An image compression and encryption method based on fractional order chaos and two-dimensional compressive sensing

Abstract

The application discloses an image compression encryption method based on fractional order chaos and two-dimensional compressed sensing, and belongs to the technical field of multimedia information security. In order to solve the problem of insufficient security of an image compression encryption scheme based on a low-dimensional integer order chaotic system, the application provides an image compression encryption method based on fractional order chaos and two-dimensional compressed sensing.A new three-dimensional fractional order chaotic system is constructed in the method, random sequences required in an image compression encryption algorithm are provided, and the complexity and key space of the algorithm are greatly expanded.The algorithm adopts a two-dimensional compressed sensing algorithm to compress an image, and a new measurement matrix is constructed according to the random sequences.Meanwhile, in order to consider the security and real-time performance of the algorithm, a combined scrambling and diffusion algorithm is constructed to be used for image encryption.Theoretical analysis and experimental results show that the algorithm has high security, and has wide application prospect and practical value.

Description

Technical Field

[0001] This invention belongs to the field of multimedia information security technology, specifically relating to an image compression and encryption method based on fractional-order chaos and two-dimensional compressed sensing. Background Technology

[0002] With the rapid development of the internet, the dissemination and storage of information have become much more convenient. Digital images, due to their vividness, have become one of the important means for people to express information. 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, image data should be compressed before transmission, storage, and encryption to improve transmission speed and reduce storage space without affecting image quality.

[0003] Many new methods have emerged in image encryption, such as DNA rules, bit-level permutations, and semi-tensor product theory. Due to the initial condition sensitivity and unpredictable trajectory of chaotic systems, they are particularly suitable for secure communication; therefore, image encryption schemes based on chaotic systems have been widely used in recent years. (Sun et al.) [1] A novel image encryption algorithm based on a seven-dimensional hyperchaotic system and row-column synchronous exchange is proposed, demonstrating good performance in both security and encryption. Compressed sensing (CS) is a novel sampling and reconstruction technique that can simultaneously perform sampling and compression. Two-dimensional compressed sensing (2D CS) samples and measures signals from two directions, further reducing the size of the compressed image and achieving better reconstruction quality. Wang et al. [2] A joint encryption and compression algorithm for 3D images based on a non-autonomous 3D chaotic system using CS is proposed. (Gan et al.) [3] A compression and encryption algorithm suitable for color images is proposed by fully utilizing information entropy and combining it with 2D CS. (Zhang et al.) [4] A novel nonlinear ETC (2DCS-ETC) compression and encryption scheme based on 2D CS is proposed. Simulation and performance analysis verify that the algorithm has good compression performance and high security.

[0004] Currently, the theory and application of fractional-order chaotic systems in the field of secure data transmission have attracted widespread attention from researchers. Using fractional-order chaotic systems for encryption offers many advantages, such as resistance to various attacks based on key space analysis, including brute-force attacks. Compared to integer-order chaotic systems, fractional-order chaotic systems exhibit higher nonlinearity and degrees of freedom, and can present more complex random sequences, thereby improving the security of cryptographic systems. [5]Researchers used two chaotic systems to encrypt images. One system utilized a piecewise composite chaotic map as the controller for Fisher-Yates scrambling, while a fractional-order 5D cellular neural network was used as the diffusion controller. This algorithm improved the encryption efficiency. Lahdir [6] Yang et al. designed a robust image compression and encryption scheme based on SPIHT coding and fractional-order discrete-time chaotic systems. [7] A team proposed an image compression and encryption scheme based on fractional-order hyperchaotic systems, two-dimensional compressed sensing, and DNA encoding. Experimental results and security analysis demonstrate the scheme's security against multiple attacks. Summary of the Invention

[0005] This invention addresses the security shortcomings of image compression and encryption schemes based on low-dimensional integer-order chaos by proposing a new method based on fractional-order chaos and two-dimensional compressed sensing. The scheme designs a novel three-dimensional fractional-order chaotic system as the intermediate key for the encryption algorithm, providing the random sequence required by the image compression and encryption algorithm, thus significantly expanding the algorithm's complexity and key space. A two-dimensional compressed sensing algorithm is used to compress the image, and a novel measurement matrix is ​​constructed based on the random sequence. Furthermore, to improve the computational efficiency and security performance of the encryption algorithm, a joint scrambling and diffusion algorithm is designed for image encryption. The designed encryption algorithm can achieve global diffusion of pixels while obfuscating their row and column positions. Experimental results show that the system possesses good statistical properties and can resist security attacks.

[0006] The design of this invention includes three aspects: a fractional-order chaotic system, the construction of a measurement matrix in two-dimensional compressed sensing, and a joint scrambling and diffusion algorithm.

[0007] 1. Fractional-order chaotic systems

[0008] Zhang [8] Others analyzed the dynamic behavior of a three-dimensional chaotic system, which can be represented as:

[0009]

[0010] In this invention, a fractional-order differential operator is used to replace the standard differential in the original chaotic system, and the new fractional-order chaotic system is designed as follows:

[0011]

[0012] Where q is the fractional order, and a, b, c, d, and e are system parameters. When the parameters of the fractional-order chaotic system are set to a = -1, b = 1, c = -80, d = -1, e = 18, and q = 0.8, the system is in a chaotic state and generates three chaotic sequences.

[0013] 2. Construction of the measurement matrix

[0014] A pseudo-random sequence is generated from a fractional-order chaotic system. To enhance the randomness of the sequence, the first 1000 values ​​of three sequences x, y, and z are discarded, and samples are taken at equal intervals. For each sequence, m / 2 bits are taken and normalized, ensuring that the values ​​of x, y, and z each account for 1 / 3 of the total measurement matrix. A block cyclic matrix z is constructed using the chaotic sequence. l Then, construct the total measurement matrix as C as follows.

[0015]

[0016] Wherein, the block cyclic matrix z l The size is m / 2 × m / 2. N columns are extracted from matrix C to form a measurement matrix Φ of size m × N. Here, N is the length of the original image, and m is the width of the image after compression. For the optimization of the measurement matrix, the QR decomposition method was chosen.

[0017] 3. Joint scrambling and diffusion algorithm

[0018] Scrambling aims to remove correlations between image pixels, while diffusion aims to change the pixel values ​​of the plaintext image. First, an index matrix and a diffusion auxiliary matrix are generated based on the chaotic sequence. [9] Then, these matrices are used to synchronously scramble and diffuse the image.

[0019] (1) Generation of index matrix and diffusion auxiliary matrix

[0020] Assume the original image P to be encrypted has a size of M×N. First, divide the fractional-order chaotic sequence into 5 random sequences. Sort the 4 random sequences in ascending order to obtain 4 index sequences seq1, seq2, seq3, and seq4, each with a length of N. seq5 is a numerical sequence with a length of M×N. Construct the index matrices Row and Col, and the auxiliary diffusion matrix Diff, using seq1 to seq5.

[0021]

[0022] After the auxiliary diffusion matrix is ​​generated, it needs to be converted into matrix form.

[0023] (2) Based on the generated matrix, the following encryption process is performed using a combination of scrambling and diffusion algorithms.

[0024]

[0025] Where F is the image depth. For example, if the image is represented as 8 bits of data, then F = 256.

[0026] 5. Security Analysis

[0027] The safety analysis in this section serves as a demonstration of the actual effects of this invention. Through actual data analysis, the beneficial effects of this invention can be clearly seen.

[0028] 5.1 Original Image and Decrypted Image

[0029] This invention uses standard images and Lena images from the USC-SIPI database for testing. The original image, encrypted image, and decrypted image are shown below. Figure 1 As shown. By Figure 1 It can be seen that the fractional-order chaotic system image compression and encryption algorithm proposed in this invention can correctly encrypt and decrypt images.

[0030] 5.2 Histogram

[0031] Histograms are one of the important standards for evaluating the security performance of image encryption algorithms. Figure 2 The test image and its corresponding encrypted and decrypted histogram are displayed. From Figure 2 It can be observed that the histogram of the encrypted image is significantly different from that of the original image, making it impossible for attackers to obtain statistical information about the original image by analyzing the histogram of the encrypted image. Therefore, the proposed image encryption algorithm can resist statistical analysis attacks.

[0032] 5.3 Correlation Analysis of Adjacent Pixels

[0033] Neighboring pixels in an image often exhibit strong correlations, and a good image encryption system should minimize this correlation as much as possible. Table 1 shows the correlation coefficients between neighboring pixels of an image and their respective encryption maps.

[0034] Table 1 Correlation coefficients of adjacent pixels

[0035]

[0036] Table 1 shows that adjacent pixels in the original image exhibit strong correlations in the horizontal, vertical, and diagonal directions, with a correlation coefficient of approximately 0.9. However, after image encryption, the correlation coefficient is significantly reduced. These results demonstrate that this method can reduce the correlation between adjacent pixels.

[0037] 5.4 Key Space Analysis

[0038] The key in this invention consists of a 256-bit initial key K and a 256-bit hash value of the original image.

[11] According to the IEEE 754-2008 standard, double-precision (binary64) data is used for storage, with eight bytes representing a double-precision number. Therefore, the key space for this encryption algorithm is 2^64 bytes. 512With current computing power, this key space is large enough to resist brute-force attacks.

[0039] 5.5 Key Sensitivity

[0040] The algorithm should be sensitive to its security key. This means that even a slight change to the encryption key will cause the algorithm to produce completely different ciphertext, and only by using the correct key can the original image be recovered. To test the key sensitivity, we obtained another key K2 by randomly changing one bit of the original key. Keys K1 and K2 are as follows:

[0041] K1='d8E0a9aA7Ba03fa37d4A5E5A462cfc9eeedE4f52d9aeA7e9CC5dcbD562Eeaebe'

[0042] K2='d7E0a9aA7Ba03fa37d4A5E5A462cfc9eeedE4f52d9aeA7e9CC5dcbD562Eeaebe'

[0043] The results of encrypting and decrypting the Lena image using the correct key and the modified key are as follows: Figure 3 As shown.

[0044] from Figure 3 As can be seen in (b), (c), and (d), encrypting the original image using K1 and K2, which differ by only 1 bit, yields two completely different encryption results with significant differences between them. Meanwhile, as... Figure 3 As shown in (e), decryption using a key that differs from the correct key by only 1 bit fails to reconstruct the original plaintext correctly. Therefore, the proposed algorithm is sensitive to its security key during encryption and decryption.

[0045] 5.6 Information Entropy

[0046] To a certain extent, information entropy reflects the randomness and unpredictability of information sources. H(m) is the information entropy of m, which can be calculated as follows:

[0047]

[0048] Wherein, P(m) i ) represents m i The probability of occurrence, N represents m i The total number of images. The ideal value of information entropy H(m) is 8. This paper calculates and compares the information entropy of four types of images, as shown in Table 2.

[0049] Table 2 Information Entropy

[0050]

[0051] As can be seen from Table 2, the entropy obtained by our algorithm is greater than that of reference

[10] , which indicates that our algorithm has good randomness.

[0052] 5.7 Differential Attack Analysis

[0053] Differential attacks are an effective and commonly used security attack. They break the plaintext without a key by comparing and analyzing the changes in the encrypted plaintext using an established connection. To test the algorithm's defense against differential attacks, we calculated the Non-Priority Change Rate (NPCR) and Uniform Average Change Intensity (UACI) for different images. If a slight change in the plaintext pixel values ​​results in a large change in the ciphertext pixel values ​​after encryption, it means the encryption scheme is effective. The calculation methods for NPCR and UACI are as follows:

[0054]

[0055] Where W and H represent the width and height of the image, respectively, and d1 and d2 are the two ciphertext images before and after changing one pixel value from the plaintext image. If d1(i,j) = d2(i,j), then D(i,j) = 0; otherwise, D(i,j) = 1. Table 3 shows the comparison results with other algorithms.

[0056] Table 3 NPCR and UACI

[0057]

[0058] The ideal expected values ​​of NPCR and UACI are known to be NPCR = 99.6094% and UACI = 33.4635%. As can be seen from Table 3, the proposed encryption scheme has better NPCR and UACI than the referenced

[10] .

[0059] 5.8 Compression performance

[0060] PSNR (Peak Signal-to-Noise Ratio) is an objective standard for evaluating images. Here, the PSNR value is used to evaluate the quality between the compressed / decompressed image R and the original image P. The formula for the PSNR value is:

[0061]

[0062] Table 4 lists the PSNR values ​​of the four test images at different compression ratios.

[0063] Table 4 PSNR values ​​under different compression ratios

[0064]

[0065] As shown in Table 4, the algorithm has good compression performance and can reduce the load on images transmitted over the network.

[0066] This invention proposes a novel chaotic image compression and encryption algorithm. The main idea is to first compress the image using compressed sensing, and then employ a joint scrambling and diffusion algorithm to achieve global pixel confusion and pixel value modification. A fractional-order chaotic system is used to provide the chaotic sequence required by the compression and encryption algorithm, enabling the algorithm to have a larger key space. Simulation results show that the algorithm has effectiveness and good statistical properties in terms of information entropy, key space, key sensitivity, and differential attack resistance. Therefore, this image compression and encryption scheme can achieve good encryption and compression effects, which is of great significance for ensuring the security of image file information content. Attached Figure Description

[0067] Figure 1 These are the original image and the reconstructed image after decryption and decompression, where (a) and (d) are the original images, (b) and (e) are the encrypted images, and (c) and (f) are the decrypted images.

[0068] Figure 2 These are histograms of the original image and the reconstructed image after decryption and decompression, wherein (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.

[0069] Figure 3 This is the key sensitivity analysis of the present invention, wherein (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|, and (e) is the image after decrypting the ciphertext C1 using K2;

[0070] Figure 4 This is a flowchart illustrating the fractional-order chaotic system and the image compression and encryption algorithm of two-dimensional compressed sensing of the present invention. Detailed Implementation

[0071] To better understand the technical solution of the present invention, the following is combined with... Figure 4 The embodiments of the present invention will be further described below.

[0072] As attached Figure 4 As shown, the image compression and encryption algorithm based on fractional-order chaos and two-dimensional compressed sensing described in this invention includes the following steps:

[0073] The first step is to calculate the initial values ​​x0, y0, and z0 of the fractional-order chaotic system using the SHA / MD5 hash value of the original image and the external key K. The 256-bit external key K is randomly generated by the communicating parties (sender and receiver) and shared through a secure channel before use, and can be represented in 8-bit decimal format as K = {k1, k2, ..., k}.32 The foreign key K and the hash value H are combined using the XOR operation (⊕) to obtain K' = {k'1, k'2, ..., k'}. 32 The initial values ​​x0, y0, and z0 of the fractional-order chaotic system are calculated based on formulas (9) and (10).

[0074]

[0075]

[0076] The second step is to input the initial values ​​from the previous step into the three-dimensional fractional chaotic equation to obtain a pseudo-random sequence, denoted as V = {x, y, z}.

[0077] The third step involves constructing the required measurement matrix based on the desired compression ratio and the pseudo-random sequence. The first 1000 values ​​of the three sequences x, y, and z are discarded, and samples are taken at equal intervals. Each of the three sequences is combined with m / 2 bits and normalized to form a block cyclic matrix z. l Based on the compression ratio, N columns are extracted from C in formula (3) to form a measurement matrix Φ of the required size m×N. Here, N is the length of the original image, and m is the width of the image after compression.

[0078] The fourth step involves image compression based on two-dimensional compressed sensing. First, a Discrete Wavelet Transform (DWT) is performed to expand the image pixel matrix, resulting in a sparse transform coefficient matrix. Then, based on the proposed measurement matrix Φ, the measured values ​​are obtained by linearly projecting the transform coefficient matrix onto the measurement matrix Φ and the orthogonal basis Ψ, thus yielding the compressed m×m image matrix. The measurement matrix is ​​optimized using the QR decomposition method.

[0079] The fifth step is to perform a quantization operation on the compressed image, so that the quantized value is an integer between 0 and 255.

[0080] The sixth step involves joint scrambling and diffusion encryption of the quantized image to obtain a compressed encrypted image. First, three auxiliary matrices are generated: an index matrix Row for generating row coordinates, an index matrix Col for generating column coordinates, and an array matrix Diff for assisting diffusion. Then, a new compressed encrypted image is constructed according to formula (5). The horizontal and vertical coordinates of each pixel in the new compressed encrypted image correspond to the values ​​of Col and Row in the index matrix of the original image. Furthermore, during the scrambling process, the array matrix Diff and the pixel values ​​undergo diffusion encryption.

[0081] Step 7: Obtain the compressed and encrypted image, completing the image compression and encryption based on fractional-order chaos and two-dimensional compressed sensing.

[0082] References

[0083] [1]Sun,S.,Y.Guo,and R.Wu."A Novel Image Encryption Scheme Based on 7DHyperchaotic System and Row-column Simultaneous Swapping."IEEEAccess(2019):28539-28547.

[0084] [2]Wang,Q.,et al."Joint encryption and compression of 3D images basedon tensor compressive sensing with non-autonomous 3D chaotic system."Multimedia Tools andApplications 77.10(2018):1-20.

[0085] [3]Gan Z,Bi J,Ding W,et al.Exploiting 2D compressed sensing andinformation entropy for secure color image compression and encryption[J].2021.

[0086] [4]Zhang B,Xiao D,Xiang Y.Robust Coding ofEncrypted Images via 2DCompressed Sensing[J].IEEE Transactions on Multimedia,2020,PP(99):1-1..

[0087] [5]Wang,X.,et al."A novel image encryption algorithm based onfractional order 5D cellular neural network and Fisher-Yates scrambling."PLoSONE 15.7(2020):e0236015.

[0088] [6]Lahdira,M.,et al."A novel robust compression-encryption of imagesbased on SPIHT coding and fractional-order discrete-time chaotic system."Optics&Laser Technology 109(2019):534-546.

[0089] [7]Yu-Guang,et al."Image compression-encryption scheme based onfractional order hyper-chaotic systems combined with 2D compressed sensingand DNA encoding."Optics&Laser Technology 119(2019):105661.

[0090] [8]Zhang,H.,et al."Chaos Entanglement:a New Approach to GenerateChaos."International Journal of Bifurcation and Chaos 23.5(2013):30014.

[0091] [9]Li T,Shi J,Zhang D.Color image encryption based on jointpermutation and diffusion[J].Journal of Electronic Imaging,2021,30(1).

[0092]

[10] Ponuma,R,and R.Amutha."Compressive sensing based imagecompression-encryption using Novel1D-Chaotic map."Multimedia Tools&Applications(2017).

[0093]

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Claims

1. An image compression and encryption method based on fractional-order chaos and two-dimensional compressed sensing, which is implemented in the following seven steps: The first step is to calculate the initial values ​​x0, y0, and z0 of the fractional-order chaotic system using the SHA / MD5 hash value of the original image and the external key K. The 256-bit external key K is randomly generated by the communicating parties (sender and receiver) and shared through a secure channel before use. In 8-bit decimal format, it can be represented as K = {k1, k2, ..., k...} 32 }; Combine the foreign key K and hash value H using the ⊕ (XOR) operation to obtain K' = {k'1, k'2, ..., k'}. 32 The initial values ​​x0, y0, and z0 of the fractional-order chaotic system are calculated based on formulas (1) and (2). The second step is to input the initial values ​​from the previous step into the three-dimensional fractional chaotic equation to obtain a pseudo-random sequence, denoted as V = {x, y, z}. The third step involves constructing the required measurement matrix based on the desired compression ratio and the pseudo-random sequence; discarding the first 1000 values ​​of the three sequences x, y, and z, and sampling at equal intervals; combining m / 2 bits of each of the three sequences and normalizing the results to form a block cyclic matrix Z. l ;Based on the compression ratio, extract N columns from C in formula (3) to form the required measurement matrix Φ of size m×N; where, N is the length of the original image, and m is the width of the image after compression. The fourth step involves image compression based on two-dimensional compressed sensing. First, a Discrete Wavelet Transform (DWT) is performed to expand the image pixel matrix, resulting in a sparse transform coefficient matrix. Then, based on the proposed measurement matrix Φ, the transformation coefficient matrix is ​​linearly projected onto the measurement matrix Φ and the orthogonal basis Ψ to obtain the measured values, thus yielding the compressed m×m image matrix. The measurement matrix is ​​optimized using the QR decomposition method. The fifth step is to perform a quantization operation on the compressed image, so that the quantized value is an integer between 0 and 255; The sixth step is to perform joint scrambling and diffusion encryption on the quantized image to obtain a compressed encrypted image. First, three auxiliary matrices are generated: an index matrix Row for generating row coordinates, an index matrix Col for generating column coordinates, and an array matrix Diff for assisting diffusion. Then, a new compressed encrypted image is constructed according to formula (4). The horizontal and vertical coordinates of each pixel in the new compressed encrypted image correspond to the values ​​of Col and Row in the index matrix of the original image. During the scrambling process, the array matrix Diff and the pixel values ​​are diffused and encrypted. Step 7: Obtain the compressed and encrypted image, completing the image compression and encryption based on fractional-order chaos and two-dimensional compressed sensing.

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