Color image encryption via compressive sensing and chaotic systems

. In this paper, we design a color image encryption algorithm based on chaotic system and block compressive sensing. Firstly, the sparse representation of the plain-image is obtained by an adaptive learning dictionary. Secondly, the key streams are produced from two excellent low-dimensional chaotic maps, where updating the initial values and parameters rely on the SHA-384 and the input image. Thirdly, three measurement matrices of R, G, B components are constructed from the chaotic sequences, respectively. Finally, a random rows and columns diffusion method is performed on the encrypted image. Experimental results and safety analysis prove that the proposed scheme has excellent performance.


Introduction
Compressive Sensing (CS) theory is one of the methods of digital images encryption, which can achieve compression and encryption simultaneously.In [1], a CS image encryption algorithm was proposed, where the measurement matrix is constructed via logistic map.The chaotic system had a pivotal role in constructing the measurement matrix because their chaotic sequences are random and deterministic signal.Subsequently, Gong et al. [2] put forward an image compression and encryption algorithm based on chaotic system, which had a good ability on resistance the known plaintext attacks.However, the measurement matrix was generated from low-dimensional chaotic systems with the simple structures, which greatly reduce the security and the sensitive of the algorithms.To solve this problem, highdimensional chaotic maps were applied in the image encryption methods [3][4][5].The Chen's hyperchaotic system was performed on the 2D CS-based image encryption algorithm in [3].Chai et al. [4] explored a new magnetic controlled memristive chaotic system to construct the circular measurement matrix and encrypt the image, which can enhance the security.Recently, Xu et al. [5] also applied a hyper-chaotic system to encrypt image, which achieved an acceptable compression effects.Unfortunately, these complex chaotic maps increased the computation complexity, and the constructed measurement matrices are single, which would reduce the security and sensibility of cryptosystem.In the current studies, most of them employed the fixed dictionary such as DWT to represent different images sparsely, and the attained dictionaries are identical and has an influence on the recovering the image.
Motivated by the above analysis, we propose an image encryption algorithm by means of two simple improved chaotic systems, the adaptive learning dictionary algorithm and a random diffusion of rows and columns process.Compared with the existing schemes, our method has a higher security and a better image reconstruction effect.Numerical experiments have verified the feasibility and validity of the proposed algorithm.

Compressive sensing
A signal with size is sparse in a transform domain , and a measurement matrix which is uncorrelated with is used to yield the measurement in linear projection [6] as follows: where is the sensing matrix, is the sparse coefficient.can be recovered by: .
Obviously, the signal reconstruction is an ill-posed inverse problem.To overcome this problem, we adopt the Iterative Reweighted Least Squares algorithm (IRLS) [7], which is based on minimum norm with weight iteration.

Chaotic systems
In this paper, two chaotic maps with excellent performance, i.e., LSS and LASM [8], are used to image encryption, which are separately described as follows: , where , , .
3 The proposed encryption scheme

Generating the sparse dictionaries
Suppose a color image with size .The R, G, B components of are denoted as , , , respectively.We divide three components , and into non-overlapping blocks, and convert the image blocks into the vectors with .After image segmentation, different image blocks have some similar properties.So we 03017 (2020) https://doi.org/10.1051/matecconf/202030903017 calculate the variance values of all blocks, and the image blocks with large variance are selected as the parameters of the Method of Optimal Directions (MOD) algorithm [9], where the number of chose blocks is .And use the MOD algorithm to get the sparse dictionaries with size , where .

Generating the key streams for pixel-level diffusion
Step1 Employ the SHA-384 and the plain image to update the initial values , , and the parameters , as follows: where denotes the hash values, .represents an exclusive OR, and returns the remainder after division.
Step 2 Use , , , and to iterate LSS and LASM for times, where , , and is the compression ratio of blocks.In order to alleviate the harmful effect of the transient, we discard the former 1000 values and get three key streams , , with length .
Step 3 To enhance the sensitivity of the system, we recombine the sequences , and by: , (10) , (11) .(12) Step 4 To get the ideal pseudo-random sequences, the chaotic sequences , and are further processed as follows: , respectively.We quantize three matrices , and into the range of , and get three matrices , and .

Encryption process
Step 1 Input a color image with , and employ MOD in Section 3.1 to generate the sparse dictionaries.
Step 2 To construct three measurement matrices of R, G, B components, we sample the chaotic sequences , and with intervals , where , and get three measurement matrices , and with size .
Step 3 The matrices are projected and measured by the measurement matrices in column to get measurements , where .And transform them into three matrices , and with , where .
Step 4 The key streams , and are applied to shuffle the positions of , and , and obtain three matrices , and .
Step 5 To further strengthen the security of the algorithm, the key streams , and are employed to modify the pixel values of , and as follows: ' 10 1 floor(10 1) Y y y ,: ,: ,: where , , the matrices , and are three components of the final encrypted image.

Simulation results
We choose the 'Lena' image with size as the test image, which is displayed in Fig. 1(a).The initial values are set as follows: , , , , , and the compression ratio of blocks is 0.4.In other words, the total compression ratio is 50%, and the encryption results are shown in Fig. 1(b).One can clearly see that the cipher image is noise-like images and the corresponding decrypted image in Fig. 1(c) is almost identical to the original image (Fig. 1(a)).Therefore, our algorithm is feasible and has high reconstruction accuracy.

Performance and security analysis 5.1 Compressive sensing analysis
The peak signal-to-noise ratio (PSNR) is an important indicator to assess image quality.PSNR is defined as follows: where denotes the maximum value, and separately represent the original image and the processed image.Generally, the human eyes cannot differentiate between the original and processed images when [10].Table 1 provides the PSNR values for the different methods.We can apparently see that our method is superior to other schemes in 5,11,12]

Key space analysis
The designed algorithmm mainly involves the secret keys , , , and .If the computational precision is , the key space of our method is , which is larger than those in [3,5].Apparently, the key space of our method is large enough to resist the brute force attacks.

Secret key sensitivity analysis
In the sensitivity analysis, a very slight change is applied to the secret key , for example, . is applied to decrypt the encrypted image Fig. 1(b).Fig. 2 gives the simulation results.From Fig. 2(a), we do not find any information of Lena, and their difference rate between Fig. 1(c) and Fig. 2(a) is more than 99%.The simulation results show that our algorithm is very sensitive to the secret keys.

Conclusion
In this paper, an encryption scheme is proposed using compressive sensing and chaotic maps.On the one hand, the scheme uses CS to reduce the size of cipher image and improve the accuracy of image reconstruction.On the other hand, the key streams are utilized to reencrypt the compressed image, which greatly improves the security of encryption algorithm.Experimental results and security analysis have demonstrated that the proposed method has a satisfactory performance.

Table 1 .
in terms of reconstructed image quality.PSNRs (dB) for the different methods.