CN117579756B - Image encryption method based on block selection Zigzag scrambling and rotary coding of wheel disc - Google Patents

Abstract

The invention provides an image encryption method based on block selection of Zigzag scrambling and rotary coding of a wheel disc, which comprises the following steps: calculating an initial value of the 4D hyper-chaotic system; carrying out iteration by taking the 4D hyper-chaotic system to obtain four chaotic sequences; selecting a coordinate value by using a value of the chaotic sequence, and scrambling a plaintext image according to a random Zigzag scrambling method to obtain a matrix P ₁; dividing the matrix P ₁ into small pixel blocks, converting pixel values into quaternary system, and scrambling according to a quaternary position scrambling method to obtain a pixel matrix P ₂; expanding the pixel matrix P ₂ into a one-dimensional sequence according to the row, and diffusing the one-dimensional sequence by using a chaotic sequence through a wheel disc rotary coding algorithm to obtain an image matrix P ₃; the chaotic sequence is converted into a chaotic matrix, the chaotic sequence is sequenced, the index sequence is converted into an index matrix, and the image matrix P ₃ is diffused according to the bidirectional nonsequential sequence of the index matrix by utilizing the chaotic matrix to obtain a ciphertext image. The invention can effectively resist various attacks, and has better encryption effect and high security.

Description

Image encryption method based on block selection Zigzag scrambling and roulette wheel rotation coding

Technical Field

The present invention relates to the technical field of digital image encryption, and in particular to an image encryption method based on block selection Zigzag scrambling and roulette rotation coding.

Background Art

With the rapid development of computer technology and multimedia communication, a large amount of information such as images, audio and video is stored, shared and transmitted through cloud services, especially in e-commerce, medicine, military and other fields. However, there are many threats and attacks on images on the Internet, such as theft, tampering and forgery. This not only causes damage to individuals, but may also lead to huge economic and political losses. Therefore, ensuring the security of digital image content has become an important issue that needs to be solved urgently. Image encryption, as an effective method for image information security protection, has a wide range of applications in data hiding, privacy protection and other fields. However, the encryption efficiency of traditional image encryption schemes is low. Therefore, it is crucial to study safe and efficient methods.

Compared with traditional image encryption methods, encryption methods based on chaotic systems have higher encryption efficiency. Therefore, many researchers have proposed image encryption schemes based on chaotic systems. In 1989, mathematician Matthews first introduced chaos into encryption systems and proposed and explained the concept of chaotic cryptography. Fairouz Belilita improved the one-dimensional Logistic map, improved the performance of the chaotic system by expanding the chaotic range of the control parameters, and applied the improved Logistic map to the encryption of grayscale images, achieving good encryption results. Ma proposed two-dimensional logical cosine cascade mapping (2D-LCCM) and three-dimensional logical cosine cascade mapping (3D-LCCM), analyzed the chaotic properties of 2D-LCCM and 3D-LCCM, combined 2D-LCCM with zigzag scrambling for scrambling, and combined 3D-LCCM with DNA coding algorithm for diffusion. The scheme has the advantages of high efficiency and strong anti-interference performance, and can be effectively applied to the image encryption process. Zhang added a nonlinear controller containing a sine function to the Lorenz system and proposed a Lorenz-Sine coupled chaotic system with multi-symmetric strange attractors and a large chaotic range; at the same time, he proposed an image encryption scheme based on the spiral rotation and random permutation of the chaotic system, which has good resistance to differential attacks. In addition, it is highly robust to noise attacks and cropping.

The application of Zigzag scrambling in image encryption is also very common. Many researchers have proposed a large number of image encryption schemes based on Zigzag scrambling. Zhang proposed a new three-dimensional transformation zigzag diffusion algorithm, which arranges the pixel values of the plaintext image into a cube shape and uses a pseudo-random matrix for zigzag diffusion. This scheme can resist different types of attacks. Wang proposed a chaotic image encryption algorithm based on extended zigzag scrambling and RNA operation, which can not only process images in non-square matrix form, but also select from four directions to determine the starting position of the encryption process; this scheme not only solves the problem that some elements always remain in the same position in each round of encryption, but also the encoding rules of the RNA matrix are completely controlled by the chaotic sequence, which makes the operation results more unpredictable; experimental simulation and performance analysis data show that the encryption algorithm has high security performance. Li et al. proposed a color image Zigzag encryption scheme based on three-dimensional chaotic mapping. Experiments show that this method has strong resistance to brute force attacks and statistical attacks, but lacks differential attack analysis.

Simple pixel-level scrambling can only change the pixel position. In order to enhance the scrambling effect, many researchers have proposed image encryption algorithms that include bit-level scrambling. Xu proposed an image encryption algorithm based on three-dimensional bit matrix and orthogonal Latin cube. Unlike most existing image encryption algorithms, this algorithm regards the original image as a three-dimensional bit matrix and uses Latin cube for scrambling and diffusion. The combination of three-dimensional bit matrix and Latin cube enables the algorithm to achieve both an ideal security level and high efficiency. Wen proposed an image encryption system based on binary bit plane extraction and multiple chaotic mappings. The system mainly includes bit-level permutation of the upper 4 bit planes and XOR diffusion of the lower 4 bit planes. It has low computational complexity, but is not very secure against chosen plaintext attacks and is easily cracked.

Diffusion is an important stage in image encryption. Common one-way diffusion is easy to crack. For this reason, some scholars have proposed encryption algorithms based on multi-directional diffusion. Liu proposed a color image encryption algorithm based on improved DNA coding and fast diffusion. In the diffusion stage, matrices and vectors are used as diffusion units, and three-dimensional six-way diffusion is performed on the three channels of the color image. This scheme has high security, but the encryption efficiency is not high. Hussain Muhammad proposed an image encryption scheme that performs scrambling diffusion along the diagonal-anti-diagonal direction. This scheme performs scrambling and diffusion operations on the pixels of a given input image in parallel, which has high security. However, these diffusions still process pixel values in sequence and are easy to be cracked.

Summary of the invention

In view of the technical problems of high computational complexity and low security of existing image encryption methods, the present invention proposes an image encryption method based on block-selected Zigzag scrambling and roulette rotation coding, which is highly sensitive to keys and plaintext images, realizes dual-reset scrambling at the pixel level and the bit level, breaks the high correlation between image pixels, and improves the security of image encryption.

In order to achieve the above object, the technical solution of the present invention is implemented as follows: an image encryption method based on block selection Zigzag scrambling and roulette rotation encoding, the steps of which are as follows:

Step 1: Use the SHA-384 algorithm to calculate the hash value H of the plaintext image P of size M×N, and calculate the initial value of the 4D hyperchaotic system according to the hash value H; bring the initial value into the 4D hyperchaotic system for iteration to obtain four chaotic sequences X, Y, Z and W;

Step 2: Select multiple groups of elements in the chaotic sequences Z and W and convert them into elements with values less than M and less than N to obtain sequences U and V, use the values of sequences U and V to select coordinate values, and scramble the plaintext image P according to the random Zigzag scrambling method to obtain the matrix P ₁ ;

Step 3: Divide the matrix _P1 into several 2×2 small pixel blocks and convert the pixel values in each small pixel block into quaternary, and perform scrambling according to the quaternary position scrambling method to obtain the scrambled pixel matrix _P2 ;

Step 4: Map the values of chaotic sequence X to the range of 1-8 to obtain sequence X′, and convert the values of chaotic sequence Y to 0 or 1 to obtain sequence Y′; expand the pixel matrix P ₂ into a one-dimensional sequence by row, and use the roulette wheel encoding algorithm to diffuse the one-dimensional sequence using the sequences X′ and Y′, and convert the diffused one-dimensional sequence back into a matrix to obtain the image matrix P ₃ ;

Step 5: Convert the chaotic sequences Z and W into sequences Z1 and W1 with values ranging from 0 to 255 respectively and then convert them into chaotic matrices Z′ and W′ respectively; sort the chaotic sequence X and convert the index sequence into an index matrix, and use the chaotic matrices Z′ and W′ to diffuse the matrix P ₃ bidirectionally and non-sequentially according to the index order of the index matrix to obtain the ciphertext image C.

Preferably, the method for calculating the initial value of the 4D hyperchaotic system is: inputting the plaintext image into the SHA-384 algorithm to output a 384-bit binary hash value H, dividing the hash value H into 48 groups of binary sequences, each group of 8 bits, to obtain H _k =h ₁ ,h ₂ ,h ₃ ,…h ₄₈ , performing operations on the 48 groups of binary sequences, and calculating the initial values x ₀ , y ₀ , z ₀ , w ₀ of the 4D hyperchaotic system as follows:

Among them, K ₁ ~ K ₈ are intermediate variables, 1≤i2≤8, is the XOR operation, and mod is the modulo function.

Preferably, the expression of the 4D hyperchaotic system is:

Among them, x, y, z, and w are state variables. are the derivatives of the state variables x, y, z, w, and a, b, c, d are the control parameters that affect the behavior of the 4D hyperchaotic system; when the control parameters a＝15.5, b＝50, c＝3.6, d＝0.2, the 4D hyperchaotic system is in a hyperchaotic state;

The initial value is brought into the 4D hyperchaotic system and iterated 1000+M×N times. The first 1000 iteration results are discarded to obtain four chaotic sequences X, Y, Z and W.

Preferably, the random Zigzag scrambling method is implemented by: obtaining L pairs of coordinate values according to the sequences U and V, determining the range of the small matrix in the plaintext image P by the coordinate values, and performing Zigzag scrambling on L groups of small pixel matrices in sequence to obtain the matrix P ₁ ;

The Zigzag scrambling starts from the upper left corner of the small pixel matrix, moves vertically first, then moves diagonally, and repeats this process until all pixels are traversed, the elements of the matrix are extracted in the traversal order and arranged into a one-dimensional sequence, and then the one-dimensional sequence is converted back into a pixel matrix of the same size as the small matrix.

Preferably, the implementation method of the random Zigzag scrambling method is: randomly select 30 groups of small matrices in the pixel matrix of the plaintext image P, and perform Zigzag scrambling on each small matrix respectively. The steps of the random Zigzag scrambling method are: determine the coordinate value: if U(i1)≠V(i1) and U(i1+1)≠V(i1+1), {min(U(i1+1),V(i1+1)),min(U(i1),V(i1))} is used as the coordinate of the upper left vertex of the small matrix, {max(U (i1+1),V(i1+1)),max(U(i1),V(i1))} as the lower right vertex coordinate of the small matrix, select the small matrix of the corresponding area of the plaintext image according to the upper left vertex coordinates and the lower right vertex coordinates and perform Zigzag scrambling; if U(i1)＝V(i1) or U(i1+1)＝V(i1+1), the sequence with 1 row or 1 column is selected, and the row or column is arranged in reverse order; repeat the above steps until the 30 selected small matrices are all scrambled; obtain the scrambled matrix _P1 ; where i1＝1,2,3……60.

Preferably, the implementation method of the quaternary position chaos method is:

1): Divide the matrix _P1 into several 2×2 small pixel blocks;

2): Convert each pixel in the small pixel block into quaternary, convert each pixel value into a four-digit quaternary number, and then arrange the four digits of each quaternary number in sequence, and convert each 2×2 small pixel block into a 4×4 pixel matrix;

3): shuffle the position of the 4×4 pixel matrix: each 4×4 pixel matrix is re-divided into four 2×2 sub-matrices, each 2×2 sub-matrix is shuffled to obtain a new small matrix, and each new small matrix obtains a new quaternary number;

4): Convert the new quaternary number back to decimal to obtain a new 2×2 small pixel block;

5): Merge the new 2×2 small pixel blocks to obtain an M×N pixel matrix P ₂ .

The method of sequential arrangement in step 2) is to arrange the four bits of the four-bit binary number from high to low in the order of upper left, upper right, lower left, and lower right;

The position disorder in step 3) is as follows: the elements of the upper left, upper right, lower left and lower right of the four sub-matrices are respectively formed into four new sub-matrices, and the four new sub-matrices are respectively placed at the upper left, upper right, lower left and lower right positions of the new small matrix to obtain a new small matrix;

The method of mapping the chaotic sequence X value to the range of 1-8 to obtain the sequence X′, converting the chaotic sequence Y value to 0 or 1 to obtain the sequence Y′, and converting the chaotic sequences Z and W to the sequences Z1 and W1 with values of 0-255 respectively is as follows:

The method of selecting multiple groups of elements in chaotic sequences Z and W and converting them into elements with values less than M and less than N to obtain sequences U and V is:

Among them, M and N are the number of rows and columns of the plaintext image, X(i1), Y(i1), Z(i1), W(i1), X′(i1), Y′(i1), Z1(i1), W1(i1), U(i1), V(i1) are the i1-th element of the chaotic sequences X, Y, Z and W, and the sequences X′, Y′, Z1, W1, U, V, respectively, and i1=1,…,M×N.

Preferably, the roulette rotation encoding algorithm is a fourth-order roulette encoding algorithm based on multi-order roulette rotation encoding, firstly, the pixel matrix _P2 is converted into a one-dimensional sequence, and the numbers in the one-dimensional sequence are divided into groups of four, and each group of numbers is taken out and the four numbers in the group are converted into eight-bit binary numbers respectively; the eight-bit binary numbers of the four numbers are arranged clockwise on the roulette points in order from high to low, and the four numbers are arranged in order from the outer ring to the inner ring of the roulette; then the sequence X′ and Y′ are used to control the rotation angle and direction of each order of the roulette, and each roulette can be rotated clockwise or counterclockwise 45×i3°, i3＝1,2,3,4,5,6,7,8, and after rotation, the bits are extracted again by column to obtain four new eight-bit binary numbers, and the extracted four binary numbers are converted into decimal; continue to extract the next group of numbers and perform the same operation until all numbers have completed the rotation operation to obtain a new pixel matrix.

Preferably, the method of diffusion of the roulette rotation encoding algorithm is:

11) Convert the pixel matrix P ₂ into a one-dimensional sequence by row expansion, divide the pixel values into groups of four, and process one group of numbers each time;

12): Take out a set of numbers and convert them into binary. Convert each number into an eight-bit binary number and arrange each bit of the binary number in order on each ring of the roulette wheel.

13): Determine the values of sequences X′ and Y′. If the element of sequence Y′ is equal to 1, rotate clockwise. If the element of sequence Y′ is equal to 0, rotate counterclockwise. The rotation angle is equal to X′×45°.

14): Select the rotated number by column to get a new binary number and convert it into decimal;

15): Repeat steps 12)-14) until all numbers are diffused; convert the diffused one-dimensional sequence into a matrix P ₃ of size M×N.

Preferably, the chaotic sequence X is arranged in ascending order to obtain an index sequence, and the index sequence is converted into a matrix of size M×N as the index matrix I. The method of bidirectional diffusion of the matrix P ₃ is:

Forward Diffusion:

Backward diffusion:

Wherein, n＝(i-1)×N+j, i∈[1,M], j∈[1,N], I _nx , I _ny are the coordinate values of the row and column corresponding to the sequence number n in the index matrix I; is an XOR operation, C ₁ (i,j), Z′(i,j), W′(i,j), and C(i,j) are the element values of the i-th row and j-th column of the intermediate matrix C ₁ , the chaotic matrix Z′, W′, and the ciphertext image C, respectively.

The beneficial effects of the present invention are as follows: a state variable is added to the existing three-dimensional chaotic system, and a new four-dimensional chaotic system is constructed; after analysis and testing, the results show that the trajectory distribution of the new four-dimensional chaotic system is uniform and has hyperchaotic characteristics, and the parameter range is wide; the pixel value of the plaintext image is used to calculate the initial value of the four-dimensional chaotic system, so that the encryption method is highly sensitive to the key and the plaintext image; in combination with the four-dimensional chaotic system, a random selection Zigzag scrambling technology and a roulette rotation diffusion technology are proposed, and the proposed random selection Zigzag scrambling is used to scramble the pixel positions, and the size of the small blocks selected in each round is uncertain, which increases the randomness of the scrambling; secondly, the pixels of the scrambled image are scrambled at the bit level, and the double reset scrambling of the pixel level and the bit level is realized, breaking the high correlation between the image pixels; in the diffusion stage, the image is first encrypted using the roulette rotation encoding algorithm, and then a non-sequential diffusion is performed, which further enhances the encryption effect. The simulation experimental results show that the key space size of the present invention is about 2 ²⁴² , and the information entropy of the ciphertext image is 7.9976 bits. In addition, the pixel number change rate and the unified average change intensity reach 99.6094% and 33.4545% respectively, and the correlation between pixels is close to zero, so it can effectively resist cropping attacks and noise attacks. These results show that the present invention has good encryption effect and high security.

Compared with the existing chaotic mapping, the trajectory of the four-dimensional chaotic system proposed in the present invention is evenly distributed, has hyperchaotic characteristics and a wide range of parameters, and is suitable for image encryption. The randomly selected Zigzag scrambling algorithm proposed in the present invention can randomly select small blocks, and each selected pixel point is controlled by a chaotic sequence, which can enhance the randomness of the scrambling.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

FIG. 1 is a flow chart of the present invention.

Figure 2 is a phase diagram of the 4D hyperchaotic system of the present invention, wherein (a) is x-y space, (b) is y-w space, (c) is z-w space, (d) is x-y-z space, (e) is x-y-w space, and (f) is y-z-w space.

FIG. 3 is a fixed parameter Lyapunov index diagram of the present invention.

Figure 4 is the Lyapunov index diagram and corresponding bifurcation diagram of the hyperchaotic system under different control parameters of the present invention, wherein (a) is the Lyapunov index diagram of the hyperchaotic system when the control parameter a is the control variable, (b) is the bifurcation diagram corresponding to (a), (c) is the Lyapunov index diagram of the hyperchaotic system when the control parameter c is the control variable, and (d) is the bifurcation diagram corresponding to (c).

Figure 5 is a schematic diagram of the initial value sensitivity test of the 4D hyperchaotic system, where (a) is the test result when _x0 is slightly changed, (b) is the test result when _y0 is slightly changed, (c) is the test result when _z0 is slightly changed, and (d) is the test result when _w0 is slightly changed.

FIG6 is a schematic diagram of standard Zigzag scrambling.

FIG. 7 is an example diagram of randomly selected Zigzag scrambling of the present invention.

FIG8 is a schematic diagram of the sub-matrix scrambling order of the present invention.

FIG. 9 is a schematic diagram of the quaternary position scrambling method of the present invention.

FIG. 10 is a schematic diagram of the roulette rotation encoding of the present invention.

FIG. 11 is a schematic diagram of the rotary diffusion of the roulette wheel of the present invention.

FIG. 12 is a schematic diagram of bidirectional non-sequential diffusion of the present invention.

Figure 13 is the simulation result of the present invention, wherein (a) is the Bridge plaintext image, (b) is the Bridge ciphertext image, (c) is the Bridge decrypted image, (d) is the Lena plaintext image, (e) is the Lena ciphertext image, (f) is the Lena decrypted image, (g) is the Baboon plaintext image, (h) is the Baboon ciphertext image, (i) is the Baboon decrypted image, (j) is the Peppers plaintext image, (k) is the Peppers ciphertext image, (l) is the Peppers decrypted image, (m) is the Girl plaintext image, (n) is the Girl ciphertext image, and (o) is the Girl decrypted image.

Figure 14 is the test result of the key sensitivity of the present invention, where (a) is the Lena plaintext image, (b) is the image encrypted using the correct key, (c) is the image decrypted using x ₀ +10 ^-12 , (d) is the image decrypted using y ₀ +10 ^-12 , (e) is the image decrypted using z ₀ +10 ^-12 , (f) is the image decrypted using w ₀ +10 ^-12 , (g) is the difference image between (c) and (d), (h) is the difference image between (c) and (e), (i) is the difference image between (c) and (f), (j) is the difference image between (d) and (e), (k) is the difference image between (d) and (f), and (l) is the difference image between (e) and (f).

Figure 15 is the plaintext image histogram and ciphertext image histogram of the present invention, wherein (a) is the Bridge plaintext image histogram, (b) is the Bridge ciphertext image histogram, (c) is the Lena plaintext image histogram, (d) is the Lena ciphertext image histogram, (e) is the Baboon plaintext image histogram, (f) is the Baboon ciphertext image histogram, (g) is the Peppers plaintext image histogram, (h) is the Peppers ciphertext image histogram, (i) is the Girl plaintext image histogram, and (j) is the Girl ciphertext image histogram.

Figure 16 is the correlation analysis result of the Lena image of the present invention in various directions, where (a) is the statistics of adjacent pixels in the horizontal direction of the ciphertext image, (b) is the statistics of adjacent pixels in the vertical direction of the ciphertext image, (c) is the statistics of adjacent pixels in the diagonal direction of the ciphertext image, (d) is the statistics of adjacent pixels in the horizontal direction of the plaintext image, (e) is the statistics of adjacent pixels in the vertical direction of the plaintext image, and (f) is the statistics of adjacent pixels in the diagonal direction of the plaintext image.

Figure 17 shows the ciphertext images and decrypted images of noise attacks of different intensities according to the present invention, where (a) is a ciphertext image with an intensity of 0.05, (b) is a ciphertext image with an intensity of 0.1, (c) is a ciphertext image with an intensity of 0.15, (d) is a ciphertext image with an intensity of 0.2, (e) is a decrypted image with an intensity of 0.05, (f) is a decrypted image with an intensity of 0.1, (g) is a decrypted image with an intensity of 0.15, and (h) is a decrypted image with an intensity of 0.2.

Figure 18 shows the ciphertext images and decrypted images of different cropping attacks of the present invention, where (a) is a 1/64 cropped ciphertext image, (b) is a 1/16 cropped ciphertext image, (c) is a 1/4 cropped ciphertext image, (d) is a 1/2 cropped ciphertext image, (e) is a 1/64 cropped decrypted image, (f) is a 1/16 cropped decrypted image, (g) is a 1/4 cropped decrypted image, and (h) is a 1/2 cropped decrypted image.

DETAILED DESCRIPTION

The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

As shown in Figure 1, an image encryption method based on block-selected Zigzag scrambling and roulette rotation coding proposes a new four-dimensional chaotic system based on the Lorentz chaotic system, and combines the four-dimensional chaotic system to perform image encryption. The present invention first uses the plaintext image and the SHA-384 algorithm to obtain the initial value of the 4D hyperchaotic system, substitutes the initial value into the 4D hyperchaotic system for iteration, and obtains four chaotic sequences; then, the plaintext image is randomly selected for Zigzag scrambling according to the chaotic sequence, and then the scrambled pixel matrix is divided into blocks for bit cross scrambling; then, the scrambled image matrix is roulette-rotated encoded and bidirectionally non-sequentially diffused to obtain the final ciphertext image. Assuming that the size of the encrypted plaintext image P is M×N, the detailed encryption steps of the present invention are as follows:

Step 1: Initial value generation: Use the SHA-384 algorithm to calculate the hash value H of the plaintext image P of size M×N, and calculate the initial values x ₀ , y ₀ , z ₀ , w ₀ of the 4D hyperchaotic system based on the hash value H.

The present invention uses the SHA-384 algorithm to generate a key, inputs a plaintext image into the SHA-384 algorithm, and outputs a 384-bit binary hash value H. The hash value H is equally divided into 48 groups of binary sequences, each group of 8 bits, that is, H _k = h ₁ , h ₂ , h ₃ , ... h ₄₈ , and these 48 groups of hash values are substituted into formula (1) to calculate the initial values x ₀ , y ₀ , z ₀ , w ₀ of the 4D hyperchaotic system, that is:

Among them, [K ₁ ~K ₈ ] are intermediate variables, 1≤i≤8, is the XOR operation. mod is the modulo function.

Step 2: Iterate the 4D hyperchaotic system 1000+M×N times and discard the first 1000 iteration results to obtain four chaotic sequences X, Y, Z and w. Perform data processing on the four chaotic sequences X, Y, Z and w to obtain sequences X′, Y′, Z′, W′ and sequences U and V.

The Lorenz chaotic system is a nonlinear dynamic model proposed by Edward Lorenz in 1963. It was originally used to describe the movement of atmospheric fluids and is also widely used in other natural sciences and engineering fields. The expression of this chaotic system is:

The present invention expands on the Lorentz chaotic system, introduces a new state variable w, and adds a linear term to the second equation, thereby forming a 4D hyperchaotic system, which is reversible and discrete, and has the ability to generate four more complex chaotic sequences at the same time, which not only expands the key space, but also enhances the resistance of the encryption algorithm to various attacks, making up for the shortcomings of the low-dimensional chaotic system with a small key space and a limited range of behavior. The expression of the 4D hyperchaotic system is:

Among them, x, y, z, and w are state variables. is the derivative of the state variables x, y, z, w, and a, b, c, d are the control parameters that affect the behavior of the 4D hyperchaotic system. When the control parameters a = 15.5, b = 50, c = 3.6, and d = 0.2, the Lyapunov exponents of the 4D hyperchaotic system are λ ₁ = 1.3208, λ ₂ = 0.2140, λ ₃ = -9.0862, and λ ₄ = -21.5302, which contain two positive Lyapunov exponents. The sum of the four Lyapunov exponents is a negative value. At this time, the 4D chaotic system is in a hyperchaotic state.

Based on the above parameters, after discretization using the fourth-order Runge-Kutta method, Figure 2 plots the phase diagrams of the hyperchaotic system in two-dimensional and three-dimensional space. From the phase diagrams of each dimension, it can be seen that the 4D hyperchaotic system has multiple attractors and is a complex hyperchaotic system.

The Lyapunov index (LE) is a key characteristic index of chaotic systems. It was proposed by Russian mathematician Alexander Lyapunov in the late 19th century. For a nonlinear dynamic system, if the value of LE is greater than 0, the system has chaotic characteristics. If there are multiple LEs greater than 0, the system belongs to a hyperchaotic system, and its chaotic characteristics are more complex and more random than ordinary chaotic systems. Figure 3 shows the Lyapunov index diagram of a 4D hyperchaotic system with control parameters a=15.5, b=50, c=3.6, d=0.2, and initial values x0=0.1, y0=0.3, z0=0.1, and w0=0.1. At this time, the 4D hyperchaotic system has two positive Lyapunov indexes.

In order to better observe how the Lyapunov exponent in the chaotic system evolves with the change of control parameters, the present invention uses control parameters a and c as control variables to record the change of Lyapunov exponent when other parameters remain unchanged. Figure 4 (a) shows the Lyapunov exponent diagram of the 4D hyperchaotic system in the range of parameter value a∈[4,16] when the control parameters b＝50, c＝3.6, d＝0.2, initial values x0＝0.1, y0＝0.3, z0＝0.1, w0＝0.1 remain unchanged, and control parameter a is used as the control variable. Figure 4 (c) shows the Lyapunov exponent diagram of the 4D hyperchaotic system in the range of parameter value c∈[0,7] when the control parameters a＝15.5, b＝50, d＝0.2, initial values x0＝0.1, y0＝0.3, z0＝0.1, w0＝0.1 remain unchanged, and control parameter c is used as the control variable. The bifurcation diagram gives a visual representation of the chaotic value relative to the parameter. Figure 4 (b) and (d) are the bifurcation diagrams when the control parameters a and c are changed within a certain range while other parameters remain unchanged, which can better understand the behavior of the chaotic system.

Chaotic systems are highly sensitive to initial values, and even a slight change may make the generated sequence completely different. In Figure 5 (a), (b), (c), and (d), the red lines are the four time series x, y, z, and _w generated by iterating the 4D hyperchaotic system using the initial values {x ₀ , y ₀ , z ₀ , w ₀ }, and the blue lines are the time series x1, y1, z1, and w1 generated by using the initial values {x ₀ +10^-14, y _{0 , z 0 ,} w ₀ }, {x ₀ , y ₀ +10^-14, z 0 , w ₀ }, {x ₀ , y ₀ , z 0 +10^-14, w ₀ }, and {x ₀ , y ₀ , z ₀ , w ₀ +10^-14}. It can be seen that although only one component in the initial value changes slightly, after multiple iterations, these small differences expand rapidly, resulting in completely different generated trajectories, which indicates that the 4D hyperchaotic system proposed in the present invention is extremely sensitive to initial conditions.

The total energy of a chaotic system decreases with time, and the phase space also shrinks during the system's motion. This is called the dissipative nature of a chaotic system and is defined by equations (3) and (4):

Where t is time, V is dissipation volume, and the control parameters a = 15.5, b = 50, c = 3.6, d = 0.2 are substituted into formula (5) to obtain When z>-100.5, At this point, the chaotic system is dissipative. As time goes by, the phase space of the 4D hyperchaotic system eventually converges to a point, forming an attraction domain.

In a chaotic system, the equilibrium point refers to the state in which the system does not change after a period of time. The equilibrium point analysis is to find the solution of the system in a stable state by determining the dynamic equation of the system. According to the calculation, the 4D hyperchaotic system has 1 equilibrium point, E1 = (0, 0, 0, -28).

The Jacobian matrix of the 4D hyperchaotic system is shown in formula (7):

From formula (7), we can calculate that the eigenvalue of the equilibrium point is -3.2, so E1 is the stable equilibrium point of the chaotic system.

The NIST test is a statistical tool consisting of multiple tests, which is used to evaluate the randomness of binary sequences generated by pseudo-random number generators. In order to evaluate the randomness of the chaotic sequence generated by the four-dimensional hyperchaotic system proposed in the present invention, a random sequence of sufficient length is generated and converted into a binary bit stream for testing. The test results are shown in Table 1. According to the test results, the generated binary stream passed all tests, indicating that the chaotic sequence generated by the 4D hyperchaotic system proposed in the present invention has ideal randomness.

Table 1 NIST test table

Substitute the initial values x ₀ , y ₀ , z ₀ , w ₀ into the 4D hyperchaotic system and iterate 1000+M×N times. Discard the first 1000 iterations to eliminate transient effects, and obtain four chaotic sequences X, Y, Z and W. Use formulas (8) and (9) to process the four chaotic sequences and obtain sequences X′, Y′, Z1, W1, U, and V for encryption and decryption, respectively.

Where M and N are the number of rows and columns of the plaintext image, X(i1), Y(i1), Z(i1), W(i1), X′(i1), Y′(i1), Z1(i1), W1(i1), U(i1), V(i1) are the i1th element of the chaotic sequences X, Y, Z and W, and the sequences X′, Y′, Z1, W1, U, and V, respectively, and i1 = 1, …, M×N. Formula (8) converts the value of the sequence X′ into 1-8, the value of the sequence Y′ into 0 and 1, and the values of the sequences Z1 and W1 into 0-255. Formula (9) converts the value of the sequence U into 1-M and the value of the sequence V into 1-N, and obtains a sequence of length 60. The sequences U and V are used to generate the vertex coordinates of the Zigzag scrambled small blocks, the sequences X′ and Y′ are used to control the rotation angle and direction of the wheel, respectively, and the sequences Z1 and W1 are used for the non-sequential diffusion process.

Step 3: Use the pixel values of the sequences U and V to select coordinate values, and scramble the plaintext image P according to the random Zigzag scrambling method to obtain the matrix P ₁ .

The random Zigzag scrambling method is: 30 pairs of coordinate values are obtained according to the sequence U and V, the range of the small matrix in the plaintext image P is determined by the coordinate values, and the 30 groups of small matrices are Zigzag-scrambled in turn to obtain the matrix P ₁ .

Zigzag scrambling is a graphic transformation technology that is widely used in the field of image encryption. The principle of Zigzag scrambling is to traverse the pixel matrix according to specific rules and rearrange them. Specifically, the traversal path starts from the upper left corner of the pixel matrix, moves vertically first, then moves diagonally, and repeats this process until all pixels are traversed, the pixel values are extracted in the traversal order and arranged into a one-dimensional sequence, and finally the one-dimensional sequence is converted back into a pixel matrix of the same size as the original image. Figure 6 is a diagram of the effect of standard Zigzag scrambling.

Traditional Zigzag scrambling is very vulnerable to known plaintext attacks and has low security. Therefore, the present invention proposes a randomly selected Zigzag scrambling method, that is, randomly selecting several small blocks in the plaintext image pixel matrix, and performing Zigzag scrambling on each small pixel matrix block. After multiple experiments, it is found that for a 256×256 image, randomly selecting 30 groups of small matrices can make the pixels involved in the scrambling account for 80% of the total pixels. Therefore, the present invention will select 30 groups of small pixel matrices. The specific scrambling steps are as follows:

Step 1: Use formula (9) to process the chaotic sequences Z and W to obtain two sequences U(i1), V(i1) (i1 = 1, 2, 3...60) to control the size of the small pixel matrix.

Step 2: Determine the coordinate value:

If U(i1)≠V(i1) and U(i1+1)≠V(i1+1), {min(U(i1+1),V(i1+1)),min(U(i1),V(i1))} are used as the coordinates of the upper left vertex of the small matrix, and {max(U(i1+1),V(i1+1)),max(U(i1),V(i1))} are used as the coordinates of the lower right vertex of the small matrix. According to the coordinates of the upper left vertex and the lower right vertex, the small matrix of the corresponding area of the plaintext image is selected and Zigzag scrambled.

If U(i1)=V(i1) or U(i1+1)=V(i1+1), the selected sequence is a sequence with 1 row or 1 column. In this case, the scrambling method is to reverse the row or column.

Step 3: Repeat step 2 until all 30 selected pixel matrices have completed the scrambling operation.

Step 4: After the scrambling is completed, the scrambled matrix P ₁ is obtained.

In order to better explain the random selection Zigzag scrambling method proposed in the present invention, an 8×8 matrix is taken as an example, and the complete scrambling process is shown in Figure 7. First, the coordinate control sequence U={3,1,2,5,7,6,8,4}, V={5,7,6,3,2,4,5,6} is obtained, and then four sets of matrix vertex coordinates are obtained from the sequences U and V: (1,3), (7,5); (3,2), (5,6); (4,2), (6,7); (4,5), (6,8). According to the first set of coordinates, a 7×3 small matrix A can be selected, A1 is the effect diagram after the small matrix A is scrambled, and the small matrix A is replaced with the small matrix A1, and the first set of scrambling is completed; according to the second set of coordinates, a 3×5 small matrix B is selected, and a Zigzag scrambling is performed on it to obtain a small matrix B1, and the small matrix B1 replaces the matrix B, and the second set of scrambling is completed; according to the third set of coordinates, a 3×6 small matrix C is selected, and a Zigzag scrambling is performed on it to obtain a small matrix C1, and the small matrix C is replaced with the small matrix C1, and the third set of scrambling is completed; according to the fourth set of coordinates, a 3×4 small matrix D is selected, and a Zigzag scrambling is performed on it to obtain a small matrix D1, and the small matrix D is replaced with the small matrix D1, and the scrambling is completed. It can be seen from Figure 7 that the positions of most pixels in the matrix have changed, indicating that the randomly selected Zigzag scrambling method proposed in the present invention can achieve a good scrambling effect.

Step 4: Divide the matrix _P1 into several 2×2 small pixel blocks and convert the pixel values in each block into quaternary, and perform scrambling according to the quaternary position scrambling method to obtain the scrambled pixel matrix _P2 .

The present invention proposes a new bit-level scrambling method, namely, a quaternary position scrambling method. The scrambling method is to first divide the pixel matrix into blocks, then convert it into quaternary, and then perform bit swapping to finally obtain a new pixel value. The specific scrambling process is as follows:

Step 1: Divide the image into several 2×2 small pixel blocks. If the image size is M×N, then (M×N)/4 small matrix pixel blocks can be obtained.

Step 2: Convert each pixel in the small pixel block into quaternary. Each pixel value can be converted into a four-bit quaternary number. Then arrange the four bits of each quaternary number in the order shown in Figure 8. Then, each 2×2 small pixel block can be converted into a 4×4 pixel matrix. In Figure 8, the four bits of the four-bit binary number are arranged from high to low in the order of upper left, upper right, lower left, and lower right.

Step 3: Disorder the pixel matrix: Each 4×4 pixel matrix can be re-divided into four 2×2 sub-matrices A, B, C, and D. The corresponding values of positions Ⅰ, Ⅱ, Ⅲ, and Ⅳ in each sub-matrix are placed in sub-matrices E, F, G, and H to obtain new quaternary numbers. That is, the elements of the upper left, upper right, lower left, and lower right of the four sub-matrices are respectively formed into four new small matrices, and the four small matrices are placed in the upper left, upper right, lower left, and lower right positions of the new matrix.

Step 4: Convert the scrambled quaternary number back to decimal.

Step 5: Merge the 2×2 small pixel blocks to regain the M×N pixel matrix.

In order to better explain the quaternary position scrambling process, a 4×4 pixel matrix is taken as an example, and its scrambling process is shown in Figure 9. It can be seen that the image is first divided into four blocks of size 2×2, and then the quaternary conversion is performed. Taking the first block as an example, the initial values are 64, 123, 98, and 101, which are converted to quaternary values of 1000, 1223, 1202, and 1211 respectively. After scrambling, the new quaternary numbers are 1111, 0322, 0201, and 0321 respectively. Finally, the scrambled quaternary numbers are converted to decimal numbers to obtain 85, 58, 33, and 57. The other block scrambling methods are similar. In the end, it can be seen that most of the pixel values have changed, indicating that this method can achieve a good scrambling effect.

Step 5: Expand the pixel matrix P ₂ into a one-dimensional sequence by row, and diffuse it using the roulette wheel coding algorithm. Convert the diffused one-dimensional sequence back into a matrix of size M×N and record it as matrix P ₃ .

Wu proposed a second-order bit compass diffusion technology. Inspired by it, the present invention proposes a fourth-order roulette encoding algorithm based on multi-order roulette rotation encoding. First, the pixel matrix is converted into a one-dimensional sequence, and the numbers in the sequence are divided into groups of four. During encryption, a group of numbers is taken out each time, and the four numbers in the group are recorded as α, β, _γ , and δ respectively; and the four numbers are converted into eight-bit binary numbers to obtain {α ₁ ,α ₂ ,α ₃ ,α ₄ ,α ₅ ,α ₆ ,α ₇ ,α ₈ }, {β ₁ ,β ₂ ,β ₃ ,β ₄ ,β 5 ,β ₆ ,β ₇ ,β ₈ }, {γ ₁ ,γ ₂ ,γ ₃ ,γ ₄ ,γ 5 ,γ ₆ ,γ ₇ ,γ ₈ } and {δ ₁ ,δ ₂ ,δ ₃ ,δ ₄ ,δ ₅ ,δ ₆ ,δ ₇ ,δ ₈ } _respectively . Arrange them on the roulette wheel points in order from high to low, where α is placed at the red roulette wheel point, β is placed at the green roulette wheel point, γ is placed at the orange roulette wheel point, and δ is placed at the yellow roulette wheel point. Then use the chaotic sequence to control the rotation angle and direction of each order of the roulette wheel. Each roulette wheel can be rotated 45×i° clockwise or counterclockwise, i＝1,2,3,4,5,6,7,8. After rotation, re-extract the bits by column to obtain four new eight-bit binary numbers. Finally, convert the extracted four binary numbers into decimal, and continue to extract the next group of numbers to perform the same operation until all numbers have completed the rotation operation to obtain a new pixel matrix. Taking the example of rotating the ring where α is located 45° clockwise, the ring where β is located 135° clockwise, the ring where γ is located 45° counterclockwise, and the ring where δ remains unchanged, four new binary numbers can be obtained after rotation, namely {α ₈ ,β ₆ ,γ ₂ ,δ ₁ ,δ ₅ ,γ ₆ ,β ₂ ,α ₄ }, {α ₁ ,β ₇ ,γ ₃ ,δ ₂ ,δ ₆ ,γ ₇ ,β ₃ ,α ₅ }, {α ₂ ,β ₈ ,γ ₄ ,δ ₃ ,δ ₇ ,γ ₈ ,β ₄ ,α ₆ }, {α ₃ ,β ₁ ,γ ₅ ,δ ₄ ,δ ₈ ,γ ₁ ,β ₅ ,α ₇ }. Figure 10 shows a schematic diagram of the roulette rotation encoding process.

The roulette wheel rotation encoding algorithm is a diffusion algorithm proposed by the present invention. The specific encryption process is as follows:

Step 1: Two sequences X′ and Y′ are obtained from formula (8). The value of sequence X′ can be 1-8, and the value of sequence Y′ can be 0 or 1, which are used to control the degree and direction of the wheel rotation encoding respectively.

Step 2: Expand the pixel matrix _P2 row by row and convert it into a one-dimensional sequence. Since the roulette wheel used in the present invention has four rings in total, the pixel values are divided into groups of four, and one group of numbers is processed each time.

Step 3: Take out a set of numbers and convert them into binary. Each number can be converted into an eight-bit binary number. Arrange each bit of the binary number in order on each ring of the roulette wheel.

Step 4: Determine the values of sequences X′ and Y′. If the element of sequence Y′ is equal to 1, rotate clockwise. If the element of sequence Y′ is equal to 0, rotate counterclockwise. The rotation angle is equal to X′×45°.

Step 5: Select the rotated numbers by column to get the new binary number and convert it to decimal.

Step 6: Repeat steps 3 and 4 until all numbers are encrypted.

Step 7: Convert the encrypted one-dimensional sequence back into an M×N matrix P ₃ .

In order to better demonstrate the roulette wheel rotation encoding algorithm, a 2×2 matrix is taken as an example below. FIG11 shows the complete encryption process. First, we get X′＝{2,3,1,6}, Y′＝{0,1,1,0} to control the rotation direction and angle respectively. Then, convert each number in the small matrix into binary and arrange them in order on each disk of the roulette wheel. The binary number corresponding to 187 is 10111011, the binary number corresponding to 98 is 01100010, the binary number corresponding to 241 is 11110001, and the binary number corresponding to 108 is 01101100. Y′(1)＝0, X′(1)＝2, so the rotation direction and angle corresponding to the outermost disk are 90° clockwise. Similarly, rotate the remaining three disks by the corresponding angles, and extract the new binary numbers by column, which are 11101110, 00010011, 11100011, and 10110001. Finally, convert them into decimal to get four new numbers, which are 238, 19, 227, and 177.

Step 6: Sort the chaotic sequence X in ascending order and convert the index sequence into an index matrix. Then use the sequences Z′ and W′ to diffuse the matrix P ₃ bidirectionally and non-sequentially according to the index order of the index matrix to achieve a global diffusion effect and obtain the ciphertext image C.

The encryption algorithm should have diffusion characteristics. Most image encryption algorithms change the pixel value by operating the current pixel with the previous pixel value. However, processing image pixels in a fixed order will lead to reduced encryption performance and provide attackers with a lot of useful information for cryptanalysis. In order to overcome this shortcoming, the encryption scheme of the present invention adopts non-sequential diffusion and uses a random access mechanism to process pixels. The processing order is determined by the chaotic sequence and is not fixed. Therefore, a pixel may be affected by any pixel in the pixel matrix. Assuming that the image size is M×N, firstly, the two Z1 and W1 are obtained by formula (8):

A chaotic sequence is obtained and converted into chaotic matrices Z and W. Z is used for forward diffusion and W is used for reverse diffusion. The chaotic sequence X is arranged in ascending order to obtain the corresponding index sequence, and the index sequence is converted into a matrix of size M×N as the index matrix I. Then, bidirectional diffusion is performed on the pixel matrix obtained above according to formulas (10) and (11):

Forward Diffusion:

Backward diffusion:

Wherein, n＝(i-1)×N+j, i∈[1,M], j∈[1,N], I _mx , I _ny are the coordinate values of the row and column corresponding to the sequence number n in the index matrix I. is an XOR operation, C ₁ (i,j), Z′(i,j), W′(i,j), and C(i,j) are the element values of the i-th row and j-th column of the intermediate matrix C ₁ , the chaotic matrix Z′, W′, and the ciphertext image C, respectively.

To better explain the non-sequential diffusion algorithm, a 2×2 matrix is taken as an example. Figure 12 shows the complete diffusion process. Among them, Z′＝{145,69,220,189}, W′＝{25,84,241,103}, X＝{0.1321,0.6245,0.0125,0.3252}. First, the sequence Z′, W′, x is converted into a 2×2 matrix, and then bidirectional diffusion is performed using formulas (10) and (11) in the order corresponding to the index matrix.

The encryption algorithm proposed in the present invention is symmetric encryption, and the decryption algorithm is the inverse process of the encryption algorithm.

A secure image encryption method should be able to successfully resist all kinds of attacks (such as statistical attacks, known plaintext attacks or known ciphertext attacks, etc.), and the security analysis of the proposed image encryption method is carried out. Different sample images are used for detailed analysis, and the specific results are as follows.

In order to verify the feasibility of the technical solution of the present invention, simulation was performed on the Matlab R2017a experimental platform. As shown in Figure 13, it can be seen that the encrypted image is a noise-like image, while the decrypted image is completely consistent with the encrypted image, indicating that the present invention can effectively hide plaintext information and restore the original image without distortion, and has a good encryption effect.

The size of the key space refers to the set of all possible keys used in the image encryption method. The size of the key space determines the ability of the encryption algorithm to resist brute force cracking. It is generally believed that the larger the key space, the higher the security of the method. When the key space is greater than 2 ¹⁰⁰ ≈10 ³⁰ , the encryption method can resist brute force attacks. The present invention uses the SHA-384 algorithm to generate keys, and its key space is 2 ^192. Assuming that the calculation accuracy of double-precision numbers is 10 ¹⁵ , the total key space of the scheme of the present invention is approximately 2 ¹⁹² ×10 ¹⁵ =2 ²⁴² , which is much larger than 2 ^100. Therefore, the encryption method of the present invention is secure.

A secure image encryption algorithm should be highly sensitive to the input key. If the key is entered incorrectly, the correct decrypted image cannot be obtained. When performing a key sensitivity test, only a small change is made to the original key and then the ciphertext image is decrypted. The greater the difference between the plaintext image and the decrypted image, the more sensitive the key is.

The test of the present invention takes the Lena image as an example. After encryption with the initial key, the initial parameters x ₀ , y ₀ , z ₀ , w ₀ are slightly changed, and the modified initial parameters are used for decryption. The result is shown in FIG14 . From FIG14 (c)-(f), it can be seen that even if a slight change is made to the key, the decrypted image is completely different. At the same time, FIG14 (g)-(l) also shows the difference between any two decrypted images, indicating that the encryption algorithm proposed by the present invention has a strong sensitivity.

The histogram is one of the most statistical features of an image. It reflects the distribution of grayscale pixel intensity levels in an image and may be used by attackers to extract image information. If the histogram of an encrypted image is uniform, it means that it does not provide any information about the image features, making statistical attacks difficult. Therefore, an effective image encryption algorithm will generate a ciphertext image with a histogram close to a uniform distribution, which is more robust to statistical attacks. Figure 15 shows the histograms of plaintext images and ciphertext images of different images. As can be seen, the encrypted image histogram is almost uniform, so the present invention can resist statistical attacks well.

The chi-square (χ2) test can further quantitatively analyze the difference between the plaintext image and the corresponding ciphertext image, which can be calculated by formula (12):

In the formula, ^χ2 represents the variance, which directly reflects the degree of deviation between the ciphertext image histogram and the theoretical histogram, _fi is the frequency of the grayscale pixel value in the ciphertext image histogram, and g is the theoretical frequency of the grayscale pixel value in the histogram, expressed as g = (M × N) / 256. When the significance level is selected as 0.05, When the variance of the test ciphertext image is less than this variance, the histogram is approximately uniformly distributed. Table 2 shows the variance values of the plaintext image and the ciphertext image. By comparison, the variance value of the ciphertext image is much smaller than the variance value of the plaintext image, which means that the distribution of the pixel values of the ciphertext image is uniform.

Table 2 Variance results of plaintext images and ciphertext images

The correlation coefficient is another well-known statistical feature of an image, which can reflect the connection between image pixels. For ordinary images, there is a strong correlation between adjacent pixel values in the horizontal, vertical, and diagonal directions. A good image encryption method should reduce this correlation as much as possible. The correlation coefficient between adjacent pixels can be calculated as

Among them, cov(x,y)=E([xE(x)][yE(y)]), R _x,y is the correlation coefficient, E(x) represents the expectation of x, D(x) represents the variance of x, cov(x,y) represents the covariance, x and y are a pair of pixel values, and N is the number of pixels in the image. The total number of pixels. If X and Y show a high correlation, the correlation coefficient is close to 1; otherwise, it is close to 0. Therefore, the smaller the correlation coefficient of two adjacent pixel sequences, the weaker their correlation. .

5000 pairs of pixels are randomly selected from the plaintext image and the ciphertext image, and the correlations between adjacent pixels in the horizontal, vertical and diagonal directions are calculated respectively. As can be seen from Figure 16, the encrypted image completely eliminates the correlation between adjacent pixels and can effectively resist statistical attacks. In addition, Table 3 gives the correlation coefficients of adjacent pixels of different encrypted images. It can be seen that the correlation coefficient of the encrypted image is close to 0, so the present invention effectively eliminates the pixel correlation. At the same time, the correlation comparison under different encryption algorithms is shown in Table 4. The results show that compared with other encryption algorithms, the encryption method proposed in the present invention is more destructive to the correlation of plaintext images. Among them, reference [1] comes from reference [Zhen, Ping, et al. "Chaos-based image encryption scheme combining DNA coding and entropy." Multimedia Tools and Applications 75(2016): 6303-6319.], reference [2] comes from reference [Zhang, Yong, and Ying jun Tang. "A plaintext-related image encryption algorithmbased on chaos." Multimedia Tools and Applications 77.6(2018): 6647-6669.], reference [3] comes from reference [Xu, Lu, et al. "A novel bit-level image encryption algorithm based on chaotic maps." Optics and Lasers in Engineering 78(2016): 17-25.], and reference [4] comes from reference [Wang, Tao, and Ming-hui Wang. "Hyperchaotic image encryption algorithmbased on bit-level permutation and DNA encoding." Optics&Laser Technology 132(2020): 106355.].

Table 3 Correlation coefficients of plaintext images and ciphertext images in various directions

Table 4 Comparison of ciphertext image correlation with other schemes

Information entropy is one of the concepts in information theory that measures the amount of information. In image processing, information entropy can be used to measure the complexity or uncertainty of an image, reflecting the distribution of pixel values in the image. The higher the information entropy, the more evenly and randomly distributed the pixels in the image are, and the higher the security is. For grayscale images, there are 256 gray levels, N = 8. Therefore, the entropy of an ideal encrypted image should be equal to 8. The closer the information entropy value is to 8, the better the encryption performance. The calculation formula for global information entropy is:

Where L represents the grayscale of the image, _mi is the i-th pixel value of the image, and P( _mi ) is the probability of occurrence of each corresponding grayscale value. Table 5 shows the entropy of the plaintext and ciphertext images. It can be seen from Table 5 that these values are indeed close to the ideal value of 8. This proves that the proposed encryption method is secure and can resist entropy attacks well. Among them, reference [5] comes from reference [He, Chen chen, et al. "An Algorithm Based on Hodgkin-Huxley Model and Latin Square for Image Encryption." IEEE Access 11(2023):34163-34174.], reference [6] comes from reference [Jun, Wang Ji, and Tan Soo Fun. "A new image encryption algorithm based on single S-box and dynamic encryption step." IEEE Access 9(2021):120596-120612.], and reference [7] comes from reference [Wang, Xing yuan, and Mao chang Zhao. "An image encryption algorithm based on hyperchaotic system and DNA coding." Optics & Laser Technology 143(2021):107316.].

For ciphertext images with uniform pixel distribution, the calculated information entropy is accurate, but some ciphertext images may have uneven pixel distribution in some areas, in which case the calculated information entropy is inaccurate. Local information entropy overcomes this shortcoming of global information entropy and further improves the accuracy of this evaluation standard. The calculation method of local information entropy is:

Wherein, _Si is the ciphertext information entropy, k is the number of selected groups, and _TB is the number of pixels in each group. When k is selected as 3, _TB is 1936, and the significance level α is 0.05, the ideal range of the local information entropy is [7.901515698, 7.903422936]. If the test result of the local information entropy is within this range, it means that the ciphertext image passes the test. In the local information entropy test, five different images are encrypted respectively using the encryption method proposed in the present invention, and then the local information entropy of their ciphertext images is obtained. The test results are shown in Table 6. It can be clearly seen that all ciphertext images pass the local information entropy test, indicating that the ciphertext images have a high degree of randomness.

Table 5 Comparison of global information entropy of plaintext image and ciphertext image with other algorithms

Table 6 Local information entropy results of ciphertext images

Noise resistance is an important indicator for testing the performance of encryption schemes. During information transmission, images may be affected by noise. Noise attacks can damage ciphertext images and reduce clarity. Common noise attacks include Gaussian noise and salt and pepper noise. The present invention adds salt and pepper noise of different intensities for testing. Figure 17 shows the decrypted image with interference from salt and pepper noise of intensities of 0.05, 0.1, 0.15, and 0.2. It can be seen that under high noise intensity, the restored image is still visible, so the encryption method proposed in the present invention has good robustness.

When digital images are transmitted on the Internet, they may be partially cropped or lost, which makes it difficult for the recipient to restore these damaged images. If the decryption algorithm is not strong in resisting cropping attacks, decryption will fail due to missing information during the decryption process. The method to test the encryption algorithm's ability to resist cropping attacks is to delete part of the pixel values in the ciphertext image and then decrypt it. If the plaintext image can be restored to a large extent, it means that the encryption method of the present invention has a strong ability to resist cropping attacks. Figure 18 shows the corresponding decrypted images after the Lena ciphertext image is cropped at a ratio of 1/64, 1/16, 1/4, and 1/2. It can be seen that the decrypted image is still recognizable, so the present invention has good resistance to cropping attacks.

The basic idea of differential attack is to select two similar plaintext images with only one bit changed, generate two different cipher images with the same algorithm and key, and then judge the difference between them. For an encryption algorithm, if only a small change is made to the plaintext image, but the encrypted image changes greatly, then the new encryption method can resist differential attack.

In order to detect the impact of plaintext changes on ciphertext changes, two parameters are measured. They are the Normalized Pixel Change Rate (NPCR) and the Uniform Average Change Intensity (UACI). NPCR detects the number of pixel changes that occur in the ciphertext when a single pixel in the plaintext changes. When the NPCR value is close to 100%, the cipher is more sensitive to changes in the plaintext. The larger the UACI value, the more sensitive the ciphertext is to changes in the plaintext, and the greater the resistance to differential attacks. The calculation formulas for NPCR and UACI are:

Wherein, M×N is the scale of the ciphertext image, C ₁ and C ₂ are two ciphertexts to be compared, and D(i,j) is used to determine whether the ciphertexts C ₁ (i,j) and C ₂ (i,j) are equal. If they are equal, it takes 0, otherwise it takes 1. The ideal values of NPCR and UACI are 99.6049% and 33.4635%, respectively. When the key remains unchanged, the encryption method of the present invention is used to encrypt two plaintext images. Tables 7 and 8 show the calculated NPCR and UACI values and the comparison results with other algorithms. It can be seen from the comparison results that the NPCR and UACI values calculated by the present invention are closer to the theoretical values than most other algorithms. Among them, reference [8] comes from reference [Cun, Qi qi, et al. "A new chaotic image encryption algorithm based on dynamic DNA coding and RNA computing." The Visual Computer (2023): 1-20.], reference [9] comes from reference [Abbasi, Ali Asghar, Mahdi Mazinani, and Rahil Hosseini. "Chaotic evolutionary-based image encryption using RNA codons and amino acid truthtable." Optics & Laser Technology 132 (2020): 106465.], reference [10] comes from reference [Vidhya, R., and M. Brindha. "A novel approach for Chaotic image Encryption based on block level permutation and bit-wise substitution." Multimedia Tools and Applications 81.3 (2022): 3735-3772.], and reference [11] comes from reference [Zhou, Minjun, and Chunhua Wang. "A novel image encryption scheme based on conservative hyperchaotic system and closed-loop diffusion between blocks." Signal Processing 171(2020):107484.], reference [12] comes from reference [Cun, Qi qi, et al. "Selective image encryption method based on dynamic DNA coding and new chaotic map." Optik243(2021):167286.].

Table 7 NPCR and UACI values of different image ciphertexts

Table 8 NPCR and UACI values of Lena ciphertext images and comparison with other algorithms

MSE and PSNR are two commonly used indicators for evaluating image quality. MSE is used to measure the difference between two images. It calculates the square of the average difference between the corresponding pixels of the two images. PSNR is the peak signal-to-noise ratio, which is used to measure the quality of the encrypted image. The larger the PSNR value, the smaller the image distortion and the clearer the image. When testing plaintext and ciphertext images, the smaller the PSNR value, the better the encryption effect. The calculation formulas are as follows (17) and (18).

Wherein, M is the width of the image, N is the length of the image, P is the plaintext image, C is the ciphertext image, b is the length of the pixel binary string, and MSE is the mean square error between the plaintext image P and the ciphertext image C. The MSE and PSNR values after encryption of different plaintext images and the comparison with other algorithms are shown in Table 9. It can be seen from the data in the table that the PSNR of the encrypted image of the present invention is very small, and through comparative analysis with other algorithms, it is also better than other algorithms, indicating that the encryption method of the present invention has high security. Among them, reference [13] comes from reference [Wang, Xing yuan, and Mao changzhao. "An image encryption algorithm based on hyperchaotic system and DNA coding." Optics & Laser Technology 143 (2021): 107316.], reference [14] comes from reference [Wang, Xing yuan, and Xuan Chen. "An image encryption algorithm based on dynamic row scrambling and Zigzag transformation." Chaos, Solitons & Fractals 147 (2021): 110962.], reference [15] comes from reference [Wang, Xing yuan, Wen hua Xue, and Ju bai An. "Image encryption algorithm based on LDCML and DNA coding sequence." Multimedia Tools and Applications 80 (2021): 591-614.], and reference [16] comes from reference [Yousif, Sura F., Ali J. Abboud, and Raad S. Alhumaima. "A new image encryption based on bit replacing, chaos and DNA coding techniques."Multimedia Tools and Applications 81.19(2022):27453-27493.].

Table 9 PSNR and MES of ciphertext images and comparison with other algorithms

The present invention proposes a new 4D hyperchaotic system, which has a wide range of chaotic and hyperchaotic characteristics. At the same time, the present invention designs an image encryption method based on randomly selected Zigzag scrambling and roulette rotation diffusion. First, multiple groups of coordinates are generated according to the chaotic sequence, and the plaintext image is selected according to the coordinates. The selected small block matrix is sequentially Zigzag scrambled, and then the scrambled image pixels are bit-level scrambled. The double reset scrambling at the pixel level and the bit level breaks the high correlation between the image pixels; then the roulette rotation diffusion algorithm is used to diffuse the scrambled image to further enhance the encryption effect, and finally the image pixel matrix is bidirectionally non-sequentially diffused to achieve the effect of global diffusion, and the ciphertext image is obtained. Simulation experiments verify that the encryption method of the present invention can encrypt different grayscale images into unrecognizable ciphertext images, and can be completely restored using the correct key, and can effectively resist selected plaintext attacks, brute force attacks, statistical analysis attacks, differential attacks, cropping attacks and noise attacks. Therefore, the encryption method proposed by the present invention has good security performance.

The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (6)

1. An image encryption method based on block selection Zigzag scrambling and rotary coding of a wheel disc is characterized by comprising the following steps:

step one: calculating a hash value H of a plaintext image P with the size of MxN by using an SHA-384 algorithm, and calculating an initial value of the 4D hyper-chaotic system according to the hash value H; carrying out iteration by taking the initial value into a 4D hyper-chaotic system to obtain four chaotic sequences X, Y, Z and W;

The expression of the 4D hyper-chaotic system is as follows:

wherein x, y, z, w is a state variable, and wherein, As a derivative of the state variable x, y, z, w, a, b, c, D is a control parameter that affects the behavior of the 4D hyper-chaotic system;

Step two: selecting a plurality of groups of elements in the chaotic sequence Z and the chaotic sequence W, respectively converting the elements into elements with values smaller than M and smaller than N to obtain a sequence U, V, selecting coordinate values by using the values of the sequence U, V, and scrambling a plaintext image P according to a random Zigzag scrambling method to obtain a matrix P ₁;

The implementation method of the random Zigzag scrambling method comprises the following steps: obtaining L pairs of coordinate values according to the sequence U, V, determining the range of a small matrix in the plaintext image P according to the coordinate values, and sequentially carrying out Zigzag scrambling on L groups of small pixel matrixes to obtain a matrix P ₁;

The Zigzag scrambling is that the Zigzag scrambling starts from the upper left corner of a small pixel matrix, moves along the vertical direction, then moves along the diagonal direction, and continuously repeats the process until all pixels are traversed, elements of the matrix are extracted and arranged into a one-dimensional sequence according to the traversing sequence, and then the one-dimensional sequence is reconverted into a pixel matrix with the same size as the small matrix;

Step three: dividing the matrix P ₁ into a plurality of small pixel blocks with the size of 2 multiplied by 2, converting the pixel value in each small pixel block into quaternary system, and scrambling according to a quaternary position scrambling method to obtain a scrambled pixel matrix P ₂;

The implementation method of the quaternary position disorder method comprises the following steps:

1): dividing the matrix P ₁ into a plurality of small pixel blocks of 2×2;

2): converting each pixel in the small pixel block into a quaternary system, converting each pixel value into a quaternary system number, then arranging the quaternary system numbers in a quaternary sequence, and converting each 2×2 small pixel block into a 4×4 pixel matrix;

3): the 4 x 4 pixel matrix is subjected to position scrambling: dividing each 4×4 pixel matrix into four 2×2 sub-matrices again, performing position disorder on each 2×2 sub-matrix to obtain a new small matrix, and obtaining a new quaternary number by each new small matrix;

4): converting the new quaternary number into decimal again to obtain a new small pixel block of 2 multiplied by 2;

5): combining the new small pixel blocks of 2×2 to obtain a pixel matrix P ₂ of m×n;

Step four: mapping the value of the chaotic sequence X into the range of 1-8 to obtain a sequence X ', and converting the value of the chaotic sequence Y into 0 or 1 to obtain a sequence Y'; expanding the pixel matrix P ₂ into a one-dimensional sequence according to rows, diffusing the one-dimensional sequence by using a rotary coding algorithm of a wheel disc by using sequences X ', Y', and converting the diffused one-dimensional sequence into a matrix again to obtain an image matrix P ₃;

The wheel disc rotation coding algorithm is a four-order wheel disc coding algorithm based on multi-order wheel disc rotation coding, firstly, a pixel matrix P ₂ is converted into a one-dimensional sequence, numbers in the one-dimensional sequence are divided into groups of four, a group of numbers are taken out each time, and four numbers in the group of numbers are respectively converted into eight-bit binary numbers; arranging eight binary digits of four digits on the wheel count clockwise in the order from the high order to the low order, and arranging the four digits in the order from the outer ring of the wheel to the inner ring of the wheel; then using sequences X 'and Y' to control the rotation angle and direction of each wheel disc, each wheel disc can rotate clockwise or anticlockwise (45 multiplied by i 3) °, i3=1, 2,3,4,5,6,7,8, extracting bits by columns again after rotation, obtaining four new eight-bit binary numbers, and converting the extracted four binary numbers into decimal numbers; continuing to extract the next group of numbers to execute the same operation until all the numbers complete the rotation operation, so as to obtain a new pixel matrix;

Step five: converting the chaotic sequences Z and W into sequences Z1 and W1 with values of 0-255 respectively and converting the sequences Z and W into chaotic matrixes Z ', W'; the chaotic sequence X is arranged in an ascending order to obtain an index sequence, the index sequence is converted into a matrix with the size of MxN to be used as an index matrix I, and a matrix P ₃ is bidirectionally and non-sequentially diffused according to the index sequence of the index matrix I by utilizing the chaotic matrices Z ', W', so that a ciphertext image C is obtained;

The method for performing bidirectional diffusion on the matrix P ₃ comprises the following steps:

forward diffusion:

back diffusion:

wherein, n= (I-1) x n+j, I e [1, M ], j e [1, N ], I _nx、I_ny is the coordinate value of the row and column corresponding to the sequence number N in the index matrix I; For the exclusive-or operation, C ₁ (i, j), Z '(i, j), W' (i, j), and C (i, j) are the element values of the intermediate matrix C ₁, the chaotic matrix Z ', W', and the ith row and jth column of the ciphertext image C, respectively.

2. The image encryption method based on block selection Zigzag scrambling and wheel disc rotation encoding according to claim 1, wherein the method for calculating the initial value of the 4D hyper-chaotic system is as follows: inputting a plaintext image into an SHA-384 algorithm, outputting 384-bit binary hash value H, equally dividing the hash value H into 48 groups of binary sequences, obtaining H _k＝h₁,h₂,h₃,…h₄₈ by 8 bits of each group, performing operation on the 48 groups of binary sequences, and calculating an initial value x ₀、y₀、z₀、w₀ of the 4D hyper-chaotic system as follows:

Wherein K ₁～K₈ is an intermediate variable, i2 is more than or equal to 1 and less than or equal to 8, Mod is an modulo function, which is an exclusive or operation.

3. The image encryption method based on block selection Zigzag scrambling and wheel rotation encoding according to claim 1 or 2, wherein when the control parameters a=15.5, b=50, c=3.6, d=0.2, the 4D hyperchaotic system is in a hyperchaotic state;

And (3) carrying out iteration 1000+M multiplied by N times in the 4D hyperchaotic system, and discarding the previous 1000 iteration results to obtain four chaotic sequences X, Y, Z and W.

4. The image encryption method based on block selection Zigzag scrambling and wheel disc rotation encoding according to claim 1, wherein the implementation method of the random Zigzag scrambling method is as follows: 30 groups of small matrixes are randomly selected from the pixel matrix of the plaintext image P, and each small matrix is subjected to Zigzag scrambling respectively, wherein the random Zigzag scrambling method comprises the following steps: judging coordinate values: if U (i 1) noteqV (i 1) and U (i 1+1) noteqV (i 1+1), { min (U (i 1+1), V (i 1+1)), min (U (i 1), V (i 1)) } as the top left vertex coordinates of the small matrix, { max (U (i 1+1), V (i 1+1)), max (U (i 1), V (i 1)) } as the bottom right vertex coordinates of the small matrix, selecting the small matrix of the corresponding area of the plaintext image according to the top left vertex coordinates and the bottom right vertex coordinates and performing Zigzag scrambling; if U (i 1) =v (i 1) or U (i1+1) =v (i1+1), a sequence with 1 row number or 1 column number is selected, and the rows or columns are arranged in reverse order; repeating the steps until the selected 30 groups of small matrixes complete scrambling operation; obtaining a scrambled matrix P ₁; wherein i1=1, 2,3 … …, 38360.

5. The image encryption method based on block selection Zigzag scrambling and rotary coding of a wheel disc according to claim 1, wherein the method of sequential arrangement in the step 2) is that four digits of quaternary numbers are sequentially arranged from high to low according to the upper left, the upper right, the lower left and the lower right;

The position disorder in the step 3) is as follows: the method comprises the steps of respectively forming elements of upper left, upper right, lower left and lower right in four submatrices into four new submatrices, and respectively placing the four new submatrices at the positions of upper left, upper right, lower left and lower right of a new small matrix to obtain a new small matrix;

Mapping the value of the chaotic sequence X to the range of 1-8 to obtain a sequence X ', converting the value of the chaotic sequence Y into 0 or 1 to obtain a sequence Y', and respectively converting the chaotic sequences Z and W into sequences Z1 and W1 with the values of 0-255, wherein the method comprises the following steps:

The method for selecting a plurality of groups of elements in the chaotic sequence Z and the chaotic sequence W and respectively converting the elements into elements with values smaller than M and smaller than N to obtain a sequence U, V comprises the following steps:

Where M and N are the number of rows and columns of the plaintext image, X (i 1), Y (i 1), Z (i 1), W (i 1), X '(i 1), Y' (i 1), Z1 (i 1), W1 (i 1), U (i 1), V (i 1) are the i1 st elements of the chaotic sequences X, Y, Z and W, the sequences X ', Y', Z1, W1, U, V, respectively, i1=1, …, mxn.

6. The image encryption method based on block selection Zigzag scrambling and wheel rotation coding according to claim 1 or 5, wherein the method for diffusing the wheel rotation coding algorithm is as follows:

11 Pixel matrix P ₂ is converted into a one-dimensional sequence by line expansion, pixel values are divided into groups of four, and a group number is processed at a time;

12): taking out a group of numbers to convert into binary numbers, converting each number into an eight-bit binary number, and arranging each bit of the binary numbers on each ring of the wheel disc in sequence;

13): judging the values of the sequences X 'and Y', rotating clockwise if the element of the sequence Y 'is equal to 1, and rotating anticlockwise if the element of the sequence Y' is equal to 0; the rotation angle is equal to X' ×45°;

14): the rotated numbers are selected according to the columns to obtain new binary numbers and are converted into decimal numbers;

15): repeating steps 12) -14) until all numbers have completed diffusion; the one-dimensional sequence obtained by diffusion is converted into a matrix P ₃ of size mxn.