Efficient joint compression and encryption approach based on compressed sensing with multiple chaotic systems

doi:10.21203/rs.3.rs-1547540/v1

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Efficient joint compression and encryption approach based on compressed sensing with multiple chaotic systems

https://doi.org/10.21203/rs.3.rs-1547540/v1

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This paper puts forward a new algorithm that utilizes compressed sensing and two chaotic systems to complete image compression and encryption concurrently. Firstly, the hash function is utilized to obtain the initial parameters of two chaotic maps, which are 2D-SLIM and 2D-SCLMS map respectively. Secondly, a sparse coefficient matrix is transformed from the plain image through discrete wavelet transform. In addition, one of the chaotic sequences created by 2D-SCLMS system performs pixel transformation on the sparse coefficient matrix. The other chaotic sequences created by 2D-SLIM are utilized to generate a measurement matrix and perform compressed sensing operations. Subsequently, the matrix rotation is combined with row scrambling and column scrambling, respectively. Finally, the bit-cycle operation and the matrix double XOR is implemented to acquire the ciphertext image. Simulation experiment analysis manifests that the compressed encryption scheme has advantages in compression performance, key space and sensitivity, and is resistant to statistical attacks, violent attacks and noise attacks.

Image encryption

Compressed sensing

Chaotic system

Bit-cycle operation

Double XOR operation

In the wake of development of the internetwork and information technique, digital images are extensively used in numerous domains [1–4]. A great quantity of information is presented in digital image form, which usually contains private and important information. When important information is falsified or leaked, it can cause acute consequences [5, 6], which makes the privacy security issue very prominent. Hence, the information security protection of digital images has aroused widespread attention [7, 8]. In this situation, multiple encryption scenarios have emerged.

In the past few years, owing to the excellent characteristics of chaotic maps [9, 10], various encryption scenarios based on chaotic maps have been created [11–14]. Wang et al. utilized parameter controlled scroll chaotic attractors for encryption [15]. Gao proposed a new 2D hyperchaotic system for image encryption [16]. In addition to this, chaotic maps combined with a variety of methods for encryption. Patel et al. combined chaotic maps and DNA coding for encryption and the results indicated that the effect is better than using chaotic map alone [17, 18]. Yu et al. combined chaotic maps and fractional Fourier transform for optical image encryption [19, 20]. Choi et al. combined chaotic maps and cellular automata for encryption [21, 22]. Although the above-mentioned these algorithms have achieved good results, none of the above algorithms are applicable due to the bandwidth constraint problem.

To satisfy bandwidth-constrained demands, the theoretical concept of compressed sensing (CS) was established [23, 24]. Soon afterwards, multifarious compressed encryption scenarios based on CS have been put forward [25, 26]. Lu et al. created an image encryption scenario [27], which compressed images by CS and encrypted images by double random phase coding technology. Although this algorithm reduces the amount of data, its method of using the metric matrix as the key takes up a large amount of storage space and bandwidth-constrained.

To resolve trouble, a new image compression encryption scenario has attracted people's attention [28–32]. This scenario combines compressed sensing with chaotic systems, which utilized compressed sensing to compress the image to meet the bandwidth demands in transmission and also can make full use of the excellent properties of chaotic system by using the initial parameters of chaotic maps as the key and using the created sequences to form the measurement matrix. This method resolves the problem that the key occupies a large storage space and limited bandwidth.

To further improve schemes security, many scenarios adopt scrambling methods [33–37]. According to the position of the scrambling in the algorithm, we divide it into two categories. One is to perform scrambling and confusion operation after measurement value is obtained by compressed sensing [36, 37]. The other is to first obtain a sparse coefficient matrix through a sparse transformation of the original image and then perform a scrambling operation on the sparse coefficient matrix [34, 35]. Both types of methods can decrease the images correlation and heighten the scenarios security, while the latter has the advantage of effectively enhancing the reconstruction quality of the decrypted image [38]. In general, there are two scrambling methods in the encryption process, which are scrambling using Arnold map [34] and scrambling of index values obtained by sorting chaotic sequences [33, 35]. Both methods have drawbacks. The Arnold map scrambling method cannot be directly used for non-square images [39]. The second scrambling method is easy to operate and its scrambling effect is determined by the randomness of the chaotic sequence [40], so it is not suitable for the chaotic system with bad randomness. Therefore, a scrambling method called pixel transformation is proposed.

To increase the security and meet the demand of limited bandwidth, a compressed encryption plan based on two chaotic systems and CS is put forward in this paper. First of all, the SHA-384 of the original image is used to calculate the initial parameters of the 2D-SLIM and 2D-SCLMS system and used as the key, which greatly heightens the relevance between the scheme and the plaintext, and can better resist known plaintext attacks and selective plaintext attacks. Secondly, the plaintext image is converted into a sparse coefficient matrix. Thirdly, to increase the reconstruction quality of the decrypted image, a new scrambling technology is created. In addition, the chaotic sequence is utilized to create the measurement matrix and implement the compressed sensing operation, which greatly meets the transmission bandwidth requirement. To further heighten the security, the encryption operation combines matrix rotation with row scrambling and column scrambling, respectively, followed by a bit-cycle operation. Finally, double XOR of the matrix is implemented to acquire the ciphertext image.

The remaining sections are planed: Section 2 presents the related work. Section 3 designs the new compression encryption scenario. Section 4 demonstrates the corresponding decryption algorithms. Section 5 gives the various performance analysis of the compression encryption scenario. Section 6 draws the conclusion.

2.1 Compressed sensing

In 2006, Donoho et al. proposed a compressed sensing formulation and processing method for signals [23, 24]. This concept smashes the restrictions of Shannon's sampling theorem by exploiting the sparsity of the natural signal itself or the sparsity of a certain transform domain, allowing the recovery of the sampled signal with a small amount of samples at lower than the Nyquist sampling rate. Compressed sensing, also known as compressive sampling, allows sampling, compression and encryption to be done concurrently [39, 41].

The pivotal elements of compressed sensing comprise sparse representation, measurement matrix and reconstruction scheme. In general, the signal is not sparse in the time domain, but in some transform domains, the signal may become sparse. Therefore, the classic sparsity representation methods comprise discrete wavelet transform (DWT), fast Fourier transform (FFT) and discrete cosine transform (DCT).

We take a 1D signal to explain the step of compressed sensing. The sparsity expression for a non-sparse signal x (N×1) in some transform domain is

$$x=\Psi s$$

In Eq. (1), Ψ (N×N) is known as normal orthogonal matrix, s (N×1) is a K sparse vector.

According to Eq. (1), the specific expression of compressed sensing is

$$y=\Phi x=\Phi \Psi s=\Theta s$$

In Eq. (2), Φ (M×N) is the measurement matrix, Θ (M×N) is the sensing matrix and y (M×1) is the measured value matrix. Especially, M < N.

Compressed sensing demands that Θ have content the restricted isometry property [42], that is to say, Φ and Ψ are uncorrelated. In addition to that, the length of y ought to content

$$M \geqslant cK\log \frac{N}{K}$$

In Eq. (3), c is a constant with a small value.

To exactly recover the s from measured value matrix y, theoretically, the problem of l₀ norm minimization should be solved

$$\begin{gathered} \hbox{min} ||s|{|_0} \hfill \\ s.t.{\text{ }}y=\Phi \Psi s \hfill \\ \end{gathered}$$

However, Eq. (4) is NP-hard problem. Therefore, in general, the problem of l₁ norm minimization is used to supersede Eq. (4)

$$\begin{gathered} \hbox{min} ||s|{|_1} \hfill \\ s.t.{\text{ }}y=\Phi \Psi s \hfill \\ \end{gathered}$$

There are many reconstruction algorithms for compressed sensing, the most common ones are the orthogonal matching tracking algorithm, subspace pursuit algorithm and the smooth l₀ norm (Sl₀) algorithm. We choose the Sl₀ algorithm for the reconstruction in this paper.

2.2 Sigmoid function

A common S-shaped function, also known as an S-shaped growth curve, is the Sigmoid function [35], whose expression is

$$y=\frac{a}{{1+{e^{ - b(x - c)}}}}$$

Where the range of y is [0, a]. We utilize the sigmoid function for quantization, so we set a = 255, b = 80/(15.518×(X_max - X_min)), c = (X_max + X_min)/2. X_max and X_min are the maximum and minimum value of X, respectively. For different images, X_max and X_min are different, i.e., the values of b and c are taken differently.

2.3 Chaotic system

2.3.1 2D-SCLMS system

2D-SCLMS map is a hyperchaotic system generated based on logistic and sine maps [43] with the expression

$$\left\{ \begin{gathered} {x_{i+1}}=\sin (4{\pi ^2}(u\sin (4\pi {x_i}(1 - {x_i})))+4u{y_i}(1 - {y_i})) \hfill \\ {y_{i+1}}=\sin (4{\pi ^2}(u\sin (4\pi {y_i}(1 - {y_i})))+4u{x_{i+1}}(1 - {x_{i+1}})) \hfill \\ \end{gathered} \right.$$

Where $u>0.1$ is the parameter. ${x_i},{y_i} \in ( - 1,1)$,$i=1,2, \cdots$.

2.3.2 2D-SLIM

2D-SLIM is a chaotic map with complex properties for image encryption [44]. Its expression is

$$\left\{ {\begin{array}{*{20}{l}} {{x_{i+1}}=\sin ({u_1}{y_i})\sin (50/{x_i})} \\ {{y_{i+1}}={u_2}(1 - 2x_{{i+1}}^{2})\sin (50/{y_i})} \end{array}} \right.$$

Where ${u_1},{u_2} \in (0,+\infty ),$${x_i},{y_i} \in ( - 1,1),$$i=1,2, \cdots$. We set ${u_1}=2\pi$,${u_2}=1$.

A new encryption scenario is created and its flow chart is presented in Fig. 1.

3.1 Key generation

The hash algorithm is utilized to create the initial parameters of the 2D-SCLMS map and the 2D-SLIM, which enhances the relevance between the ciphertext image and the original image. First, SHA-384 hash function generates a binary sequence composed of 384 bits and then this sequence is separated into blocks every 8 bits, i.e. 48 decimal numbers h₁, h₂, ..., h₄₈. The 2D-SCLMS system is mainly used for pixel transformation, row scrambling and column scrambling, which initial values and parameters are calculated as

$$\left\{ {\begin{array}{*{20}{c}} {{x_0}=\bmod (\sum\limits_{{i=1}}^{{10}} {{k_i}} ,256)/256} \\ {{y_0}=\bmod (\sum\limits_{{i=11}}^{{20}} {{k_i}} ,256)/256} \\ {u=\bmod (\sum\limits_{{i=21}}^{{30}} {{k_i}} ,256)/256+\alpha } \end{array} } \right.$$

2D-SLIM is mainly utilized to establish the measurement matrix and perform bit-cycle, which initial values are calculated as

$$\left\{ {\begin{array}{*{20}{c}} {a=\bmod (\sum\limits_{{i=43}}^{{48}} {{k_i}} ,256)/2560} \\ {{z_0}=\bmod (\sum\limits_{{i=31}}^{{36}} {{k_i}} ,256)/256+a} \\ {{w_0}=\bmod (\sum\limits_{{i=37}}^{{42}} {{k_i}} ,256)/256} \end{array} } \right.$$

3.2 Encryption process

The proposed encryption algorithm is depicted as follows.

Step 1

The initial parameters (x₀, y₀, u) obtained in section 3.1 are entered into the 2D-SCLMS map for 500 + N² iterations. The first 500 values are removed to acquire the sequences X, Y. Sequence X₁ is obtained

$${X_1}=\bmod (round(X \times 10\^8),4)$$

X ₁ is divided equally into 4 sequences and each sequence is transformed into an $\frac{N}{2} \times \frac{N}{2}$ matrix named X₁₁, X₁₂, X₁₃, X₁₄.

The sequence X is transformed into a matrix X₂ (N×N) and X₂ is divided into X₂₁, X₂₂ by rows, so the matrices A, B are obtained according to Eqs. (14) and (15), respectively.

$${X_{21}}={X_2}(1:N \times CR,:)$$

$${X_{22}}={X_2}((1 - CR) \times N+1:end,:)$$

$$A=mod(floor({X_{21}} \times {10^{10}}),256)$$

$$B=mod(floor({X_{22}} \times {10^{10}}),256)$$

The sequence Y is transformed into a N×N matrix and then divided into two parts Y₁, Y₂ according to the number of rows. The matrix Y₁ is arrayed in descending order by columns and the matrix Y₂ is arrayed in ascending order by rows to get the index matrix L₁, L₂ respectively.

$${Y_1}=Y(1:N \times CR,:)$$

$${Y_2}=Y((1 - CR) \times N+1:end,:)$$

$$[\sim ,{L_1}]=sort({Y_1},2,'descend')$$

$$[\sim ,{L_2}]=sort({Y_2})$$

Step 2

The plaintext image P (N×N) generates a discrete coefficient matrix P₁ through DWT and then matrix P₁ is divided equally into 4 small matrices P₁₁, P₁₂, P₁₃ and P₁₄.

$${P_1}=\Psi \times P \times {\Psi ^{\rm T}}$$

Step 3

Perform pixel transformation on P₁₁, P₁₂, P₁₃, P₁₄ using matrices X₁₁, X₁₂, X₁₃, X₁₄. Take X₁₁ as an example for illustration.

$$\begin{array}{*{20}{c}} {If~{X_{11}}(i,j)=0,then} \\ {temp={P_{11}}(i,j)} \\ {{P_{11}}(i,j)={P_{12}}(i,j)} \\ {{P_{12}}(i,j)=temp} \end{array}$$

$$\begin{array}{*{20}{c}} {If~{X_{11}}(i,j)=1,then} \\ {temp={P_{11}}(i,j)} \\ {{P_{11}}(i,j)={P_{13}}(i,j)} \\ {{P_{13}}(i,j)=temp} \end{array}$$

$$\begin{array}{*{20}{c}} {If~{X_{11}}(i,j)=2,then} \\ {temp={P_{11}}(i,j)} \\ {{P_{11}}(i,j)={P_{14}}(i,j)} \\ {{P_{14}}(i,j)=temp} \end{array}$$

$$\begin{array}{*{20}{c}} {If~{X_{11}}(i,j)=3,then} \\ {temp={P_{11}}(i,j)} \\ {{P_{11}}(i,j)={P_{11}}(\tfrac{N}{2}+1 - i,\tfrac{N}{2}+1 - j)} \\ {{P_{11}}(\tfrac{N}{2}+1 - i,\tfrac{N}{2}+1 - j)=temp} \end{array}$$

Similarly, pixel transformation is performed again based on the values of ${X_{12}},{X_{13}},{X_{14}}$, respectively. When the pixel transformation is over, the four matrices are combined to acquire P₂.

Step 4

The initial values (z₀, w₀) created in section 3.1 and parameters are entered into the 2D-SLIM iterating 500 + d×M×N times to produce two chaotic sequences. The first 500 values of the two sequences are removed to obtain the chaotic sequence Z, W. M = CR×N, wherein, CR is the compression rate and d is the sampling distance.

Sequence Z₁ is acquired by sampling from sequence Z according to the sampling distance d. The measurement matrix $\Phi$(M×N) is generated.

$${Z^{\prime}_i}=1 - 2{Z_{1+id}},i=1,2, \cdots ,MN$$

$$\Phi =\sqrt {\frac{2}{M}} reshape({Z^{\prime}_i},M,N)$$

Take MN values from the sequence W and transform it into a matrix W₁. According to Eqs. (23) and (24), W₂ and C can be generated as (27) and (28).

$${W_2}=\bmod (floor({W_1} \times {10^6}),8)$$

$$C=\bmod (floor({W_1} \times {10^6}),256)$$

Step 5

Compress P₂ to obtain the measurement results P₃.

$${P_3}=\Phi \times {P_2}$$

Step 6

Quantize P₃ according to the sigmoid function introduced in Section 2.2 and round the quantized result to obtain P₄.

$${P_4}=\frac{a}{{1+{e^{ - b({P_3} - c)}}}}$$

Step 7

Rotate P₄ counterclockwise by 180° and then scramble the columns according to the index matrix L₁ to get P₅.

$${P_{41}}=rot90(rot90({P_4}))$$

$${P_5}(i,j)={P_{41}}(i,{L_1}(i,j))$$

Step 8

Rotate P₅ counterclockwise by 180° and then scramble the rows according to the index matrix L₂ to get P₆.

$${P_{51}}=rot90(rot90({P_5}))$$

$${P_6}(i,j)={P_{51}}({L_2}(i,j),j)$$

Step 9

Rotate P₆ counterclockwise by 180° and then perform the bit-cycle operation according to W₂. If W₂(i, j) = 1, then P₆₁(i, j) is shifted left by 1 bit. If W₂(i, j) = 2, then P₆₁(i, j) is shifted left by 2 bits. Similarly, if W₂(i, j) = 7, then P₆₁(i, j) is shifted left by 7 bits and finally P₇ is obtained.

$${P_{61}}=rot90(rot90({P_6}))$$

$${P_7}={P_6}(bit - cycle)$$

Step 10

The final ciphertext image P₈ is obtained by double XOR of P₇.

$${P_8}=bitxor(\bmod (bitxor({P_7},C)+A,256),B)$$

The specific decryption method is demonstrated as below and its flow chart is presented in Fig. 2.

The initial parameters are brought into the two chaotic systems. The specific method is the same as Step 1 and 4 of the section 3.2.

Perform the reverse operation of the double XOR on the ciphertext image P₈ to obtain P₇, then perform the inverse operation of the bit cycle and rotate 180° counterclockwise to obtain P₆.

$${P_7}=IXOR({P_8})$$

$${P_{61}}={P_7}(Ibit - cycle)$$

$${P_6}=rot90(rot90({P_{61}}))$$

Perform the inverse scrambling operation on the rows of P₆ according to the index matrix L₂ and then rotate 180° counterclockwise to obtain P₅.

$${P_{51}}({L_2}(i,j),j)={P_6}(i,j)$$

$${P_5}=rot90(rot90({P_{51}}))$$

Perform the inverse scrambling operation on the columns of P₅ according to the index matrix L₁ and then rotate 180° counterclockwise to obtain P₄.

$${P_{41}}(i,{L_1}(i,j))={P_5}(i,j)$$

$${P_4}=rot90(rot90({P_{41}}))$$

Perform inverse quantization on P₄ according to the sigmoid function introduced in section 2.2 to obtain P₃.

$${P_3}= - \log (\frac{a}{{{P_4}}} - 1) \times \frac{1}{b}+c$$

Smooth l₀ norm method is used to reconstruct P₂.

$${P_2}=S{L_0}({P_3},\Phi )$$

Divide P₂ into 4 blocks on average and perform inverse pixel transformation to obtain P₁.

$${P_1}=IPT({P_2})$$

Perform the reverse DWT on P₁ to acquire the decrypted image P.

$$P={\Psi ^T} \times {P_1} \times \Psi$$

Multiple experiments are tested to prove the performance of the newly presented compressed encryption scenario. The operating system for all experiments is Window 10 Ultimate with AMD Ryzen 2.00GHz CPU, 8G RAM and 1TB hard disk and the operating software is MATLAB R2020a.

5.1 Simulation results

Figure 3 displays the original images, compressed encrypted images and decrypted images of ''Lena'', ''Cameraman'', ''Cattle'', "Einstein", "Boat" and "Couple" of size 512×512. All experiments are verified with a compression ratio of 0.5 as an example.

The ciphertext images are similar to noise and are smaller than the original images in Fig. 3, which indicates that this scheme has a good compression and encryption effect. Besides, the decrypted images are of high quality and are the same size as the plaintext images, which manifests that the scenario has good reconstruction and decryption effect.

5.2 Compression performance analysis

5.2.1 Peak Signal to Noise Ratio (PSNR)

PSNR [45] is utilized for assessment of the compression performance. It is expressed as

$$MSE=\frac{1}{{N \times N}}\sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1}}^{N} {{{(X(i,j) - Y(i,j))}^2}} }$$

$$PSNR=10 \times {\log _{10}}(\frac{{255 \times 255}}{{MSE}})$$

In Eq. (49), X and Y are respectively the plaintext and the decrypted image. The larger the PSNR value, the better compression performance. Figure 4 appears the simulation of ''Lena'' under different CRs and their corresponding PSNR values. It can be concluded that even if CR = 0.25, the PSNR exceeds 30db. Table 1 lists the PSNR of different images (512×512). The PSNR of all the tested images exceed 30db, which indicates that the compression characteristic of scenario is excellent and stable. Table 2 compares the PSNR of different compression encryption algorithms for ''Lena'' (256×256). The PSNR of our algorithm is 32.6176, which is higher than other scenarios, which manifests that the newly proposed scenario is better.

Table 1

PSNR (db) of the proposed algorithm for different images (512×512)
Image	PSNR
Lena	33.9387
Cameraman	32.3925
Boat	31.1828
Couple	31.1743
Einstein	32.4941

Table 2

PSNR (db) comparison for several schemes
Image	Ref.[28]	Ref.[44]	Ref.[46]	Ours
Lena (256×256)	30.71	29.23	31.2302	32.6176

5.2.2 Structural similarity index measurement (SSIM)

A momentous indicator to survey of the similarity of two images is SSIM and its range is [0, 1]. The larger the SSIM [47], the greater the similarity of the two images. SSIM expression is

$$SSIM(X,Y)=\frac{{(2{\mu _X}{\mu _Y}+{{({K_1}L)}^2})(2{\sigma _X}_{Y}+{{({K_2}L)}^2})}}{{(\mu _{X}^{2}+\mu _{Y}^{2}+{{({K_1}L)}^2})(\sigma _{X}^{2}+\sigma _{Y}^{2}+{{({K_2}L)}^2})}}$$

In Eq. (51), X and Y are the plaintext and reconstructed image. µ_X and$\sigma _{X}^{2}$are the mean value and the variance of X respectively. µ_Y and$\sigma _{Y}^{2}$are mean value and the variance of Y respectively. σ_XY is the covariance of X and Y. M is the total number of windows. L = 255. K₁ = 0.01, K₂ = 0.03. We tested the SSIM values for multiple images, displayed in Table 3. The SSIM of all images (512×512) are near 1, which indicates that the plaintext image and reconstructed image have high similarity, i.e., the reconstruction algorithm achieves good results. Table 4 compares the SSIM of different compression encryption algorithms for ''Lena'' (256×256). The SSIM calculated by the newly proposed scenario is larger, which shows that the image reconstructed by the new scenario is more similar to the plaintext image.

Table 3

SSIM for different images (512×512)
Image	SSIM
Lena	0.9001
Cameraman	0.8323
Boat	0.8447
Couple	0.8313
Einstein	0.8704
Cattle	0.8021

Table 4

SSIM comparison for different algorithms
Image	Ref.[44]	Ref.[46]	Ours
Lena (256×256)	0.7129	0.6475	0.7337

5.3 Key space analysis

The key space of a scenario must be larger than 2¹⁰⁰ to ensure that the algorithm is good and secure enough against brute force attacks [48]. The new algorithm has a key$\alpha$and utilizes the hash-384 algorithm, assuming that the computer has a computational precision of 10^− 14, the entire key space is 10¹⁴+2³⁸⁴. It is much larger than 2¹⁰⁰. Thus the scenario can resist violent attacks.

5.4 Sensitivity analysis

This part is designed to prove the sensitivity analysis of the new scenario, which includes key sensitivity analysis and original image sensitivity analysis.

5.4.1 Key sensitivity analysis

A good encryption scenario is sensitive to the key, that is, even though the key alters very little, the encrypted image has great difference.

5.4.2 Original image sensitivity analysis

An excellent encryption scenario is sensitive to the plaintext image, in other words, even though the original image has very small changes, the encrypted image can be completely different.

The number of pixel change rate (NPCR) and the unified average change intensity (UACI) test the sensitivity of the scenario. They are expressed as

$$NPCR=\frac{1}{{M \times N}}\sum\limits_{{i=1}}^{M} {\sum\limits_{{j=1}}^{N} {|Sign({C_1}(i,j) - {C_2}(i,j))|} }$$

$$UACI=\frac{1}{{M \times N}}\sum\limits_{{i=1}}^{M} {\sum\limits_{{j=1}}^{N} {\frac{{|{C_1}(i,j) - {C_2}(i,j)|}}{{255}}} }$$

Where C₁ and C₂ are two different cipher images. Table 5 lists the NPCR and UACI for multiple images (512×512).

Table 5

NPCR and UACI (512×512)
Image	Key		Plaintext image
Image	NPCR	UACI	NPCR	UACI
Lena	99.5911%	33.4233%	99.5941%	33.4078%
Einstein	99.6094%	33.5026%	99.6460%	33.5236%
Couple	99.6117%	33.4125%	99.5987%	33.4989%
Cattle	99.6017%	33.3956%	99.5781%	33.4013%
Boat	99.6056%	33.4358%	99.6185%	33.4173%
Cameraman	99.5834%	33.5923%	99.6048%	33.4373%
Mean	99.6005%	33.46035%	99.6067%	33.4477%
The NPCR and UACI regarding both the key and plaintext image are close to 99.6094% and 33.4635%, which indicates that the scenario is sensitive to both. At the same time, the scenario is resistant to differential attack.

5.5 Statistical attack analysis

5.5.1 Histogram analysis

A momentous index to appraise encryption scenarios performance is the histogram. Figure 5 displays the histogram of multiple plaintext images and ciphertext images.

The histograms of the plaintext images are uneven, but those of the cipher images are similar to the uniform distribution, which illustrates that the scenario resists statistical attacks.

In addition, we utilizes histogram variance to survey the effectiveness of this algorithm.

$$Var(Z)=\frac{1}{{{{256}^2}}}\sum\limits_{{i=0}}^{{N - 1}} {\sum\limits_{{j=0}}^{{N - 1}} {\frac{{{{({z_i} - {z_j})}^2}}}{2}} }$$

In Eq. (54), z_i and z_j represent the number of pixel values correspond to i and j. The histogram variance of the plaintext images are very large, with an average of 1.24×10⁶, while those of the ciphertext images are smaller, with an average of 506.99 in Table 6. This illustrates that the histogram of the ciphertext images are flatter.

Table 6

Histogram variance of multiple images (512×512)
Image	Original image	Encrypted image
Lena Couple Cameraman Boat Einstein Cattle Average	1.0827×10⁶ 1.1955×10⁶ 1.6741×10⁶ 1.5359×10⁶ 1.1987×10⁶ 7.5077×10⁵ 1.2396×10⁶	461.9766 515.0078 584.1484 558.7188 455.4297 466.6719 506.9922

Table 7 compares the histogram variance of ''Lena'' (256×256) with different algorithms. The histogram variance of the new scenario is smaller, explaining that the histogram is flatter. That is, the newly proposed algorithm is more resistant to statistical attacks.

Table 7

Histogram variance comparison of Lena (256×256) of different algorithms
Plain image	Ref.[46]	Ref.[49]	Ours
3.0665×10⁴	121.4063	181.7109	105.6328

5.5.2 Correlation analysis

The encryption scenario is to break the correlation of the original image. The evaluation index to assess the effectiveness of the scenario is the correlation coefficient, which expression is

$${\rho _{xy}}=\frac{{cov(x,y)}}{{\sqrt {D(x)D(y)} }}$$

$$\operatorname{cov} (x,y)=\frac{1}{N}\sum\limits_{{i=1}}^{N} {({x_i} - \frac{1}{N}\sum\limits_{{i=1}}^{N} {{x_i}} )({y_i} - \frac{1}{N}\sum\limits_{{i=1}}^{N} {{y_i}} )}$$

$$D(x)=\frac{1}{N}\sum\limits_{{i=1}}^{N} {{{({x_i} - \frac{1}{N}\sum\limits_{{i=1}}^{N} {{x_i}} )}^2}}$$

In Eqs. (55)-(57), x and y are the image adjacent pixels. Table 8 enumerates the correlation coefficients of different original images (512×512) and ciphertext images. The comparison values of "Lena" (256×256) with several encryption scenarios are enumerated in Table 9.

Table 8

Correlation coefficients for different images (512×512)
Image	Horizontal	Vertical	Diagonal
Lena Cameraman Boat Couple Einstein Cattle	0.9840 -0.0032 0.9853 -0.00014 0.9833 -0.0015 0.9624 0.0037 0.9687 -0.0014 0.8573 -0.0057	0.9835 -0.0037 0.9870 -0.0017 0.9727 0.0058 0.9650 0.00045 0.9644 -0.00043 0.9103 -0.0016	0.9717 -0.00085 0.9765 0.0030 0.9617 -0.0018 0.9386 0.0014 0.9548 -0.00021 0.8423 -0.0015

Table 9

Correlation coefficients of Lena (256×256) for different schemes
Direction Plain image Ref.[28] Ref.[44] Ref.[46] Ref.[49] Ours	Horizontal 0.9746 0.0009 -0.0015 0.0076 -0.0160 0.0012	Vertical 0.9651 -0.0062 0.0041 -0.0066 -0.0044 -0.0041	Diagonal 0.9539 -0.0087 0.0069 -0.0035 -0.0052 0.0032

The correlation coefficients of the plaintext images are near 1, but those of the ciphertext images are about 0 in Table 8, and that value of our scenario is smaller in Table 9, which manifests that the scenario has better resistance to statistical attacks.

The correlation of the ''Lena'' is presented in Fig. 6 for clearly observation. The correlation of the plaintext image in three directions is diagonal, but those of the ciphertext image is interspersed over the whole range. This manifests that the encryption scenario effectively abate the correlation.

5.5.3 Information entropy (IE)

The quota to assess the overall randomness of images is the IE and its expression is

$$H(s)= - \sum\limits_{{i=0}}^{M} {p({s_i}){{\log }_2}p({s_i})}$$

In Eq. (59), $M=255$.$p({s_i})$is the probability of ${s_i}$.

Table 10 lists the IE of several original (512×512) and ciphertext images. Table 11 lists comparison value of ''Lena'' (256×256) with different algorithms.

Table 10

IE of several images (512×512)
Image	Original image	Encryption image
Lena Couple Cameraman Boat Einstein Cattle	7.3920 7.2010 7.0480 7.1914 7.2655 7.3579	7.9987 7.9986 7.9987 7.9985 7.9987 7.9987

Table 11

Information entropy of Lena (256×256) of different algorithms
Plain image	Ref.[44]	Ref.[46]	Ref.[49]	Ours
7.5683	7.9935	7.9946	7.9944	7.9954

The IE of the original image is only a little more than 7, but the IE of the ciphertext image is extremely near 8 in Table 10. The IE of the newly encryption algorithm is higher than the other algorithms in Table 11. Therefore, this algorithm has very good results. The encrypted image has stronger randomness and is resistant to statistical attacks.

5.5.4 Local information entropy (LIE)

A momentous metric for analyzing the randomness of the local image is the LIE. Some non-overlapping image blocks are randomly selected and the LIE can be obtained by calculating the IE of each block and then taking the average value. The expression is

$$\overline {{L{H_{k,{T_B}}}}} (P)=\sum\limits_{{i=1}}^{k} {\frac{{H({S_i})}}{k}}$$

Where $H({S_i})$ is the IE of sub-block ${S_i}$. Let $k=30$,${T_B}=1936$for calculation. When the confidence level is 0.05, the range of LIE is [7.901901305,7.903037329] [29]. Table 12 lists the LIE of different images (512×512). The LIE of all test images has passed the experiment, which illustrates that the local image has good randomness.

Table 12

LIE (512×512)
Image	LIE	Result
Lena Cameraman Boat Cattle Einstein Couple	7.902316286 7.902787296 7.902113612 7.902520981 7.902336151 7.902842150	Pass Pass Pass Pass Pass Pass

5.5.5 Chi-square analysis

To appraise the performance of the new scenario to resist statistical attacks, this paper utilizes the chi-square [50] to examine, the expression is

$${\chi ^2}=\sum\limits_{{i=0}}^{{{2^8} - 1}} {\frac{{{{({u_i} - {u_0})}^2}}}{{{u_0}}}}$$

In Eq. (60), u_i is the frequency of value i. u₀ = MN/2⁸. Table 13 enumerates the chi-square results of multiple images (512×512). The values for five images are less than 293.2478, which manifests that this algorithm has good effects and can resist statistical attack. Table 14 compares results of several scenarios for ''Lena'' (256×256). The chi-square value of the newly proposed encryption scenario is the smallest, which manifests that scenario is more resistant to statistical attacks.

Table 13

Chi-square (512×512)
Image	Lena	Couple	Einstein	Cattle	Boat
Chi-square	230.9883	257.5039	227.7148	233.3359	279.3594
Chi-square	270681.8	298865.2	299672.0	187692.2	383969.7

Table 14

Chi-square of Lena (256×256) of different algorithms
Plain image	Ref.[46]	Ref.[49]	Ours
30665.7	242.8125	253.3125	211.2656

5.6 Anti-noise attack analysis

It is subject to various noise interference during transmission. An excellent encryption scenario should resist noise attacks. The salt and pepper noise is tested at intensities of 0.005%, 0.05%, and 0.1% in ''Lena'' in Fig. 7.

Even though the added noise intensity is 0.1%, the cipher image can be decrypted and information can be viewed. This manifests that the scenario resists noise attacks.

The paper proposes a new image compression and encryption scenario based on CS and two chaotic maps. The pixel transform operation is performed before the compressed sensing firstly, which is beneficial to increase the image reconstruction quality. In the quantization process, we make full use of the performance of the sigmoid function to quantize the matrix to the interval [0, 255]. In the scrambling process, we combine rotation with row and column scrambling, which tremendously reduces the correlation. Finally, the cipher image is created by double XOR after the bit-cycle operation. After a series of tests and experimental analysis, the new scenario has a tremendous key space and is sensitive to the key and original image. In addition, the scenario resists statistical attacks, differential attack, violent attacks and noise attacks.

Acknowledgements
This work is supported by the Postdoctoral Research Foundation of China (2018M631914) and the Heilongjiang Provincial Postdoctoral Science Foundation (CN) (LBH-Z17042). Also, this work was supported by the EIAS Data Science Lab, College of Computer and Information Sciences, Prince Sultan University, Riyadh, Saudi Arabia.

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding: not applicable

Author Contribution:

Jingya Wang: Conceptualization, Software, formal analysis, methodology, investigation, writing—original draft preparation

Xianhua Song: Supervision, Data curation, project administration, visualization, writing—original draft preparation

Ahmed A. Abd El-Latif: formal analysis, methodology, investigation, visualization, writing—review and editing

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No competing interests reported.

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Efficient joint compression and encryption approach based on compressed sensing with multiple chaotic systems

Status:

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Abstract

Figures

1 Introduction

2 Related Works

2.1 Compressed sensing

2.2 Sigmoid function

2.3 Chaotic system

2.3.1 2D-SCLMS system

2.3.2 2D-SLIM

3 Image Encryption Process

3.1 Key generation

3.2 Encryption process

4 Decryption Process

5 Simulation Experiment And Performance Analysis

5.1 Simulation results

5.2 Compression performance analysis

5.2.1 Peak Signal to Noise Ratio (PSNR)

5.2.2 Structural similarity index measurement (SSIM)

5.3 Key space analysis

5.4 Sensitivity analysis

5.4.1 Key sensitivity analysis

5.4.2 Original image sensitivity analysis

5.5 Statistical attack analysis

5.5.1 Histogram analysis

5.5.2 Correlation analysis

5.5.3 Information entropy (IE)

5.5.4 Local information entropy (LIE)

5.5.5 Chi-square analysis

5.6 Anti-noise attack analysis

6 Conclusion

Declarations

References

Competing Interests

Status:

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