is compressed from both directions simultaneously using a two-dimensional cosine-type logistic map and then the image is discretized with the help of the generator matrix of a two-dimensional metacellular automaton. In a study by Khan et al.,22 the DCT data of the original image are segmented, the sparse data are disambiguated and compressed, the non-sparse data are disambiguated, and finally merged and processed to get highly obfuscated. In another study by Ye et al.,24 the image is first compressed using Bernoulli observation matrix and then the compressed image is encrypted using Henon-like map. However, these algorithms do not fully balance the quality of image reconstruction as well as the security of image encryption.

Based on the aforementioned analysis, this paper proposes an image encryption algorithm that combines one-dimensional chaotic map with compression-aware techniques. The scheme is mainly divided into two stages, in the first stage the compression perception and chunking techniques are utilized to realize the compression and encryption of the image to obtain a noise-like secret image. In the second stage image embedding is realized by discrete wavelet transform (DWT) and singular value decomposition (SVD) techniques to obtain a visually meaningful cryptographic image. The innovations are as follows:(1) Considering the phenomenon of feature degradation when the chaotic model is realized on the computer, it no longer meets the requirements of cryptographic security. For this reason, this paper proposes a chaotic model that introduces time-varying delay, which can effectively inhibit the phenomenon of degradation of the dynamics of digital chaotic systems with good security and complexity.

(2) Using the chaotic sequence generated by the chaotic model combined with the construction method of Toplitz matrix, the measurement matrix in the compressed sensing process is constructed, which satisfies the constrained isometric property (RIP).27 The measurement matrix is tested to have high-compressive reconstruction performance. The chaotic sequences generated by the chaotic model are used in the disruption and diffusion process of encryption, which increases the ability of the algorithm to resist plaintext attacks.

(3) Combining DWT and SVD techniques to realize the embedding of secret images and improve the robustness of existing embedding algorithms.

The rest of the paper is organized as follows. In Section a new chaotic model, we introduce a new chaotic model with anti-dynamic degradation properties and analyze the performance of this chaos model. A new chaotic measurement matrix is constructed in Section construction of chaotic measurement matrix. The detailed procedure of the image encryption algorithm is described in Section image encryption algorithms. Simulation results and performance analysis of the experiment are shown in Section experimental results and safety analysis. The conclusion of the paper is provided in section conclusion.

Results and discussion

A new chaotic model

Logistic map

Among all chaotic maps, logistic map can be said to be one of the most typical chaotic maps, and it is widely used in image encryption. The formula for logistic map can be written:(Equation 1) xi+1=f(xi)=μxi(1−xi)

In the aforementioned equation, f is the iterative function, μ is the control parameter, and xi is the state variable. However, the logistic map is not necessarily in a chaotic state, which is related to the value of μ. Research has shown that logistic maps only exhibit chaotic properties when 3.5699456<μ<4. The restricted interval obtained by map is [0,1]. Moreover, the state of xi+1 is determined by xi. Once an initial value x0 is given, the corresponding pseudo random sequence {xi} can be generated. However, an ideal chaotic map should have higher complexity and better randomness.

Chaotic model based on time-varying delay

Due to the limited accuracy of computers, the state of chaotic maps will eventually enter a cycle, which we call the dynamic degradation of digital chaotic systems. To solve this problem, this paper introduces time-varying delay to control parameters. Using the state variable xi to perturb parameter μ,Δμ, its mathematical model can be described as:(Equation 2) h(xi)=μ+(4−μ−Δμ)∗xi−floor(i/p)

Where h represents the parameter control function. Δμ is used here as a variable that acts as a stabilizer for the whole parametric control function. Under the influence of the finite precision of the computer, for the logistic map, the chaotic performance is stable when the value of the control parameter is close to 4 but not equal to 4. At this point, Δμ ensures that the model still has good chaotic performance even when the control parameter μ=4.

Compounding Equations 1 and 2 generates an improved and chaotic digital chaotic system. The mathematical formula is as follows:(Equation 3) xi+1=FL(f(h(xi),xi),p)=FL(h(xi)∗xi∗(1−xi),p)

(Equation 4) FL(x)=⌊x∗2p⌋2p

There, FL is the accuracy control function, expression as in Equation 4, where p denotes the computer precision. If xi+1 is only affected by the current state variable xi, the security of the encryption algorithm constructed using it will be greatly reduced. Therefore, in this study, we introduce a delayed state in the parameter control function xi−floor(i/p). It is well known that the occurrence of xp=xq in chaotic sequences cannot be avoided, and all we have to ensure is that xp+1≠xq+1 when that situation occurs. In the model proposed in this study, since the delayed state introduced is time-varying, it is guaranteed to jump out of the loop when a occurs in the next state, that is, xq+1, thus increasing the complexity of the generated sequence.

Performance analysis of chaotic models

In this section, we will analyze the performance of the model based on the following numerical parameters. Unless otherwise specified, these parameters are selected as μ=3.98, Δμ=0.01, p=10, x1=0.1234.

Lyapunov exponent

As is well known, the Lyapunov exponent is a typical analysis method for determining whether a system has chaotic characteristics and the degree of chaos. For chaos, there must be a Lyapunov exponent λ that is positive. Therefore, it is possible to determine whether the system is chaotic by calculating the maximum Lyapunov exponent. In this section, the variation of Lyapunov exponents of Equations 1 and 3 under the influence of control parameter μ is tested and compared with the literature.28,29 The parameters of the cited literature28 are set as a=μ, Δa=0.02, k=25.25−6.25×a, the parameter p=0.78 in literature.29 And the maximum Lyapunov exponent spectra are shown in Figure 1. From the figure, it can be seen that the Lyapunov exponent of the new chaotic model has always been greater than 0 and is higher than the chaos model proposed in the literature. Thereby, it can be judged that the new model has good chaos.Figure 1 Diagram of the Lyapunov exponent: comparison of Equations 1 and 3,28,29 under the influence of the parameter μ.

Trajectory and phase space

Trajectories and phase space are the most intuitive methods to show the ergodicity property and pseudo randomness of chaotic maps. In this test, the precision is set to p=10. The trajectory map of the original logistic map and the new chaotic model is shown in Figure 2. It can be seen from the figure that the trajectory of the original logic map enters the cycle without more than 50 iterations, while the newly constructed chaotic model remains chaotic after 500 iterations. Figure 3 shows the phase space of the original logistic map and the new chaotic model. As can be seen from the figure that the phase space of the new chaotic model is much larger than the original logistic map, and the phase space of the new chaotic model is still a parabola, which indicates that the new chaotic model does not destroy the phase space structure of the original logistic map.Figure 2 Trajectory diagrams

(A) Trajectory diagram of Logistic map.

(B) Trajectory diagram of the new chaotic model.

Figure 3 Phase space

(A) Phase diagram of Logistic map.

(B) Phase diagram of the new chaotic model.

Auto-correlation analysis

Auto-correlation is the standard for evaluating the randomness of a sequence. An ideal chaotic sequence should have an incremental auto-correlation function. Figure 4 provides the auto-correlation function between the original logistic map and the new chaotic model. From Figure 4, we can easily observe that the auto-correlation function of the new chaotic model rapidly decreases with the increase of interval and remains stable around 0, while the original logistic map is not. This also provides further evidence that the sequence generated by the new chaotic system is a well-random sequence.Figure 4 Auto-correlation function

(A) Autocorrelation analysis diagram of Logistic map.

(B) Autocorrelation analysis diagram of the new chaotic model.

Period analysis

A digital chaotic map ends up in a cyclic state within the limits of computer precision. And the number of iterations before the map enters the cycle is an important metric to study the resistance to dynamic degradation. In this section, the periodicity of Equations 1 and 3 and the new chaotic model is evaluated from two main aspects, respectively the length of the cycle and the number of iterations for the first entry into the cycle. In this test, we set the length of the generated sequence to 105 and varied the computer precision from 2−6 to 2−18, and the results are shown in Table 1. From the table, we can see that the cycles of the model constructed in this study cannot be detected at an accuracy of 2−8, which fully verifies that the model represented by Equation 3 has some resistance to kinetic degradation. Furthermore, compared to the original logistics map, the cycle length and the number of iterations for the first entry into the cycle of the improved chaos model are much larger than the original map as the accuracy increases. Therefore, it can be determined that the improved map has superior resistance to dynamic degradation and better chaos.Table 1 Period analysis of the original logistic map and the new chaotic model (U indicates undetected)

Precision	Period of Equation 1	Period of Equation 3	Number of iterations when entering a cycle (Equation 1)	Number of iterations when entering a cycle (Equation 3)	
2–6	11	11	5	5	
2–7	6	6	4	4	
2–8	4	U	11	U	
2–9	23	U	8	U	
2–10	3	U	1	U	
2–11	46	U	15	U	
2–12	17	U	33	U	
2–13	11	U	87	U	
2–14	12	U	114	U	
2–15	93	U	195	U	
2–16	104	U	67	U	
2–17	227	U	46	U	
2–18	684	U	328	U	

Bifurcation diagram

The chaotic behavior of chaotic maps is determined by their control parameters, and bifurcation diagrams can observe the chaotic behavior of chaotic maps within different parameter ranges. In general, the larger the parameter range for generating chaotic behavior, the larger the key space and the more resistant to brute force attacks. Figure 5 shows the bifurcation diagram of the original logical map and the new chaotic model. As can be seen from Figure 5A, the original logic map only appears when the control parameter μ>3.5699, while the new chaotic model shown in Figure 5B shows chaotic behavior between [1.52,4]. From this, it can be judged that the new chaotic model has a wider key space.Figure 5 Bifurcation diagram

(A) Bifurcation diagram of Logistic map.

(B) Bifurcation diagram of the new chaotic model.

Sensitivity to initial conditions

The sensitivity of the initial condition means that any small change in initial conditions that can cause significant differences in the future through the continuous amplification of chaotic systems. In this test, we set control parameter μ=3.98, Δμ=0.01, precision p=10, and initial value x1=0.12. From Figure 6, it can be seen that the control parameter and initial value x1 only change by 2−12, which can generate two completely different sequences. This proves that the new chaotic model has satisfactory sensitivity to initial conditions.Figure 6 Sensitivity analysis of initial conditions

(A) Control parameter μ changed.

(B) Control parameter Δμ changed.

(C) Initial value x1 changed.

Complexity analysis

The complexity of chaotic models is usually measured by two indicators: permutation entropy (PE) and approximate entropy (ApEn). The more random the sequence, the higher is its complexity and corresponding entropy value. For an ideal random sequence, its PE value should be infinitely close to 1. Refer the section “lyapunov exponent” for parameter settings in literature,28,29 and the parameters of the literature by Bryc et al.30 are set to x1=0.1234, x2=0.5122, y=0.3086, y2=0.4107, r1=3.98, a=0.99, as shown in Figures 7 and 8. It can be observed that with changes in precision p, the PE value of the new chaotic system approaches 1, and the PE and ApEn of this system outperform the original logistics map, even when compared to the models in the literature.28,29,31 This can prove that the sequence generated by the improved map is complex.Figure 7 Permutation Entropy diagram: comparison of Equations 1 and 3,2829 under the influence of the Precision p.

Figure 8 Approximate Entropy diagram: comparison of Equations 1 and 3,28,30 under the influence of the Precision p.

Correlation dimension

The association dimension is a measure of the complexity of phase space attractors. The larger the correlation dimension is, the higher the complexity of the corresponding chaotic system is, and the lower the similarity of the generated sequence is. Figure 9 shows the correlation dimensions of Equations 1 and 3 and literature by Liu L. et al. and Zhang S and Liu L.28,29 It can be seen from the figure that with the increase of precision p, the correlation dimension of the new chaotic model is always greater than the original logistic map. This indicates that the new chaotic model has higher complexity.Figure 9 Correlation dimension: comparison of Equations 1 and 3,2829 under the influence of the Precision p.

Information entropy analysis

Information entropy, in layman’s terms, is the expected value of all possible events. For a chaotic system, the more situations occur, the greater the expected value, and the greater the corresponding information entropy. Therefore, information entropy can be used as a measure of the complexity of chaotic systems. In this section, the information entropy of Equations 1 and 3 and the cited literature28,31 is calculated, and the results are shown in Figure 10. From the figure, it can be seen that the information entropy of the chaotic model proposed in this paper is always above the literature with the change of computer accuracy, from which it can be concluded that our chaotic system has a high time complexity.Figure 10 Information Entropy diagram: comparison of Equations 1 and 3,28,30 under the influence of the Precision p.

Construction of chaotic measurement matrix

Construction of measurement matrix based on new chaos model

Compressed sensing mainly includes three parts: signal sparse representation, measurement matrix construction, and reconstruction algorithm design. Among them, the construction effect of the measurement matrix directly affects the amount of information obtained after compressed sampling. Therefore, the measurement matrix, as a compressed sampling system for the original signal, plays a crucial role in the entire process. This section will apply a new chaotic model to generate chaotic sequences and use the randomness of the chaotic sequences to construct the Toeplitz measurement matrix.

The Toeplitz matrix, also known as the T-type matrix, was proposed by Bryc, Dembo, and Jiang in 2006.30 As a measurement matrix in compressed sensing algorithms, the advantage of Toeplitz matrix is that it requires very few elements to be stored or transmitted. The Toeplitz measurement matrix is composed of +1 and −1, and a random n×n Toeplitz matrix Tn can be written as:Tn=[X0X1X2⋯Xn−2Xn−1X−1X0X1Xn−2X−2X−1X0⋱⋮⋮⋱X2X2−nX0X1X1−nX2−n⋯X−2X−1X0]

It can be easily observed that the elements on the main diagonal of the Toeplitz matrix are equal, and the elements on lines parallel to the main diagonal are also equal. The Toeplitz matrix can be directly determined by the elements on the first row and column. That is to say, for a matrix of m×n, only m+n−1 elements need to be transmitted.

The measurement matrix in compressed sensing should meet the RIP restricted isometric principle. Wang, Liu, Zhang et al.32 verified that the Toeplitz matrix meets the RIP principle in 2010. The traditional Toeplitz matrix generation method requires the transfer of the first row and first column elements in the matrix, but this will occupy a large bandwidth. In order to improve this issue, this article uses a chaotic model to assist in the generation of the Toeplitz matrix. During transmission, only the initial value of the chaotic system and the value length of the chaotic sequence need to be used as the key, greatly reducing the key length. The process of constructing a measurement matrix for m×n (m<n) using a new chaotic model is as follows:

Calculate the hash value of the plaintext image and use it as the initial value of the chaotic system. The 256-bit hash value of the plaintext images was generated using the SHA-256 function, divide them into 32 groups in groups of 8 bits, denoted as {ki}, where i=1,2,...,32. Then convert {ki} to decimal, denoted as {Ki}. Calculate x1 according to Equation 4. Here, t is a custom external value, and the mod operation can limit the initial value to the interval (0,1).(Equation 5) x1=mod(t+(K1+K2+……+K32)/256,1)

(1) Input the initial value x1 of the chaotic model, iterate the new chaotic model 2×N+100 times, remove the first 100 iterative values of the random sequence, and obtain a random sequence S={s1,s2,...,s2n−1} with a length of 12×N−.

(2) The sequence S generated by the chaotic model has a value range of (0,1). The random sequence S is binarized by Equation 6, and the sequence S1 composed of +1, −1 is obtained.

(Equation 6) S1={1,mod(s∗10,2)>θ−1,else

(3) Use the generated sequence S1 to construct a Toeplitz measurement matrix of size N×N, and intercept the first M rows of the matrix to obtain the measurement matrix of M×N.

Performance analysis of measurement matrix based on new chaos model

To verify the compression and reconstruction performance of the Toeplitz measurement matrix generated by the aforementioned method, observing the quality of the reconstructed image and calculating the peak signal-to-noise ratio (PSNR) of the reconstructed image are the most intuitive methods. First of all, we use discrete wavelet basis to sparsely transform the original image. Then, the image is compressed sensing using the generated measurement matrix. In the end, use the OMP algorithm to reconstruct the image. The parameters setting during the generation of the measurement matrix are as follows: θ=0.95, t=0.35416, and refer to the section “performance analysis of chaotic models” for the chaotic system parameter settings, the sampling rate selected by the compressed sensing algorithm is CR=0.75, CR=0.50, CR=0.25. Then perform reconstruction tests on the images Lena and Peppers, and the implementation results are shown in Figures 11 and 12. Visually, it can be clearly seen that the reconstructed image still has the obvious characteristics of the original image, which indicates that it is feasible to use the measurement matrix in the compressed sensing algorithm. In order to further verify the superior compression and reconstruction performance of this measurement matrix, this section compares it with classical measurement matrices and compares the PSNR values of reconstructed images. The results are shown in Tables 2 and 3.Figure 11 Lena reconstructed image

(A) Compression ratio CR=0.75.

(B) Compression ratio CR=0.50.

(C) Compression ratio CR=0.25.

Figure 12 Peppers reconstructed image

(A) Compression ratio CR=0.75.

(B) Compression ratio CR = 0.50.

(C) Compression ratio CR=0.25.

Table 2 Comparison of PSNR (db) of Lena reconstructed from different measurement matrices

Sampling Rate	0.75	0.50	0.25	
Matrix of this article	32.7221	27.7145	21.3857	
Sparse Random Matrix	32.1105	27.3567	20.8853	
Bernoulli measurement matrix	32.2151	27.3171	21.1960	
Gaussian measurement matrix	32.0442	27.2250	21.2360	
Toeplitz matrix	32.7062	27.5687	21.3348	

Table 3 Comparison of PSNR (db) of Peppers reconstructed from different measurement matrices

Sampling Rate	0.75	0.50	0.25	
Matrix of this article	34.3174	29.0518	22.3592	
Sparse Random Matrix	34.0472	28.8324	22.1771	
Bernoulli measurement matrix	33.9622	28.8995	22.3361	
Gaussian measurement matrix	33.9395	28.5884	22.1760	
Toeplitz matrix	34.1676	28.9007	22.3549	

The reconstruction quality of the image is directly proportional to the PSNR value. The higher the PSNR value, the better the image quality. It is generally believed that the image quality is unacceptable when the PSNR value is below 20 db. Through analysis, it can be found that the reconstruction quality of the Toeplitz measurement matrix based on the new chaotic model proposed in this article is basically higher than that of other measurement matrices. Moreover, it can be clearly noticed during the measurement process that the reconstruction quality of other measurement matrices is not stable, in contrast to the construction method proposed in this study, which does not change its reconstruction quality as long as the initial value remains unchanged. This shows that the measurement matrix is practical in compressed sensing engineering.

Image encryption algorithms

The encryption process proposed in this article mainly includes image compression, scrambling, and embedding based on DWT-SVD; the decryption process includes DWT-SVD based extraction, inverse scrambling, and image reconstruction. The detailed description is as follows.

Encryption process

The proposed encryption scheme is shown in Figure 13, which mainly includes three processes: key generation, image compression, and image embedding.Figure 13 Flowchart of encryption process.

Key generation

The SHA-256 function is used to generate the initial value of the new chaos model, and the association between the encryption algorithm and the original image is established. Refer to the section “construction of measurement matrix based on new chaos model” for the specific key generation process.

Compression process based on new chaos model

Step 1: Before encryption, divide the color image of N×N into three channels: red, green, and blue, denoted as R, G, and B.

Step 2: Input the obtained initial value x1 into Equation 3, iterate 2×N+99 times, and obtain a chaotic sequence. Remove the first 100 sequence values to obtain a random sequence S with a length of 2×N−1.

Step 3: Using Equation 6 to binarize the random sequence, S1 is obtained. According to the property of Toplitz matrix, the sequence S1 is transformed into an N×N matrix, and the measurement matrix diameter of M×N is obtained by intercepting the first M rows of the matrix. In this experiment, N=2×M.

Step 4: Sparse representation of the image. A sparse basis Ψ with size N×N based on discrete wavelet change (DWT) is defined, and R, G, and B are transformed by discrete wavelet. The method is shown in Equation 7, and sparse matrices R1, G1, and B1 with size N×N are obtained.(Equation 7) {R1=full(ψ×sparse(R)×ψT)G1=full(ψ×sparse(G)×ψT)B1=full(ψ×sparse(B)×ψT)

Step 5: Image measurement. Using the measurement matrix constructed in step 4, the sparse transform matrices R1, G1, and B1 are compressed and measured, and the matrix R2, G2, and B2 of size M×N are obtained after compression. The formula is as follows:(Equation 8) {R2=ΦR1G2=ΦG1B2=ΦB1

Step 6: Normalization process. To facilitate the hiding of the compressed image, each pixel value of the image is restricted to the range of the interval [0.255].(Equation 9) {R3(i,j)=round(255×R2(i,j)−RminRmax−Rmin)G3(i,j)=round(255×G2(i,j)−GminGmax−Gmin)B3(i,j)=round(255×B2(i,j)−BminBmax−Bmin)

Step 7: Scrambling, diffusion processing. To further improve the security of the secret image, R, G, and B of image P1 are uniformly divided into a block matrix of size b×b and Arnold transform is applied to each block. And iterated n times to obtain image P2. Then, the pixel values of image P2 are diffused as a whole to ensure the uniform distribution of pixel values after diffusion to obtain the final secret image P3.

Embedding based on DWT-SVD

Step 1: Carrier image preprocessing. Firstly, the color carrier image Pc of size N×N is separated into three channels, noted as Rc, Gc, and Bc. The image is then decomposed using the DWT and the decomposition structure is extracted. After the DWT is applied to the image, each channel is decomposed into four subbands. Since DWT sampling needs to satisfy Nyquist’s theorem and generally uses two times the separation points, the length and width of all four subbands after decomposition become 1/2 of the original. That is, the low frequency information (LL) of the image; the high frequency information (LH) in the horizontal direction direction of the image; the high frequency information (HL) in the vertical direction of the image; and the high frequency information (HH) in the diagonal direction of the image. In the experiment, two subbands of LL and HH of the carrier image were selected as the carriers of information hiding, noted as RLL, RHH, GLL, GHH, BLL, and BHH.

Step 2: Block processing. In order to facilitate the embedding of the images, the carrier image and the secret image need to be blocked. The six subbands in step 1 are divided evenly into blocks of a×a as embedding carriers, block matrix as RLL{i,j}, RHH{i,j}, GLL{i,j}, GHH{i,j}, BLL{i,j}, BHH{i,j}. At the same time, divide the three channels of the secret image P3 into blocks of the same size, denoted as R3{i,j}, G3{i,j}, B3{i,j}.

Step 3: Embedding process based on SVD. The block matrices of carrier image and secret image are decomposed by SVD respectively, as shown in Equation 10. Where U, V are orthogonal matrices and S are diagonal matrices. In these three matrices, the number of elements in S is so large that making small changes to them is visually difficult to detect. Thus, we choose matrix S for embedding. According to the physiological theory of color, the embedding intensity of different channels will vary with the perceived intensity of the human eye. Therefore, we follow the rules of Equation 11 for image embedding. where α1, α2, α3 denote the embedding strength of different channels, respectively. Where, α1>α2>α3, embed according to Equation 11. Where SRL, SRH, SGL, SGH, SBL, SBH represent the S matrix after carrier image decomposition, SRL1, SRH1, SGL1, SGH1, SBL1, SBH1 represent the S matrix after embedding, SR3, SG3, SB3 and represent the S matrix of the secret image.(Equation 10) [U,S,V]=svd(A)

(Equation 11) {SRL1{i,j}=SRL{i,j}+α2×SR3{i,2j−1}SRH1{i,j}=SRH{i,j}+α2×SR3{i,2j}SGL1{i,j}=SGL{i,j}+α3×SB3{i,2j−1}SGH1{i,j}=SGH{i,j}+α3×SB3{i,2j}SBL1{i,j}=SBL{i,j}+α1×SG3{i,2j−1}SBH1{i,j}=SBH{i,j}+α1×SG3{i,2j}

Step 4: First, the S-matrix embedded in the secret image is combined with U and V of the original carrier image to realize the inverse process of singular value decomposition, as shown in Equation 12. The composite block matrix RLL1, RHH1, GLL1, GHH1, BLL1, BHH1 is obtained. Then the combination of the block matrix is realized to obtain the subbands LL1 and HH1 embedded in the secret image. Each layer subband is re-fused, and the formula is shown as Equation 13 to obtain the final cryptographic image Pe.(Equation 12) A=U×S×VT

(Equation 13) Pe=idwt2(LL1,LH,HL,HH1)

Decryption process

The decryption process is the reverse process of encryption, and the flow chart is shown in Figure 14. It can be seen that the decryption process mainly includes the extraction based on DWT-SVD and image reconstruction, and the initial value used in the chaos model is obtained from the key. Any errors in the parameters in the key cannot result in the correct decrypted image.Figure 14 Flowchart of decryption process.

Extraction based on DWT-SVD

Step 1: Separate the RGB three channels of the cryptographic image and the carrier image, and perform discrete wavelet transform on them. The LL and HH subbands of the cryptographic image are obtained, denoted as RLL2, RHH2, GLL2, GHH2, BLL2, BHH2. The subbands of the cryptographic image and carrier image are divided into blocks of size a×a, and represent the carrier image with the encrypted image as a matrix block.

Step 2: Extraction process based on SVD. First, the singular value decomposition is carried out for each subband of the carrier image Pc and the cryptographic image. The S-matrix of each subband is obtained by following the Equation 10. The cryptographic image is denoted as SRL2, SRH2, SGL2, SGH2, SBL2, SBH2. The S-matrix of secret image is extracted by Equation 14. SR4,SG4,SB4 represent the S-matrix of the secret image. Then, S matrix of the obtained secret image is combined with U and V matrix of the original secret image (U and V matrix of the original secret image are used as key transmission), and matrix blocks of each channel of the secret image are obtained by using the method shown in Equation 12.(Equation 14) {SR4{i,2j−1}=[SRL2{i,j}−SRL{i,j}]/α2SR4{i,2j}=[SRH2{i,j}−SRH{i,j}]/α2SG4{i,2j−1}=[SBL2{i,j}−SBL{i,j}]/α1SG4{i,2j}=[SBH2{i,j}−SBH{i,j}]/α1SB4{i,2j−1}=[SGL2{i,j}−SGL{i,j}]/α3SB4{i,2j}=[SGH2{i,j}−SGH{i,j}]/α3

Step 3: Block composition. The secret image P3 is obtained by combining the matrix blocks of each channel of the secret image.

Reconstruction process

Step 1: Reverse the scrambling process. The RGB three-channel of secret image P3 is separated and divided into a block matrix of size b×b. Arnold inverse transformation is performed on each block to obtain the image P4 to be reconstructed.

Step 2: The pixel values in image P4 are inversely normalized as in Equation 15. Where, R4, G4, B4 denote the three-channel matrix of image P4 and R5,G5,B5 denote the pixel matrix after the inverse normalization process.(Equation 15) {R5(i,j)=R4(i,j)×(Rmax−Rmin)255+RminG5(i,j)=G4(i,j)×(Gmax−Gmin)255+GminB5(i,j)=B4(i,j)×(Bmax−Bmin)255+Bmin

Step 3: Extract the 256-bit hash value in the key and calculate the initial value x1 of the chaos model with the external key t. The Toeplitz measurement matrix Φ based on chaos model is generated by referring to the measurement matrix construction method in the section “compression process based on new chaos model”. The sparse matrix R6, G6, and B6 are reconstructed by OMP algorithm.

Step 4: Inverse discrete wavelet transform is realized through Equation 16, and image three channels are combined to obtain the decrypted image P5 after decompression.(Equation 16) {R7=full(ΨT×sparse(R6)×Ψ)G7=full(ΨT×sparse(G6)×Ψ)B7=full(ΨT×sparse(B6)×Ψ)

Experimental results and safety analysis

In this section, we conduct a series of simulation experiments to verify the feasibility of our proposed encryption algorithm. In the simulation test, the parameters are selected as follows: the parameters in Equations 2 and 3 are set to μ=3.98,Δμ=0.01,p=10, the external value t=0.35416. The size of the original image and the carrier image is selected as 256×256, and the compression ratio is set to CR=0.5. In section compression process based on new Chaos model, the block size of Arnold transformation process is 16×16. In section “embedding based on DWT-SVD”, the block size of carrier image and secret image during information hiding process is 8×8, and the embedding intensity is set as α1=0.08,α2=0.05,α3=0.02.

Encryption and decryption tests

Two color images, “Lena” and “Peppers”, were used as the original test images, and two color images, “Butterfly” and “Boats”, were used as the carrier images. Figure 15 shows the test results, with the first to fifth rows indicating the original image, the carrier image, the secret image, the cipher image, and the decrypted image, respectively. As can be seen from the figure, the secret image is compressed to one-half of the original image and a noise-like image is formed, which protects image privacy while also reducing the bandwidth occupied during transmission. The high similarity between the cryptographic image and the original image makes the image less likely to attract the attention of hackers during transmission. In addition, the decrypted images are similar to the respective original images, verifying the feasibility of the encryption algorithm.Figure 15 Simulation results of this algorithm

(A) Original image of Lena.

(B) Original image of Peppers.

(C) Original image of Lena.

(D) Original image of Peppers.

(E) Carrier image of Butterfly.

(F) Carrier image of Butterfly.

(G) Carrier image of Boats.

(H) Carrier image of Boats.

(I) Encrypted image of Lena.

(J) Encrypted image of Lena.

(K) Encrypted image of Peppers.

(L) Encrypted image of Peppers.

(M) Cipher Image (Lena & Butterfly).

(N) Cipher Image (Peppers & Butterfly).

(O) Cipher Image (Lena & Boats).

(P) Cipher Image (Peppers & Boats).

(Q) Decrypted image (Lena & Butterfly).

(R) Decrypted image (Peppers & Butterfly).

(S) Decrypted image (Lena & Boats).

(T) Decrypted image (Peppers & Boats).

In addition, in order to further objectively compare the similarity between two images, this paper calculates the PSNR between the original and decrypted images, and between the carrier image and the cipher image, which are denoted by PSNRO and PSNRC, respectively. The results are shown in Table 4. The data in the table indicates that the distortion caused by the image after compression and reconstruction is acceptable and that the embedding of the image does not cause a significant impact on the carrier image.Table 4 PSNR value of simulation results (unit: dB)

Original Image	Carrier Image	PSNRO	PSNRC	
Lena	Butterfly	26.7208	33.1642	
Peppers	Butterfly	28.3209	33.2288	
Lena	Boats	26.7208	33.1171	
Peppers	Boats	28.3209	33.1845	

Key space

The key space is the sum of the keys that can be used in an encryption algorithm and is usually considered to be greater than 2128. In the encryption algorithm of this paper, the key keys include the control parameter μ and Δμ of the chaotic system, the external key t, the 256-bit hash value generated by the SHA-256 function, the image sampling rate CR, the normalization processing parameter Rmax, Rmin, Gmax, Gmin, Bmax, Bmin, and the U, V matrices in the image embedding process. Assuming a computer accuracy of 10−14 and disregard the 256-bit hash, the result is 2460, which is much larger than 2128. Table 5 records the results of the comparison of the key space of this paper’s encryption algorithm with other algorithms,33,34,35 and it can be seen that the key space of this paper’s algorithm is larger than that of other algorithms, and it has a high level of security.Table 5 Key Space comparison

Algorithms	Our	Ahuja et al.34	Chai et al.35	Wang et al.33	
Key Space	2460	2398	2279	2177	

Key sensitivity

A good encryption scheme should have very strong key sensitivity, that is, the correct decrypted image cannot be deciphered during decryption, even though the key has changed only slightly. In our algorithm, only μ, Δμ as well as t mentioned in the section “key space” are selected for sensitivity testing. By changing μ, Δμ, and t by 2−12 and keeping the other keys constant, the decrypted image is shown in Figure 16. From the figure, it can be seen that as long as the key is changed, it will cause the decryption algorithm to be unable to restore the correct plaintext image, which fully demonstrates that the algorithm in this paper has a strong key sensitivity.Figure 16 Sensitivity analysis of keys

(A) Decrypted image when the key is correct.

(B) The decrypted image corresponding to a change in the control parameter μ.

(C) The decrypted image corresponding to a change in the control parameter Δμ.

(D) The decrypted image corresponding to a change in the initial value x1.

Histogram analysis

Histogram analysis reflects the pixel distribution of an image. An ideal encryption algorithm should have an even distribution of pixels in the encrypted image and the corresponding histogram should be smoother than before encryption. The histograms of the original image, the carrier image, the secret image, the cipher image, and the decrypted image are shown in Figure 17 in the first to fifth rows, respectively, and correspond to Figure 15 according to numbering. As can be seen from the figure, the histogram of the original image has significant fluctuations, while the histogram of the secret image is very smooth. Moreover, the decrypted images hidden in different carriers are different from the original images, but the general distribution has not changed, which also shows the feasibility of the algorithm. On the other hand, the histogram of the cipher image is basically similar to that of the carrier image, and the histogram of the carrier image embedded with different information is also basically similar. This shows that the embedding algorithm does not destroy the original pixel value distribution of the carrier image and has strong security.Figure 17 Histogram analysis

(A) Original image of Lena.

(B) Original image of Peppers.

(C) Original image of Lena.

(D) Original image of Peppers.

(E) Carrier image of Butterfly.

(F) Carrier image of Butterfly.

(G) Carrier image of Boats.

(H) Carrier image of Boats.

(I) Encrypted image of Lena.

(J) Encrypted image of Lena.

(K) Encrypted image of Peppers.

(L) Encrypted image of Peppers.

(M) Cipher Image (Lena & Butterfly).

(N) Cipher Image (Peppers & Butterfly).

(O) Cipher Image (Lena & Boats).

(P) Cipher Image (Peppers & Boats).

(Q) Decrypted image (Lena & Butterfly).

(R) Decrypted image (Peppers & Butterfly).

(S) Decrypted image (Lena & Boats).

(T) Decrypted image (Peppers & Boats).

Pixel correlation analysis

In the original image, there is always a strong correlation between neighboring pixels. Therefore, removing strong correlations between neighboring pixels is crucial for a good encryption algorithm. In this section, we perform pixel correlation analysis on the original Lena image and its corresponding secret image, the carrier image Butterfly and its corresponding cipher image, and the results are shown in Figure 18, where red, green, and blue correspond to the R, G, and B components of the image, respectively. From Figures 18A–18C, it can be seen that the pixel distribution of the original image is close to the diagonal in all three directions, which means that there is a strong correlation between neighboring pixels. While the distribution of pixel pairs of the secret image is random, as shown in Figures 18D–18F, which means that there is no strong correlation between their pixels, which also verifies the security of this encryption algorithm. Comparison of the analyzed maps of the carrier image and the cipher image reveals that the pixel correlation has not been corrupted, indicating that the hiding algorithm is highly applicable.Figure 18 Pixel correlation analysis

(A) R component of the original image Lena in horizontal, vertical, and diagonal directions.

(B) G component of the original image Lena in horizontal, vertical, and diagonal directions.

(C) B component of the original image Lena in horizontal, vertical, and diagonal directions.

(D) R component of the secret image in horizontal, vertical, and diagonal directions.

(E) G component of the secret image in horizontal, vertical, and diagonal directions.

(F) B component of the secret image in horizontal, vertical, and diagonal directions.

(G) R component of the carrier image Butterfly in horizontal, vertical, and diagonal directions.

(H) G component of the carrier image Butterfly in horizontal, vertical, and diagonal directions.

(I) B component of the carrier image Butterfly in horizontal, vertical, and diagonal directions.

(J) R component of the cipher image in horizontal, vertical, and diagonal directions.

(K) G component of the cipher image in horizontal, vertical, and diagonal directions.

(L) B component of the cipher image in horizontal, vertical, and diagonal directions.

In addition, we applied the following methods to quantitatively analyze the correlations:(Equation 17) C=∑i=1N(xi−1N∑i=1Nxi)(yi−1N∑i=1Nyi)∑i=1N(xi−1N∑i=1Nxi)2×∑i=1N(yi−1N∑i=1Nyi)2

where xi, yi denote the pixel values of a certain pair of neighboring pixels and N is the length of the randomly selected sequence. If the value of the correlation coefficient C is closer to 0, it means that this pixel sequence is less correlated; and vice versa. From Table 6, it can be seen that the correlation coefficient values of the original, carrier and cipher images are all close to 1, while the correlation coefficient of the secret image is close to 0, which indicates that the present algorithm effectively removes the correlation between neighboring pixels. Based on the same Lena graph, the test results of this algorithm are compared with other algorithms, as shown in Table 733,36 It can be seen that our algorithm is superior in resisting statistical analysis.Table 6 Analysis of correlation coefficients

Image	Horizontal	Vertical	Diagonal	
R	G	B	R	G	B	R	G	B	
Plain	0.9511	0.9345	0.9145	0.9751	0.9684	0.9484	0.9267	0.9109	0.8885	
Secreat	0.0001	−0.0014	−0.0010	−0.0092	0.0026	−0.0021	−0.0016	0.0112	−0.0025	
Carrier	0.9262	0.9234	0.8742	0.9205	0.9164	0.8638	0.8936	0.8895	0.8179	
Cipher	0.9178	0.9202	0.8447	0.9124	0.9133	0.8354	0.8875	0.8868	0.7978	

Table 7 Comparison of correlation coefficients

Algorithms	Horizontal	Vertical	Diagonal	
R	G	B	R	G	B	R	G	B	
Our	0.0001	−0.0014	−0.0010	−0.0092	0.0026	−0.0021	−0.0016	0.0112	−0.0025	
Wang et al.36	−0.0098	−0.1900	−0.0079	0.0033	−0.0036	0.0034	0.0004	−0.0149	0.0158	
Wang et al.33	0.0029	−0.0025	−0.0063	−0.0033	−0.0059	0.0046	0.0019	0.0013	−0.0036	

Analysis of anti-differential attack

Differential attack is an attack method to crack a ciphertext image and is a selective plaintext attack. It refers to an attacker who makes minor changes to the original image and encrypts the image before and after the change separately using an encryption algorithm, compares the difference between the two encrypted images, finds out the relationship between the original image and the encrypted image, and utilizes this relationship to decrypt the ciphertext image. A good encryption algorithm to be able to resist differential attacks usually evaluates the sensitivity to plaintext in terms of the pixel change rate (NPCR) and the uniform average change intensity (UACI), which are calculated as follows:(Equation 18) NPCR=∑iM∑jNW(i,j)M×N

(Equation 19) UACI=∑iM∑jN|C1(i,j)−C2(i,j)|M×N×Q

Where M and N denote the number of rows and columns of the image, respectively, and C1(i,j) and C2(i,j) denote the pixel values before and after image modification. W(i,j) is noted as 1 if the pixel value changes, otherwise it is noted as 0. Q denotes the maximum pixel value allowed in the image. The ideal values of NPCR and UACI are 0.9960 and 0.3346, respectively. In this section, we utilize Lena image for testing by modifying the pixel values of the original image to a new image, encrypting it with the same algorithm, generating two cryptographic images C1 and C2 and calculating their NPCR and UACI, and the test results are shown in Table 8. It can be seen from the table that both NPCR and UACI are close to the ideal values, which indicates that the encryption algorithm is effective in resisting differential attacks. In addition, in order to evaluate the performance of our proposed compression-aware encryption algorithm more comprehensively, we specifically add the NPCR and UACI values of the compression-aware encryption algorithms proposed in other studies in35,36 Table 8.Table 8 Comparison of NPCR and UACI values obtained by different methods

Image	NPCR	UACI	
R	G	B	Average	R	G	B	Average	
Proposed	0.9963	0.9959	0.9962	0.9961	0.3347	0.3348	0.3347	0.3347	
Wang et al.36	0.9962	0.9960	0.9960	0.9961	0.3346	0.3343	0.3348	0.3346	
Chai et al.35	0.9957	0.9956	0.9964	0.9959	0.3342	0.3344	0.3348	0.3345	

Robustness analysis

In the network transmission, the image will inevitably be attacked by noise or lose some image information. However, a good image encryption algorithm should have some resistance to noise attacks and data loss. Therefore, we carry out noisy and clipping attacks on the cryptographic images to verify the robustness of the encryption algorithm. Column (a) in Figure 18 represents the cipher image with 0.003, 0.006, 0.01 salt and pepper noise added, respectively, and its corresponding decrypted image is listed in Figure 18B. In addition, the cryptographic images are cropped to different degrees, and the sizes of the cropped pixels are 8×8, 16×16, and 32×32, as in Figure 18C, and the corresponding decrypted images are in Figure 19D. It can be seen from the figure that the correct decrypted image can still be obtained after the attack on the image, which indicates that the encryption algorithm has strong robustness.Figure 19 Robustness analysis

(A) Cipher images were added with 0.3%, 0.6%, and 1% salt and pepper noise respectively.

(B) Corresponding decrypted images after adding 0.3%, 0.6%, and 1% salt and pepper noise, respectively.

(C) Cipher image cut out blocks of size 8×8, 16×16, and 32×32.

(D) Cut off the decrypted images corresponding to blocks of size 8×8, 16×16, and 32×32.

Efficiency of encryption and decryption

For a cryptographic algorithm, ensuring security is crucial, but at the same time, efficiency is a factor that cannot be ignored. Runtime, as one of the important indicators for evaluating the performance of an algorithm, is directly related to the time and resource consumption required for the encryption process. Table 9 details the exact time required for the encryption and decryption process when different sizes of images are compressed with different ratios using our proposed algorithm. This test aims to comprehensively evaluate the efficiency characteristics of the algorithm for different image sizes and compression ratios. In addition, Table 10 specifically compares our algorithm with other existing compression-aware encryption algorithms.37,38,39 Under the same testing conditions, the comparison results show that the algorithm proposed in this paper exhibits greater advantages in several key metrics.Table 9 Encryption and decryption times for different color images

Image	CR	Encryption Time(s)	Decryption Time(s)	
Lena(256 × 256)	0.25	0.0455	4.2844	
	0.50	0.0712	5.6493	
	0.75	0.0987	6.7236	
Milk(256 × 256)	0.25	0.0469	4.1458	
	0.50	0.0708	5.7905	
	0.75	0.1014	6.6572	
Milk(512 × 512)	0.25	0.1741	11.3719	
	0.50	0.2488	13.8490	
	0.75	0.3564	16.6496	
Baboon(256 × 256)	0.25	0.0363	3.9821	
	0.50	0.0595	5.6222	
	0.75	0.0890	6.7522	
Baboon(512 × 512)	0.25	0.1509	11.1471	
	0.50	0.2482	13.7302	
	0.75	0.3497	16.9407	

Table 10 Encryption and decryption time of different algorithms based on Lena gray scale image

Scheme	CR	Encryption Time(s)	Decryption Time(s)	
Ours	0.25	0.0677	3.9700	
	0.50	0.0984	4.6344	
	0.75	0.1265	5.7524	
Chai et al.37	0.25	1.1090	5.3639	
	0.50	1.1363	16.4861	
	0.75	1.1555	51.8935	
Xu et al.38	0.25	0.4597	3.2198	
	0.50	0.8916	6.1583	
	0.75	1.2401	15.8179	
Luo et al.39	0.25	0.5614	3.2781	
	0.50	1.2157	7.8459	
	0.75	1.4081	19.6279	

Conclusion

In this paper, a color image encryption algorithm based on time-varying delay chaotic model and compression-awareness is proposed. To start with, a time-varying delayed chaos model is constructed and its performance is tested, which confirms that the chaos model has better chaotic dynamics characteristics than the original logistics model. Then a construction method for the measurement matrix is designed based on the new chaotic model, which is tested to have a higher quality of compressive reconstruction than the classical measurement matrix. Finally, an innovative image hiding algorithm is added to further improve the security of this encryption algorithm by using the DWT-SVD algorithm to hide the secret image in the meaningless carrier image.

By analyzing the simulation experiments, we verify that our encryption algorithm has high security and can compete with other image encryption algorithms. In future research, we will focus on improving the problem of the long time spent on the compressive reconstruction process, aiming to achieve high reconstruction quality even at lower sampling rates.

Resource availability

Lead contact

Any further information and requests for resources and materials should be directed to and will be fulfilled by the lead contact, Prof. Lingfeng Liu (vatanoilcy@163.com).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• Data reported in this paper will be shared by the lead contact upon request.

• This paper does not report original code.

• Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

This work is supported by 10.13039/501100001809 National Natural Science Foundation of China (62262039 & 62262023 ), Major Project of Jiangxi Provincial Natural Science Foundation (20232ACB202009 ), Outstanding Youth Foundation of Jiangxi Province (20212ACB212006 ), the Jiangxi Provincial Key Laboratory of Data Security Technology (2024SSY03181 ), the Finance Science and Technology Special "Contract System" Project of Jiangxi Province (ZBG20230418014 ); Jiangxi Provincial Postgraduate Innovation Project in 2023 (YC2023-S011 ).

Author contributions

Conceptualization, methodology, and writing – original draft: B.J.; formal analysis and validation: L.F.; validation and investigation: B.Z.; validation and data curation: R.L.; writing – review and editing and funding acquisition: L.L.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER	
Software and algorithms	
	
MATLAB R2022a	The MathWorks	RRID: SCR_001622	

Method details

Image scrambling and diffusion

In this study, the scrambling algorithm is based on the Arnold transform, which divides the R, G, and B of the image P1 uniformly into chunked matrices of size b×b and performs the Arnold transform on each chunk, and Equation 20 is the process of transforming the matrix, iterated n times. Where (x,y) denotes the position of the original pixel and (x1,y1) denotes the position of the pixel point after one iteration. The transformed matrix block is reorganised to obtain the disordered image P2.(Equation 20) (x1y1)=(1112)(xy)modb

The purpose of diffusion is to change the pixel values of the image with the aim of destroying the statistical properties of the original image. Equations 21 and 22 are the specific implementations of diffusion. Where S denotes the sequence generated by the chaotic system and P3 denotes the image after encryption.(Equation 21) S′=(floor(S×1014))mod256

(Equation 22) P3=P2⨁S′

Experimental measurements

The design and execution of the experiments, as well as the subsequent testing and validation work, were done in the MATLAB R2022a environment. We selected two carrier images and two to-be-encrypted images as test objects respectively, and finally formed four sets of experimental data to achieve the compression and hiding of images.
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