CN113746997A

CN113746997A - Image compression encryption method based on compressed sensing and fractional order chaos

Info

Publication number: CN113746997A
Application number: CN202111036301.6A
Authority: CN
Inventors: 佟晓筠; 魏京锦; 张淼
Original assignee: Harbin Institute of Technology Weihai
Current assignee: Harbin Institute of Technology Weihai
Priority date: 2021-09-06
Filing date: 2021-09-06
Publication date: 2021-12-03

Abstract

The invention discloses an image compression encryption method based on compressed sensing and fractional order chaos, and belongs to the technical field of multimedia information security. The invention provides an image compression encryption method based on compressed sensing and fractional order chaos, aiming at the problem of insufficient security of an image compression encryption scheme based on a low-dimensional integer order chaos system. The method constructs a new three-dimensional fractional order chaotic system to provide a random sequence required by an image compression encryption algorithm, and greatly expands the complexity and the key space of the algorithm. The image is compressed by adopting a compressed sensing algorithm, and a novel measurement matrix is constructed according to a random sequence. Meanwhile, the scheme follows a classical confusion and diffusion structure, and utilizes the principle of the Josephus problem and annular diffusion as scrambling and diffusion algorithms of the image respectively. Theoretical analysis and experimental results show that the algorithm of the invention has high safety and wide application prospect and practical value.

Description

Image compression encryption method based on compressed sensing and fractional order chaos

Technical Field

The invention belongs to the technical field of multimedia information security, and particularly relates to an image compression encryption method based on compressed sensing and fractional order chaos.

Background

With the rapid development of the internet, the information is more convenient to spread and store. Digital images have become one of the important means for people to express information because of their vividness. Image encryption is an effective tool for providing security protection for data, but image data often has high redundancy and strong correlation between pixels. Therefore, it should be considered to compress image data before transmission, storage, and encryption so as not to affect image quality while increasing transmission speed and reducing storage space.

Many new methods, such as DNA rule, bit-level arrangement, half tensor product theory, etc., appear in the process of image encryption. Since the chaotic system has initial value sensitivity and track unpredictability, and is particularly suitable for secure communication, the image encryption scheme based on the chaotic system is widely applied in recent years. Sun^[1]The like proposes an image encryption algorithm based on a seven-dimensional hyper-chaotic system and line-column synchronous exchange, which is excellent in security and encryption performance. Compressed Sensing (CS) is a new sampling reconstruction technique that can perform sampling and compression simultaneously. Wang (Wang)^[2]The patent refers to the field of 'transmission of digital information'. Xu (Xu)^[3]Et al propose an image compression encryption algorithm based on compressed sensing and two-dimensional SLIM mapping, which includes permutation, compression, and diffusion processes based on galois field multiplication. Simulation and performance analysis verify that the algorithm has good compression performance and high safety.

At present, the theory and application of the fractional order chaotic system in the field of secure data transmission have attracted extensive attention of researchers. The encryption by using the fractional order chaotic system has many advantages, such as resistance to various attacks based on key space analysis, such as brute force cracking. Compared with an integer order chaotic system, the fractional order chaotic system has higher nonlinearity and degree of freedom, and can present a more complex random sequence, thereby improving the safety of the cryptosystem. Wang (Wang)^[4]Two chaotic systems are used for encrypting images, wherein a segmented composite chaotic map is used as a Fisher-Yates scrambling controller, and a fractional order 5D cell neural network is used as a diffusion controller, so that the encryption efficiency is improved. Lahdir^[5]The image robust compression encryption scheme based on the SPIHT coding and the fractional order discrete time chaotic system is designed by the inventor. Yang^[6]The patent refers to the field of 'pictorial communication,'And (4) compressing the encryption scheme. The experimental results and the security analysis show the security of the scheme to the multiple attacks.

Disclosure of Invention

The invention provides an image compression encryption method based on compressed sensing and fractional order chaos, aiming at the problem of insufficient security of an image compression encryption scheme based on a low-dimensional integer order chaos system. According to the scheme, a new three-dimensional fractional order chaotic system is constructed to serve as an intermediate key of an encryption algorithm, a random sequence required by an image compression encryption algorithm is provided, and the complexity and the key space of the algorithm are greatly expanded. The image is compressed by adopting a compressed sensing algorithm, and a novel measurement matrix is constructed according to a random sequence. Meanwhile, the scheme follows a classical confusion and diffusion structure, and utilizes the principle of the Josephus problem and the annular diffusion as scrambling and diffusion operations of an algorithm respectively. Designed Josephus scrambling based on two-dimensional images can simultaneously confuse the row and column positions of global pixels, and the circular diffusion technology can realize the global diffusion of the original images. Experimental results show that the system has good statistical properties and can resist security attacks.

The design of the invention comprises four aspects, namely the design of a fractional order chaotic system, the construction of a measurement matrix, and the Josephus scrambling and annular diffusion algorithm based on a two-dimensional image.

1. Fractional order chaotic system

Zhang^[7]Et al analyzed the dynamic behavior of a three-dimensional chaotic system, which can be expressed as:

Eshaghi^[8]and the other person replaces the standard differential in the original chaotic system with a fractional order differential operator to describe the fractional order chaotic system. According to the system, a new three-dimensional fractional order chaotic system is constructed as follows:

where q is the fractional order and a, b, c, d and e are the system parameters. When the parameters of the fractional order chaotic system are set as a-1, b-1, c-80, d-1, e-18 and q-0.8, the system is in a chaotic state, and 3 chaotic sequences are generated.

2. Construction of measurement matrices

And generating a pseudo-random sequence by the fractional order chaotic system. To enhance the randomness of the random sequence, the first 1000 values of the 3 sequences x, y, z are discarded and sampled at equal intervals. Taking N/4 bits for each sequence and carrying out normalization processing, and ensuring that the proportion of x, y and z values in the measurement matrix respectively accounts for 1/3. Constructing a block circulant matrix z using chaotic sequences_lThe total measurement matrix is then constructed as C as follows.

Wherein the block circulant matrix z_lThe size is N/4 XN/4. And (5) extracting m rows from the matrix C to form a measurement matrix phi with the size of m multiplied by N. In the aspect of optimization of the measurement matrix, a QR decomposition method is selected.

3. Josephus scrambling algorithm based on two-dimensional image

The scrambling is to remove the correlation between the image pixels. Hua (Hua)^[9]Et al have devised a two-dimensional Josephus scrambling algorithm, where the method is improved. First, initial parameters are generated from the chaotic sequence. These parameters are then used to generate a plurality of two-dimensional Josephus sequences to order the rows and columns.

(1) Generation of initial parameters

Assume that the size of an original image to be encrypted is M × N. We generate the following 4 parameters from the chaotic key X,

where MI and NI denote the position of the starting row and starting column, respectively, and mist and NIstep denote the number of steps per row and column movement, respectively.

(2) Based on the generated parameters, the Josephus scrambling procedure is as follows,

step 1: the josephsus line sequence rs of length M is generated as follows: 1) initializing a vector a to be 1: M; 2) setting rs (1) ═ MI, and removing rs (1); 3) starting from the previously removed position, move the step of MIstep in a end-to-end and assign the next value to rs (2); 4) MIstep + k, where k is the number of rounds in the cycle; 5) repeat 3) and 4) M times until all values in a have been assigned to rs.

Step 2: set the row index i to 1.

And step 3: the generation of a Josephus column sequence cs of length N is as follows: 1) initializing a vector b as 1: N; 2) setting cs (1) to NI, and removing cs (1); 3) starting from the previously removed position, moving the step NIstep in end-to-end b assigns the next value to cs (2); 4) NIstep + k is the number of rounds of the cycle; 5) operations 3) and 4) are repeated N times until all values in b have been assigned to cs.

And 4, step 4: pixels on the ith row of the image are sequentially replaced according to the sequence of { (rs (cs (1)) + i, cs (1)) }, { (rs (cs (2)) + i, cs (2)) }, …, { (rs (cs (n)) + i, cs (n)) }. If rs (cs (k)) + i (k ═ 1 to N) is greater than M, the value of rs (cs (k)) + i modulo M is used as the result.

And 5: and (5) repeating the steps 2 to 4 when i is 2-M.

4. Circular diffusion algorithm

The main idea of circular diffusion is to combine random values in a chaotic sequence with the previously encrypted pixel value to change the plaintext value^[10]. Assuming that the total length of the image is L, the chaos sequence is mapped by fractional order chaos S ═ x₁，x₂，…，x_LAnd (5) composition. The replacement process for each pixel is then defined as,

where F is the image depth. For example, if an image is represented as 8-bit data, F is 256.

Meaning a rounding operation.

5. Security analysis

The safety analysis in this section is used as the embodiment of the actual effect of the invention, and the beneficial effect of the invention can be seen visually through the actual data analysis.

5.1 original image and decrypted image

The invention adopts the standard image and the Lena image in the USC-SIPI database for testing. The original image, the encrypted image, and the decrypted image are shown in fig. 1. As can be seen from FIG. 1, the fractional order chaotic system image compression encryption algorithm provided by the invention can correctly complete encryption and decryption of images.

5.2 histogram

Histograms are one of the important criteria for evaluating the security performance of image encryption algorithms. Fig. 2 shows a test image and its corresponding encrypted and decrypted histogram. It can be seen from fig. 2 that the histogram of the ciphertext image is significantly different from the histogram of the original image, which makes it impossible for an attacker to obtain statistical information about the original image by analyzing the histogram of the ciphertext image. Thus, the proposed image encryption algorithm can resist statistical analysis attacks.

5.3 correlation analysis of neighboring pixels

Neighboring pixels of an image tend to have strong correlation, and a good image encryption system should reduce this correlation as much as possible. Table 1 shows the correlation coefficients between neighboring pixels of an image and the respective histograms.

TABLE 1 neighboring Pixel correlation coefficients

As can be seen from table 1, the correlation between the adjacent pixels of the original image in the horizontal, vertical and diagonal directions is strong, and the correlation coefficient between the adjacent pixels is about 0.9. After the image is encrypted, the correlation coefficient of the image is greatly reduced. The results show that this scheme can reduce the correlation of neighboring pixels.

5.4 Key space analysis

The key in the invention is composed of 256-bit initial key K and 256-bit hash value of original image^[12]. According to the IEEE 754-. Thus, the key space of the encryption algorithm is 2⁵¹². With current computing power, this key space is large enough to resist exhaustive attacks.

5.5 Key sensitivity

The algorithm should be sensitive to its security key. This is manifested in that slight variations in the encryption key will cause the algorithm to produce completely different ciphertexts and only use the correct key to recover the original image. To test key sensitivity, we obtained 1 additional key K2 by randomly changing 1 bit over the original key. Keys K1 and K2 are as follows:

K1＝'Fc608C8B2fBb1B85c093Cc19236a65b564A98daA7ADA8FC8ad85Fb5eb9E2b1d8'

K2＝'Fc608C8B2fBb1B85c093Cc19236a65b564A98daA7ADA8FC8ad85Fb5eb9E2b1d7'

the result of encrypting and decrypting Lena images with the correct key and the modified key is shown in fig. 3.

As can be seen from fig. 3(b) (c) (d), encrypting the artwork using K1 and K2 with only 1-bit difference, 2 completely different encryption results are obtained, with a large difference between them. Meanwhile, as shown in fig. 3(e), the original plaintext cannot be reconstructed correctly by using the 1-bit key different from the correct key for decryption. The proposed algorithm is therefore sensitive to its security keys during encryption and decryption.

5.6 entropy of information

To some extent, the entropy reflects the randomness and unpredictability of the information source. H (m) is the information entropy of m, which can be calculated as,

wherein, P (m)_i) Represents m_iProbability of occurrence, N denotes m_iThe total number of (c). The ideal value of the information entropy h (m) is 8. The information entropies of the 4 images were calculated and compared as shown in table 2.

TABLE 2 entropy of information

From Table 2, it can be seen that the entropy obtained by our algorithm is greater than that of reference [11], indicating that our algorithm is well randomized.

5.7 differential challenge analysis

Differential attacks are an effective and commonly used security attack. It analyzes the change of specific plaintext after encryption by comparison, and uses the established connection to break the plaintext without key. To test the algorithm's ability to defend against differential attacks, we calculated the pixel change rate (NPCR) and the uniform mean change strength (UACI) of the different images. If a slight change to the plaintext pixel values can cause a large change to the ciphertext pixel values after encryption, it means that the encryption scheme works well. The calculation method of NPCR and UACI is as follows:

wherein W and H represent the width and height of the image, respectively, and d₁And d₂Is 2 ciphertext images before and after the plaintext image changes by 1 pixel value. If d is₁(i，j)＝d₂(i, j), D (i, j) is 0, otherwise D (i, j) is 1. Table 3 shows the results of comparison with other algorithms.

TABLE 3 NPCR and UACI

Given the ideal expectation for NPCR and UACI of 99.6094% NPCR and 33.4635% UACI, it can be seen from table 3 that the proposed encryption scheme is superior to the quote [11 ].

5.8 compressibility

PSNR (peak signal-to-noise ratio) is an objective criterion for evaluating images. Here, the PSNR value is used to evaluate the quality between the compression-decrypted image and the original image. The formula for the PSNR value is:

table 4 lists PSNR values for different compression ratios for 4 test pictures.

TABLE 4 PSNR values under different compression ratios

From table 4, it can be seen that the algorithm has good compression performance, and can reduce the load of images transmitted in the network.

The invention provides a novel chaotic image compression and encryption algorithm. The main idea of the algorithm is to compress an image by using compressed sensing, then use Josephus to confuse image pixels quickly, and finally use a ring diffusion algorithm to realize the change of a global pixel value. The fractional order chaotic system is used for providing a chaotic sequence required by a compression encryption algorithm, so that the algorithm has a larger key space. Simulation results show that the algorithm has effectiveness and good statistical properties in the aspects of information entropy, key space, key sensitivity, differential attack and the like. Therefore, the image compression and encryption scheme can achieve a good encryption and compression effect and has important significance for ensuring the information content safety of the image file.

Drawings

FIG. 1 is an original image and decrypted decompressed reconstructed image of the present invention, where (a) and (d) are original images, (b) and (e) are encrypted images, and (c) and (f) are decrypted images;

FIG. 2 is a histogram of an original image and a decrypted decompressed reconstructed image of the present invention, where (a) and (d) are histograms of the original image, (b) and (e) are histograms of the encrypted image, and (c) and (f) are histograms of the decrypted image;

FIG. 3 is a key sensitivity analysis of the present invention, where (a) is the original image, (b) and (C) are images C1 and C2 encrypted using K1 and K2, respectively, (d) is the difference between the encrypted images | C1-C2|, (e) is the image after the ciphertext C1 is decrypted using K2;

FIG. 4 is a schematic flow chart of an image compression and encryption algorithm of the fractional order chaotic system according to the present invention.

Detailed Description

In order to better understand the technical solution of the present invention, the following describes the embodiment of the present invention with reference to fig. 4.

As shown in fig. 4, the fractional order chaotic system-based image compression encryption algorithm of the present invention sequentially comprises the following steps:

in the first step, initial values x0, y0 and z0 of the fractional order chaotic system are calculated through SHA/MD5 hash values of an original image and an external key K. The 256-bit external key K is randomly generated by the correspondent (sender and receiver) and shared over the secure channel before use, and may be expressed as K ═ { K ═ K in an 8-bit decimal format₁,k₂,…,k₃₂}. Combining the external key K and the hash value H by [, that is, an XOR operation, to obtain K '═ K'₁,k’₂,…,k’₃₂}. Initial values x0, y0 and z0 of the fractional order chaotic system are calculated based on the formula (9) and the formula (10).

And secondly, inputting the initial value of the previous step into a three-dimensional fractional order chaotic equation to obtain a pseudorandom sequence, and recording the pseudorandom sequence as V ═ x, y, z }.

Third, according to the required compression ratio cr andpseudo-random sequence, constructing the required measuring matrix. The first 1000 values of the 3 sequences x, y, z are discarded and sampled at equal intervals. Taking N/4 bit combination for each of the 3 sequences and normalizing to form a block circulant matrix z_l. The cr × M rows are extracted from C of the formula (3) according to the compression ratio to form a measurement matrix Φ of a desired size (cr × M) × N. Where M and N are image sizes.

And fourthly, performing image compression based on compressed sensing on the image. The Discrete Wavelet Transform (DWT) is firstly carried out on the image to expand the pixel matrix of the image, and a sparse transform coefficient matrix is obtained. And according to the proposed measurement matrix phi, linearly projecting the transformation coefficient matrix to the measurement matrix phi and the orthogonal basis psi to obtain a measurement value, and obtaining a compressed image matrix. Wherein the measurement matrix is optimized using QR decomposition.

And fifthly, finishing quantization operation on the compressed image to enable the quantized value to be an integer between 0 and 255.

And sixthly, performing 2 rounds of Josephus scrambling and annular diffusion encryption operations based on the two-dimensional image on the quantized image to obtain a compressed encrypted image. For Josephus scrambling based on two-dimensional images, first 4 initial parameters are generated: the initial row position MI, initial column position NI, row moving step MIstep and column moving step NIstep are generated to generate a Josephus row sequence rs with the length of M and a Josephus column sequence cs with the length of M and the M wheels are N, and the image pixels are replaced in turn according to the sequence of { (rs (cs (1)) + i, cs (1)) }, { (rs (cs (2)) + i, cs (2)) }, …, { (rs (cs (N)) + i, cs (N)) }. Wherein i is the number of image lines in the ith round of scrambling. For circular diffusion, assuming that the total length of an image is L, a chaotic sequence is mapped by fractional order chaos S ═ x₁，x₂，…，x_LAnd (5) composition. The pixel values are then replaced according to equation (5), where

Meaning a rounding operation.

And seventhly, obtaining the compressed and encrypted image, and completing the image compression and encryption based on the fractional order chaotic system.

Reference to the literature

[1]Sun,S.,Y.Guo,and R.Wu."A Novel Image Encryption Scheme Based on 7D Hyperchaotic System and Row-column Simultaneous Swapping."IEEE Access(2019):28539-28547.

[2]Wang,Q.,et al."Joint encryption and compression of 3D images based on tensor compressive sensing with non-autonomous 3D chaotic system."Multimedia Tools and Applications 77.10(2018):1-20.

[3]Xu,Q.,et al."An effective image encryption algorithm based on compressive sensing and 2D-SLIM."Optics and Lasers in Engineering 134(2020):106178.

[4]Wang,X.,et al."A novel image encryption algorithm based on fractional order 5D cellular neural network and Fisher-Yates scrambling."PLoS ONE 15.7(2020):e0236015.

[5]Lahdira,M.,et al."A novel robust compression-encryption of images based on SPIHT coding and fractional-order discrete-time chaotic system."Optics&Laser Technology 109(2019):534-546.

[6]Yu-Guang,et al."Image compression-encryption scheme based on fractional order hyper-chaotic systems combined with 2D compressed sensing and DNA encoding."Optics&Laser Technology 119(2019):105661.

[7]Zhang,H.,et al."Chaos Entanglement:a New Approach to Generate Chaos."International Journal of Bifurcation and Chaos 23.5(2013):30014.

[8]Eshaghi,S.,R.K.Ghaziani,and A.Ansari."Hopf bifurcation,chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function."Mathematics and Computers in Simulation(MATCOM)172(2020).

[9]Hua,Z.,et al."Image Encryption Using Josephus Problem and Filtering Diffusion."IEEE Access(2019):1-1.

[10]Zhou,Y.,et al."Cascade Chaotic System With Applications."IEEE Transactions on Cybernetics(2015):2001.

[11]Ponuma,R,and R.Amutha."Compressive sensing based image compression-encryption using Novel 1D-Chaotic map."Multimedia Tools&Applications(2017).

[12]Zefreh,E.Z.."An image encryption scheme based on a hybrid model of DNA computing,chaotic systems and hash functions."Multimedia Tools and Applications 5187(2020).

Claims

1. An image compression encryption method based on fractional order chaos and compressed sensing is realized by the following seven steps:

in the first step, initial values x0, y0 and z0 of the fractional order chaotic system are calculated through SHA/MD5 hash values of an original image and an external key K. The 256-bit external key K is randomly generated by the correspondent (sender and receiver) and shared over the secure channel before use, and may be expressed as K ═ { K ═ K in an 8-bit decimal format₁,k₂,…,k₃₂}. Combining the external key K and the hash value H by [, that is, an XOR operation, to obtain K '═ K'₁,k’₂,…,k’₃₂}. Initial values x0, y0 and z0 of the fractional order chaotic system are calculated based on the formula (1) and the formula (2).

And thirdly, constructing a required measurement matrix according to the required compression ratio cr and the pseudorandom sequence. The first 1000 values of the 3 sequences x, y, z are discarded and sampled at equal intervals. Taking N/4 bit combination for each of the 3 sequences and normalizing to form a block circulant matrix z_l. The cr × M rows are extracted from C of the formula (3) according to the compression ratio to form a measurement matrix Φ of a desired size (cr × M) × N. Wherein M and N are image sizes。

And sixthly, performing 2 rounds of Josephus scrambling and annular diffusion encryption operations based on the two-dimensional image on the quantized image to obtain a compressed encrypted image. For Josephus scrambling based on two-dimensional images, first 4 initial parameters are generated: the initial row position MI, initial column position NI, row moving step MIstep and column moving step NIstep are generated to generate a Josephus row sequence rs with the length of M and a Josephus column sequence cs with the length of M and the M wheels are N, and the image pixels are replaced in turn according to the sequence of { (rs (cs (1)) + i, cs (1)) }, { (rs (cs (2)) + i, cs (2)) }, …, { (rs (cs (N)) + i, cs (N)) }. Wherein i is the number of image lines in the ith round of scrambling. For circular diffusion, assuming that the total length of an image is L, a chaotic sequence is mapped by fractional order chaos S ═ x₁，x₂，…，x_LAnd (5) composition. The pixel values are then replaced according to equation (4), where

Meaning a rounding operation.