A Class of Multi-Wing Chaotic Attractors and Application Method in Image Encryption

Abstract

The present invention discloses a class of multi-wing chaotic attractors and an application method in image encryption, which relates to the technical field of image encryption. The algorithms that need to be implemented include a pixel position scrambling algorithm, an improved Arnold scrambling algorithm, and a diffusion algorithm based on GF ( 257 ). The diffusion algorithm is characterized by establishing a new multi-wing chaotic system. The scheme includes two different scrambling methods and one diffusion method: the first scrambling uses the pixel position scrambling algorithm, and the second scrambling uses the improved Arnold scrambling algorithm. Then, the pixel values are diffused through multiplication operations in the Galois field ( GF ). The simulation results and the security analysis results prove the effectiveness and superiority of the scheme.

Description

Technical Field

[0001] The present invention relates to the technical field of image encryption, and specifically to a class of multi-wing chaotic attractors and an application method thereof in image encryption. Background Technique

[0002] In recent years, network systems and distributed multimedia systems have been greatly developed in communication technologies. While the development of information transmission and reception devices is relatively complete, it also provides convenience for the illegal access and acquisition of communication data. In addition, with the development of 5G technology and the Internet of Things, due to the visual visibility characteristics of images, images have been more widely used in daily communication, commerce, military, education, and healthcare. Therefore, image information security has become an important and urgent issue. Currently, a large number of encryption schemes based on chaotic systems have been proposed in the field of image encryption. In the design of these image encryption algorithms, chaotic sequences generated by chaotic systems are usually combined to scramble and diffuse image pixels. The Arnold scrambling algorithm can quickly scramble image pixels and has broad application prospects in image encryption. In addition, significant encryption effects can be obtained by diffusing image pixels using the Galois field (GF) theory, realizing the hiding of image information.

[0003] In order to improve the complexity of chaotic signals in image encryption, constructing more complex chaotic systems has become a research highlight. A large number of typical chaotic systems have been found in previously published research. For example, the Lorenz system can generate a double-wing attractor similar to a butterfly, and the Chua's system can generate a double-vortex attractor. In recent years, many other types of chaotic systems have been successively proposed, such as hyperchaotic systems, memristive chaotic systems, fractional-order chaotic systems, and multi-scroll chaotic systems. However, there are few reports on multi-wing chaotic systems, mainly because the generation of multi-wing attractors requires breaking the balance of the original system, and there are not many systems that can produce such threshold changes. Therefore, it is necessary to further construct a new multi-wing chaotic system. Summary of the Invention

[0004] The purpose of the present invention is to provide a class of multi-wing chaotic attractors and an application method thereof in image encryption to solve the problems raised in the above background technique.

[0005] To achieve the above purpose, the present invention provides the following technical solutions, including algorithms that need to be used: pixel position disorder scrambling algorithm, improved Arnold scrambling algorithm, and diffusion algorithm based on GF(257), a class of multi-wing chaotic attractors and an application method thereof in image encryption. A class of multi-wing chaotic attractors and an application method thereof in image encryption are characterized in that the steps of establishing a new multi-wing chaotic system are as follows:

[0006] Step 1: Obtain a saddle-focus equilibrium point with index 1 and two saddle-focus equilibrium points with index 2 using the new three-dimensional double-wing chaotic system, thereby determining a double-wing attractor;

[0007] Among them, the new three-dimensional double-wing chaotic system is described as follows:

[0008]

[0009] Step 2: Then, by introducing a non-linear control term change the state variable , and then transform in the second equation into , constructing a new multi-wing chaotic system.

[0010] Among them, the non-linear function is described as:

[0011]

[0012] Furthermore, the steps of encrypting an image using the new multi-wing chaotic system are as follows:

[0013] Step 1: Read the original color image with size to obtain a matrix with size ;

[0014] Step 2: Select the parameters and initial values of the new multi-wing chaotic system to make it in a chaotic state, and iterate times to obtain the pseudo-random matrices , , ;

[0015] Step 3: Process all elements in the pseudo-random sequence obtained in Step 3 through the following formula;

[0016]

[0017] Step 4: Divide the matrix into two matrices, respectively labeled as , , where the number of rows and columns of each matrix is and . Similarly, decompose the matrices and to obtain the pseudo-random matrices , , , , where the number of rows and columns of each matrix is also and ;

[0018] Step 5: The first scrambling adopts a pixel position scrambling algorithm. Using the chaotic matrix generated in Step 4 , arrange the values in each column of the matrix in descending order to obtain an index sorting matrix ;

[0019] Step 6: Combine the index sorting matrix in Step 5 , and scramble the pixel positions of the matrix according to the following formula to obtain a scrambled matrix ;

[0020]

[0021] where, ;

[0022] Step 7: The second scrambling adopts an improved Aronld scrambling algorithm. Convert the pseudo-random matrices in Step 4 and into one-dimensional vectors respectively and perform modulo processing; use the formulas , and to obtain the vector ;

[0023] Step 8: Convert the matrix into a one-dimensional row vector, and exchange the elements of the row vector obtained in Step 7 and , so as to obtain the matrix after scrambling ;

[0024] Step 9: Adopt a diffusion algorithm based on GF(257) to change the values of image pixels. Obtain the multiplication lookup table of GF(257) from the following formula:

[0025]

[0026] Step 10: Convert the pseudo-random matrix obtained in Step 4 into a one-dimensional row vector. Combine the vector , and perform forward diffusion on according to the following equation to obtain the vector .

[0027]

[0028] where, , .

[0029] Step 11: The pseudo-random matrix obtained in Step 4 Convert it into a one-dimensional row vector. Combine the vectors , and perform backward diffusion on according to the following equation to obtain the vector .

[0030]

[0031] Among them, , .

[0032] Step 12: Convert the vector into a matrix with a size of to obtain the final encrypted image.

[0033] Furthermore, the image encryption method adopts a symmetric encryption scheme, and all algorithms used in the encryption are reversible.

[0034] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0035] 1. A new three-dimensional Lorenz-like chaotic system with two quadratic terms and four linear terms is proposed. The system has two saddle-focus equilibria with index 2 and one saddle-focus equilibrium with index 1, thus obtaining a double-wing attractor.

[0036] 2. Based on the above three-dimensional Lorenz-like chaotic system, by introducing a sign function and another non-linear state feedback controller to transform the non-linear terms and expand the equilibria with index 2, a new multi-wing chaotic system is obtained. The dynamic analysis of the new multi-wing system shows that the system has two positive Lyapunov exponents within a large parameter range. Obviously, the system has strong chaos and is very suitable for image encryption.

[0037] 3. A color image encryption algorithm based on the constructed multi-wing chaotic system is proposed. The scheme includes two different scrambling methods and a diffusion method: the first scrambling adopts a pixel position disorder scrambling algorithm, the second scrambling adopts an improved Arnold scrambling algorithm, and then the pixel values are diffused through multiplication operations in the Galois field (GF). The simulation results and security analysis results prove the effectiveness and superiority of the scheme. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 is the phase diagram of the double-wing chaotic attractor of the present invention;

[0039] Figure 2 is the phase diagram of the chaotic attractor of the multi-wing chaotic system of the present invention;

[0040] Figure 3 is the Lyapunov exponent spectrum of the present invention under different parameter ranges;

[0041] Figure 4 is the encryption and decryption flow chart of the present invention;

[0042] Figure 5 are the encryption and decryption effect diagrams of images of different sizes according to the present invention;

[0043] Figure 6 are the histogram analysis results of images of different sizes according to the present invention;

[0044] Figure 7 are the correlation analysis results of images of different sizes according to the present invention;

[0045] Figure 8 are the test results of different shear attacks according to the present invention;

[0046] Figure 9 are the test results of noise attacks on images Boat(3×256×256) and Camera(3×512×512) according to the present invention. Detailed implementation manners

[0047] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0048] Please refer to Figures 1 - 9 , in the embodiments of the present invention, a new three-dimensional double-wing chaotic system is described as follows:

[0049]

[0050] Wherein, , , , are positive real parameters, , , are the state variables of the system. The parameters of the system are respectively considered as , , , , and the initial conditions are considered as , , . Using this set of parameter values and initial conditions, a double-wing chaotic attractor is generated in Figure 1 . At this time, the equilibrium point is calculated as , and the corresponding eigenvalues are respectively calculated as , This indicates that the system has a saddle-focus equilibrium point with index 1 and two saddle-focus equilibrium points with index 2 at this time. According to the Silnikov theorem, each saddle-focus equilibrium with index 2 can generate a unique wing in the attractor phase diagram.

[0051] Based on the above system, a non-linear control term is introduced into the third equation Changing the state variables , and then the in the second equation is transformed into , and a new multi-wing chaotic system is constructed.

[0052]

[0053] Among them, the non-linear function is described as:

[0054]

[0055] In the function , the parameter can control the number of wings in the chaotic attractor generated by the multi-wing chaotic system. Let , , , , , and the initial values are , , . At this time, the multi-wing chaotic system can generate wing attractors. The numerical simulation results are as shown in Figure 2 , and the corresponding parameters are shown in Table 1.

[0056] In addition, when the system parameter , the Lyapunov dimension of the multi-wing chaotic system is , indicating that the multi-wing chaotic system is in the fractal dimension. At the same time, the Lyapunov exponents can also prove that the system is in a chaotic state. Figure 4 The Lyapunov exponent spectra in different parameter ranges are shown in . It should be noted that two of the three Lyapunov exponents in different parameter ranges are always greater than zero, indicating that the chaos of the chaotic system is very strong, while the other is always less than zero, so it is not shown in the figure.

[0057] Table 1. Corresponding parameters for generating multi-wing attractors

[0058]

[0059] The algorithms that need to be used are:

[0060] 1. Pixel position disorder scrambling algorithm

[0061] The pixel position scrambling algorithm can quickly scramble the pixel positions of an image in both the row and column directions, destroy the correlation between adjacent pixels of the image, and make the original image information inaccessible after encryption, with low time complexity and high security. The specific operation steps of this algorithm are as follows:

[0062] Step 1: Read the grayscale plaintext image , and set its size to .

[0063] Step 2: Use the chaotic matrix with size generated by the new multi-wing chaotic system, sort the values of each column in descending order to obtain the index sorting matrix . .

[0064] Step 3: Combine the index sorting matrix , and according to the following formula, destroy the matrix value positions to obtain the scrambled matrix .

[0065]

[0066] 2. Improved Arnold Scrambling Algorithm

[0067] The essence of the Arnold scrambling transformation is a one-to-one mapping of pixel positions. The specific transformation matrix is as follows:

[0068]

[0069] That is , .

[0070] It should be noted that the elements at the pixel positions and can be swapped with the pseudo-random variables and , which means that the role of is not considered. Therefore, the improved Arnold algorithm only considers a part of the above equation, regards as a new random number, still denoted as , then the improved Arnold scrambling algorithm is defined as the following formula:

[0071]

[0072] The specific steps of this algorithm are described as follows:

[0073] Step 1: Read the grayscale plaintext image with size , and convert it into a one-dimensional vector 。

[0074] Step 2: Iterate the new multi-wing chaotic system times to generate a chaotic matrix and , and then convert it into a one-dimensional row vector.

[0075] Step 3: Vectors and are processed through the formulas and , and thus a one-dimensional row vector is obtained through the equation .

[0076] Step 4: Reorder the index values of the vector for the vector to disturb the pixel positions of the plaintext image.

[0077] Step 5: Convert the one-dimensional row vector obtained in Step 4 into a matrix to obtain the permuted matrix .

[0078] 3. Diffusion algorithm based on GF(257)

[0079] Compared with pixel scrambling, the diffusion operation changes the pixel values of the image without changing the pixel positions, hiding the pixels of the original image in as many ciphertext pixels as possible. Galois field theory (GF) is a set containing a finite number of elements, with definitions of addition, subtraction, multiplication, and division operations and satisfying certain rules. It is also called a finite field GF(p), where p is a prime number and is an important field in cryptography. In the present invention, the multiplication operation in GF(257) is used to diffuse the image pixels, and the division operation can be used to restore the original pixel values of the image. In addition, to reduce the information loss caused by the number 0 in the multiplication operation, the number 0 is eliminated in the multiplication operation.

[0080] The specific steps of this algorithm are described as follows:

[0081] Step 1: Read a grayscale plaintext image of size and convert it into a one-dimensional vector denoted as . .

[0082] Step 2: Iterate the new multi-wing chaotic system times to generate a pseudo-random matrix and , and then convert them into one-dimensional vectors.

[0083] Step 3: The multiplication lookup table of GF(257) is obtained from the following formula:

[0084]

[0085] Step 4: Combine the vector , and perform forward diffusion on the vector according to the following equation to obtain the vector .

[0086]

[0087] Among them, , , and the operations in the equation follow the multiplication rules in GF(257).

[0088] Step 5: Combine the vector , and perform backward diffusion on the vector according to the following equation to obtain the vector .

[0089]

[0090] Among them, , , and the operations in the equation follow the multiplication rules in GF(257).

[0091] Step 6: Convert the vector into a matrix of to obtain the diffused digital image.

[0092] Then, the picture is encrypted through the following steps:

[0093] Step 1: Read a grayscale image with a size of to obtain a matrix with a size of .

[0094] Step 2: Select the parameters and initial values of the new multi-wing chaotic system to make it in a chaotic state. Randomly take three integers between 0 and 255, denoted as , , Let the new multi-wing chaotic system iterate times to eliminate the transient effect, and then iterate the system times to obtain the chaotic sequence .

[0095] Step 3: Convert all elements in the sequence through the following formula to obtain the pseudo-random matrix , , .

[0096]

[0097] Among them, represents the largest integer not greater than , is a remainder function, that is, the remainder of two digital expressions after division. , , , , , are expressed as follows:

[0098]

[0099]

[0100] Step 4: Matrix is divided into two matrices, respectively labeled as , , where the number of rows and columns of each matrix is and . Similarly, matrices and are decomposed to obtain pseudo-random matrices , , , , where the number of rows and columns of each matrix is also and ;

[0101] Step 5: The first scrambling adopts the pixel position scrambling algorithm. Using the chaotic matrix generated in Step 4, the values of each column in the matrix are arranged in descending order to obtain the index sorting matrix .

[0102] Step 6: Combining the index sorting matrix in Step 5, the pixel positions of matrix are scrambled respectively according to the following formula to obtain the scrambled matrix .

[0103]

[0104] Among them, ;

[0105] Step 7: The second scrambling adopts the improved Aronld scrambling algorithm. The pseudo-random matrices and in Step 4 are respectively transformed into one-dimensional vectors and modulo operations are performed, such as formula , , to obtain one-dimensional vectors and . From the formula a one-dimensional row vector is obtained.

[0106] Step 8: Convert the matrix into a one-dimensional row vector. By swapping the positions of the elements of the row vector obtained in Step 7 and . Thus, the scrambled matrix is obtained. .

[0107] Step 9: Adopt a diffusion algorithm based on GF(257) to change the values of image pixels. The multiplication lookup table of GF(257) is obtained from the following formula:

[0108]

[0109] Step 10: Convert the pseudo-random matrix obtained in Step 4 into a one-dimensional row vector. Combine the vector , and perform forward diffusion on respectively according to the following equation to obtain the vector .

[0110]

[0111] where , .

[0112] Step 11: Convert the pseudo-random matrix obtained in Step 4 into a one-dimensional row vector. Combine the vector , and perform backward diffusion on respectively according to the following equation to obtain the vector .

[0113]

[0114] where , .

[0115] Step 12: Convert the vector into a matrix with a size of to obtain the final encrypted image.

[0116] where

[0117] The present invention adopts a symmetric encryption scheme, and all algorithms used in encryption are reversible. Therefore, the decryption algorithm is the inverse process of encryption.

[0118] To verify the feasibility of the encryption and decryption of the present invention and the security performance of the algorithm, experiments were conducted using a personal computer equipped with 12 GB of memory, an Intel Core i7-7500U (2.7 GHz / L3 4M) processor, and a Windows 10 operating system. MATLAB R2018b was used as the experimental software. In addition, simulation experiments were carried out using grayscale images Boat(3×256×256) and Camera(3×512×512) of different sizes.

[0119] And the experiments were analyzed as follows:

[0120] Key space analysis:

[0121] The keys of the image encryption system used mainly include the system parameters of the multi-wing chaotic system , , , , , , , , , , , , , , , and the initial values of the system , , as well as the iterative random numbers . Among them, the calculation accuracies of the parameters , , , , , , , , , , , , , and the initial values , , are all . The calculation accuracies of the parameters , are all . Therefore, the calculated key space value of this algorithm is greater than , which far exceeds the theoretical key space value . Therefore, this scheme can effectively resist brute-force attacks.

[0122] Histogram analysis:

[0123] A digital image is composed of pixels with different gray values, and the distribution of these pixels is an important feature of the image. A good encryption algorithm can make the pixels in the ciphertext image evenly distributed. The histogram of an image can directly and accurately reflect the distribution of pixel values and is usually used to detect whether the encrypted image can resist statistical attacks. By numerical simulation, the histograms of the original experimental image and the related encrypted image are obtained respectively, as Figure 6 shown. It can be seen from the comparison that the pixel distribution characteristics of the plaintext image histogram are obvious, while the pixel distribution of the ciphertext image is approximately uniform, indicating that the designed image encryption scheme can effectively resist statistical analysis attacks.

[0124] Analysis of pixel - to - pixel correlation:

[0125] The correlation between pixels represents the degree of similarity between adjacent pixels. Generally speaking, there is a strong correlation between adjacent pixels in the plaintext image in the horizontal, vertical, and diagonal directions. For the encrypted image, the lower the correlation between pixels, the higher the security of the encryption algorithm and the better the encryption effect. Therefore, a good encryption algorithm should minimize the correlation coefficient between adjacent pixels in the image.

[0126] Table 2. Calculation results of correlation coefficients for images of different sizes

[0127]

[0128] Analysis of information entropy:

[0129] Information entropy is used to describe the degree of chaos of information and can reflect the randomness of image information. It is an important indicator to test whether the encryption algorithm is secure. The more chaotic the information in the image, the larger the value of the information entropy. For an image with a gray scale of , the theoretical value of the information entropy is . The calculation results of the information entropy values for images of different sizes are shown in Table 3. It can be seen that the information entropy of the ciphertext image is closer to the theoretical value than that of the plaintext image.

[0130] Table 3. Calculation results of information entropy values for images of different sizes

[0131]

[0132] Robustness analysis:

[0133] In the actual transmission process, digital images will inevitably be affected by factors such as noise interference and data loss. Therefore, an excellent encryption algorithm should also have sufficient resistance to shear attacks and noise attacks. To test the ability of the designed algorithm to resist shear attacks, the ciphertexts of images Boat(3×256×256) and Camera(3×512×512) were sheared to different degrees, and the test results are as Figure 8 shown. In addition, to test the noise attack resistance of the scheme, salt-and-pepper noise with intensities of 0.01, 0.06, and 0.1 and Gaussian noise with intensities of 0.000001, 0.000002, and 0.000003 were respectively used. The noise attack test results of images Boat(3×256×256) and Camera(3×512×512) are as Figure 9 shown. Obviously, when the ciphertext image has data loss to different degrees or is attacked by noise, the decrypted image can still recover the main information in the plaintext image. Therefore, the algorithm has strong robustness against data loss and noise interference attacks.

[0134] For those skilled in the art, it is obvious that the present invention is not limited to the details of the above exemplary embodiments, and without departing from the spirit or basic characteristics of the present invention, the present invention can be implemented in other specific forms. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-restrictive. The scope of the present invention is defined by the appended claims rather than the above description. Therefore, all changes falling within the meaning and scope of the equivalent elements of the claims are intended to be encompassed by the present invention, and any reference signs in the claims should not be regarded as limiting the claimed rights.

[0135] In addition, it should be understood that although this specification is described according to embodiments, not every embodiment only contains an independent technical solution. This narrative way of the specification is only for clarity. Those skilled in the art should regard the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A class of multi-wing chaotic attractors and an application method in image encryption, including algorithms to be used such as a pixel position disorder scrambling algorithm, an improved Arnold scrambling algorithm, and a diffusion algorithm based on GF(257), characterized in that, The steps to establish a new multi-wing chaotic system are as follows: Step 1: Use the new three-dimensional two-wing chaotic system to obtain a saddle-focus equilibrium point with index 1 and two saddle-focus equilibrium points with index 2, thereby determining a two-wing attractor; Among them, the new three-dimensional two-wing chaotic system is described as follows: Step 2: Then, by introducing a non-linear control term change the state variables , and then convert the in the second equation into to construct a new multi-wing chaotic system; Among them, the non-linear function is described as: 。 2. The method for a class of multi-wing chaotic attractors and their application in image encryption according to claim 1, characterized in that The steps to encrypt an image using the new multi-wing chaotic system are as follows: Step 1: Read the original color image with a size of to obtain a matrix with a size of ;​ Step 2: Select the parameters and initial values of the new multi-wing chaotic system to make it in a chaotic state and iterate times to obtain a pseudo-random matrix , , ; Step 3: Process all elements in the pseudo-random sequence obtained in Step 3 through the following formula; Step 4: Divide the matrix into two matrices, respectively labeled as , , where the number of rows and columns of each matrix is and respectively. Similarly, decompose the matrices and to obtain the pseudo-random matrices , , , , where the number of rows and columns of each matrix is also and ; Step 5: The first scrambling adopts a pixel position scrambling algorithm and uses the chaotic matrix generated in Step 4 , and arranges the values in each column of the matrix in descending order to obtain an index sorting matrix ; Step 6: Combine the indexing and sorting matrix in Step 5 , and scramble the pixel positions of the matrix according to the following formula to obtain a scrambled matrix ; Among them, ; Step 7: The second scrambling uses the improved Arnold scrambling algorithm to convert the pseudo-random matrices in Step 4 and into one-dimensional vectors respectively and perform modulo processing; using the formulas , and to obtain the vector ; Step 8: Convert the matrix into a one-dimensional row vector, and swap the positions of the elements of the row vector obtained in Step 7 with and to obtain the scrambled matrix for ; Step 9: Adopt a diffusion algorithm based on GF(257) to change the values of image pixels, and obtain the multiplication lookup table of GF(257) from the following formula: Step 10: Convert the pseudo-random matrix obtained in Step 4 into a one-dimensional row vector, and combine it with the vector . According to the following equation, perform forward diffusion on to obtain the vector ; Among them, , ; Step 11: Convert the pseudo-random matrix obtained in Step 4 into a one-dimensional row vector, and combine it with the vector . According to the following equation, perform backward diffusion on to obtain the vector ; Among them, , ; Step 12: Convert the vector into a matrix of size to obtain the final encrypted image.

3. The multi-wing chaotic attractor of a kind according to claim 2 and the application method thereof in image encryption, characterized in that: The image encryption method adopts a symmetric encryption scheme, and all algorithms used in the encryption are reversible.

Images