CN110086601A - Based on the associated Josephus traversing of pixel value and hyperchaotic system image encryption method - Google Patents

Abstract

The invention proposes one kind to be based on the associated Josephus traversing of pixel value and hyperchaotic system image encryption method, the steps include: the initial value that original image input hash function union is obtained to piecewise linear maps and Chen hyperchaotic system；Pseudo-random sequence is obtained by piecewise linear mapsU, four pseudo-random sequences are obtained by Chen hyperchaotic system；Pseudo-random sequenceUAs index value, line shuffle is carried out to original image with pixel value associated Joseph's disorder method and obtains ciphertext image C₁；Four pseudo-random sequences are respectively to the ciphertext image C of equal part₁Ciphertext blocks carry out pixel replacement obtain ciphertext image C₂；Pseudo-random sequenceUAs index value, with the associated Joseph's disorder method of pixel value to ciphertext image C₂It carries out column scramble and obtains ciphertext image C₃；Using key pixel to ciphertext image C₃Pixel is done to spread to obtain encrypted ciphertext image.The present invention is strong to the sensibility of key and plaintext, improves the connection between the safety and pixel of ciphertext image.

Description

Joseph traversal and image encryption method for hyperchaotic systems based on pixel value association

technical field

The invention relates to the technical field of digital image encryption, in particular to a Joseph traversal and hyperchaotic system image encryption method based on pixel value association.

Background technique

With the development of the economy, the progress of the society, the improvement of the level of informatization, and the rapid development of technologies such as the Internet and the Internet of Things, human beings are entering the era of big data and artificial intelligence. Image information has been widely used in many fields because of its image, intuition, vividness, and huge amount of information, such as multimedia technology, digital office, mobile payment, etc. The application of image information is changing people's lifestyle and living habits. Image information is filling every corner of human activities. However, with the wide application of image information, its security problems are gradually revealed. Digital image encryption technology is an important means to effectively protect digital image transmission. Some traditional text encryption algorithms such as DES, AES, etc. are not suitable for the encryption of image information: first, the encryption efficiency is low, and second, the encryption effect is general. For example, Silva et al. noticed that when using the DES algorithm to encrypt an image with a large area of the same gray level, some areas with low encryption quality will appear in the ciphertext image. Due to the wide application of image information in the military, political, and economic fields, the leakage of these information will have huge influence and destructive power. Therefore, it is imminent to design a more reliable image encryption algorithm, and image encryption is becoming a hot research field.

Image encryption includes pixel scrambling and pixel replacement. Common pixel scrambling methods include: graphics-based scanning pattern scrambling method, pseudo-random sequence-based pixel rearrangement method, etc. These scrambling methods produce an encryption effect by changing the position of pixels in the image. Common pixel replacement methods include: encryption method based on Hill matrix, cipher stream encryption method based on chaotic system, etc. These displacement methods achieve the effect of encryption by changing the value of pixels. Among the encryption schemes proposed in recent years, chaotic cryptography has attracted much attention. As a complex nonlinear dynamical system, chaotic system has the characteristics of sensitive initial parameters, unpredictable trajectory, and strong state ergodicity, and is usually used as a pseudo-random number generator. Applying the chaos system to the scrambling and permutation process of image encryption can improve the security of the encryption system and make up for the shortcomings of traditional encryption algorithms. However, with the improvement of cryptanalysis technology, chaotic encryption technology has also exposed the low sensitivity to keys and other problems. In order to solve these problems, some researchers proposed that fusion with other methods is an effective way to improve the security of chaotic image encryption technology. The Joseph question evolved from the question raised by the famous ancient Roman historian Josephus. Its use has played a deterrent role in various illegal activities such as malicious tampering and copyright infringement. Use Joseph traversal for data scrambling, which increases data security.

SUMMARY OF THE INVENTION

Aiming at the technical problems that the existing image encryption method is less sensitive to the key and the data security is poor, the present invention proposes a Joseph traversal and hyper-chaotic system image encryption method based on pixel value association, which has a very large key Sensitivity, good pseudo-randomness, and security analysis results show that it can effectively resist various classic attacks.

In order to achieve the above object, the technical solution of the present invention is achieved in that a kind of Joseph traversal based on pixel value association and hyperchaotic system image encryption method, its steps are as follows:

Step 1: Input the original image with a size of Height×Width into the SHA-384 hash function to obtain a 384-bit binary sequence H; divide the 384-bit binary sequence H into 48 parts, and perform operations in groups of eight to obtain a piecewise linear map and the initial value of Chen hyperchaotic system;

Step 2: Bring the initial value of the piecewise linear map into the piecewise linear map for iteration to obtain a pseudo-random sequence U of length Height*Width, and bring the initial value of the Chen hyperchaotic system into the Chen hyperchaotic system for iteration to obtain four A pseudo-random sequence x, y, z, w with a length of Height*Width/4;

Step 3: Use the first Height pseudo-random numbers of the pseudo-random sequence U as index values, and use the Joseph scrambling method associated with the pixel values to perform row scrambling operations on the original image to obtain the ciphertext image C ₁ ;

Step 4: Divide the ciphertext image C ₁ into 4 equal parts to obtain ciphertext blocks, use the pseudo-random sequence x, y, z, w to perform pixel replacement on the pixels in the 4 ciphertext blocks and reassemble the replaced ciphertext blocks , get the ciphertext image C ₂ ;

Step 5: Use the last Width pseudo-random numbers of the pseudo-random sequence U as index values, and use the Joseph scrambling method associated with the pixel value to perform column scrambling operations on the ciphertext image C ₂ to obtain the ciphertext image C ₃ ;

Step 6: Use two key pixels Pixel ₁ and Pixel ₂ to perform a pixel diffusion operation on the pixels in the ciphertext image C ₃ to obtain the encrypted ciphertext image C.

The method for calculating the initial value of the piecewise linear map and the Chen hyperchaotic system in the step 1 is: divide the 384-bit binary sequence H into 48 parts to obtain the 8-bit binary sequence h ₁ , h ₂ , ..., h ₄₈ , use The calculation method of the piecewise linear map and the initial value of the Chen hyperchaotic system is:

Among them, u ₁ and p are the initial values of the piecewise linear map, x ₁ , y ₁ , z ₁ , and w ₁ are the initial values of the four-dimensional Chen hyperchaotic system, Indicates that XOR operation is performed on corresponding position elements in two 8-bit binary sequences, u′ ₁ , x′ ₁ , y′ ₁ , z′ ₁ , and w′ ₁ are the given initial values.

The iterative equation of the piecewise linear mapping is:

Among them, p is the parameter of the piecewise linear mapping, and the value range is (0,0.5); u(i+1)∈[0,1) represents the pseudo-random number generated by the ith iteration of the piecewise linear mapping, i= 1,2,...,Height+Width-1; u(i)∈[0,1) is the pseudo-random number or initial value u ₁ of the previous iteration used in the i-th iteration;

The dynamic equation of the Chen hyperchaotic system is:

in, and respectively represent the reciprocal of the dynamic parameters x', y', z', w', a, b, c, d, k are the parameters of Chen's hyperchaotic system respectively, when a=36, b=3, c=28, When d=16 and -0.7≤k≤0.7, the Chen hyperchaotic system is in a hyperchaotic state.

The implementation method of the pseudo-random sequence U is: the piecewise linear mapping utilizes the initial value u ₁ to iterate Height+Width-1 times, and the initial value u ₁ and the obtained Height+Width-1 pseudo-random numbers form a length of Height The chaotic sequence u of +Width; reconstruct the chaotic sequence u into a pseudorandom sequence U:

Among them, u(:) is the element in the chaotic sequence u, and mod(,) represents the remainder function;

The method of obtaining the pseudo-random sequence x, y, z, w is as follows: bring the initial values x ₁ , y ₁ , z ₁ , w ₁ into the four-dimensional Chen hyperchaotic system, and iterate the Chen hyperchaotic system Height×Width/4 +999 times, the value of the first 999 iterations is discarded, and four pseudo-random sequences x, y, z, and w of length Height×Width/4 are obtained.

The implementation method of the Joseph scrambling method associated with the pixel value in the step 3 is as follows: the element of the pseudo-random sequence U(j) is used as the initial step size InitialValue, and the element of the jth row of the original image is brought into the expansion as the element of the sequence S Joseph function, the obtained sequence S' is the jth row element of the scrambled ciphertext image C ₁ ; where, j=1,2,...,Height;

The implementation method of the Joseph scrambling method associated with the pixel value in the step 5 is: the element of the pseudo-random sequence U(Height+q) is used as the initial step size _InitialValue , and the qth column element of the ciphertext image C2 is used as the sequence S The elements of are brought into the extended Joseph function, and the obtained sequence S' is the jth row element of the scrambled ciphertext image C ₁ ; where, q=1,2,...,Width.

The extended Joseph function is: S'=F(S, InitialValue);

Among them, F(,) is the Joseph function, S is the sequence to be scrambled, S′ is the sequence after scrambling, and InitialValue is the initial step size;

The implementation method of the extended Joseph function is as follows: the step size during the initial traversal is InitialValue, and the step size during the nth cycle is S' _n-1 +1; where 2≤n≤N, N is the sequence S The number of elements; the specific implementation method is:

Step 1: traverse the Joseph ring composed of sequence S with the step size as InitialValue, and obtain S′ ₁ ;

Step 2: traverse the Joseph ring with a step size of S′ ₁ +1 to obtain S′ ₂ ;

Step 3: Cycle Step 2: Traversing the Joseph ring with a step size of S′ _n-1 +1 to obtain S′ _n ;

Step 4: traverse the Joseph ring with a step size of S′ _N-1 +1 to obtain S′ _N ;

Step 5: The sequence composed of S' ₁ , S' ₂ , ..., S' _n , ..., S' _N is the scrambled sequence S'.

The method for performing pixel replacement in step 4 is:

Step S1: Reconstruct the four pseudo-random sequences x, y, z, and w to obtain four pseudo-random matrices X, Y, Z, and W with a value range of [0,255];

Step S2: Divide the ciphertext image C ₁ whose size is Height×Width into ciphertext blocks Block ₁ , Block ₂ , Block ₃ , and Block ₄ whose size is Height×Width/4, and use pseudorandom matrices X, Y, Z , W and ciphertext blocks Block ₁ , Block ₂ , Block ₃ , and Block ₄ perform bitwise XOR operations respectively to obtain four matrices Block′ ₁ , Block′ ₂ , Block′ ₃ , and Block′ ₄ ;

Step S3: Utilize the pseudo-random matrix X, Y, Z, W to calculate the control word matrix ControlMatrix ₁ and the control word matrix ControlMatrix ₂ respectively;

Step S4: using the control word matrix ControlMatrix ₁ , matrix Block' ₁ and matrix Block' ₄ are interleaved to obtain crossed matrix Block" ₁ and matrix Block"₄; using ControlMatrix ₂ as the control word matrix, matrix Block' ₂ Carry out cross operation with matrix Block ' ₃ and obtain matrix Block " ₂ and matrix Block " ₃ after crossing;

Step S5: Recompose the crossed matrices Block″ ₁ , Block″ ₂ , Block″ ₃ and Block″ ₄ into a ciphertext image C ₂ .

The reconstruction method of the pseudo-random matrix is:

Among them, x(:), y(:), z(:), w(:) are elements in the sequence x, y, z, w respectively, is the rounding down symbol, mod( , ) is the remainder function, and the reshape function reshape(a1,b1,c1) means to rearrange the array a1 into an array of size b1×c1 in the order of column priority;

The matrix Block' ₁ =bitxor(Block ₁ ,X), Block' ₂ =bitxor(Block ₂ ,Y), Block' ₃ =bitxor(Block ₃ ,Z), Block' ₄ =bitxor(Block ₄ ,W) , wherein, bitxor(·,·) is a binary bit XOR operation; the control word matrix ControlMatrix ₁ =bitxor(X,Y), and the control word matrix ControlMatrix ₂ =bitxor(Z,W). The method of using the control word matrix to carry out the interleaving operation is as follows: the elements at the corresponding positions in the two matrices to be intersected are respectively converted into eight-bit binary numbers A and binary numbers B, and the elements at the corresponding positions in the control word matrix are identified Convert it into an eight-bit binary number to obtain the control word C. If the value of a certain control bit in the control word C is 1, the binary characters in the binary number A and the binary number B corresponding to the control bit are exchanged; when the control word C When the value of a certain control bit of is 0, the binary characters corresponding to the control bit in binary number A and binary number B do not perform any operation.

The method for performing pixel diffusion in step 6 is:

Convert the ciphertext image C ₃ into a one-dimensional pixel sequence PixelSequence whose length is Height*Width, and use the key pixel Pixel ₁ to diffuse the one-dimensional pixel sequence PixelSequence backward to obtain the one-dimensional pixel sequence PixelSequence':

Use the key pixel Pixel ₂ to diffuse backward to obtain a one-dimensional pixel sequence PixelSequence'forward diffusion to obtain a one-dimensional pixel sequence PixelSequence":

The encrypted ciphertext image C is obtained by reconstructing the one-dimensional pixel sequence "PixelSequence" whose length is Height*Width into a two-dimensional matrix whose size is Height*Width; where bitxor(·,·) is a binary bit XOR operation.

Beneficial effects of the present invention: the Joseph scrambling method associated with pixel values associates the Joseph problem, the chaotic system, and the pixel values, with the help of the sensitivity and pseudo-randomness of the chaotic map to initial conditions, combined with the advantages of Joseph ergodic scrambling, increasing The sensitivity to the plaintext is reduced, and the number of iterations of the chaotic system in the scrambling process is reduced; the image is divided into blocks, combined with a high-dimensional chaotic system, the XOR and cross operations between the blocks are performed to replace the pixels, increasing The randomness of the ciphertext image is improved and the security of the ciphertext image is also improved; all pixels in the ciphertext image are affected by other pixels by using pixel diffusion operation, and the connection between pixels is improved. Simulation results and security analysis show that the invention has strong sensitivity to keys, can effectively resist attack operations such as statistical analysis and exhaustive analysis, and can be used for image encryption.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only These are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to these drawings without creative efforts.

Fig. 1 is a flowchart of the present invention.

Fig. 2 is the phase orbit diagram of the Chen chaotic system of the present invention, wherein, (a) is x-y, (b) is x-z, (c) is x-w, (d) is y-z, (e) is y-w, (f) is z-w .

Fig. 3 is the original image of the present invention and the image scrambling using the Joseph problem associated with the pixel value, wherein, (a) is the original image of Lena, (b) is the Lena image after scrambling, (c) is the original image of Cameraman, (d) is the Cameraman image after scrambling.

Fig. 4 is a schematic diagram of the operation of pixel replacement in the present invention, wherein (a) is a schematic diagram of interleaving operation, and (b) is a flow chart of image segmentation, bitwise XOR operation and interleaving operation.

Fig. 5 is original image and ciphertext image of the present invention, wherein, (a) is the Cameraman original image of 128 * 128, (b) is the Lena original image of 256 * 256, (c) is the Cameraman original image of 256 * 256 , (d) is the original Brain image of 512×512, (e) is the Cameraman ciphertext image of 128×128, (f) is the Lena ciphertext image of 256×256, (g) is the Cameraman ciphertext image of 256×256 Image, (h) is a 512×512 Brain ciphertext image.

Fig. 6 is the 256×256 Lena original image of the present invention, the ciphertext image and the decrypted image when the key changes slightly, wherein, (a) Lena original image, (b) is the correct decrypted image, (c) is the decrypted image after changing u1 by 10 ^-15 , (d) is the decrypted image after changing p by 10 ^-15 , (e) is the decrypted image after changing x ₁ by 10 ^-15 , (f) is changing y ₁ by 10 ^-15 (g) is the decrypted image after z ₁ is changed by 10 ^-14 , (h) is the decrypted image after w ₁ is changed by 10 ^-15 .

Fig. 7 is the original image of the present invention, the histogram of original image and the histogram of ciphertext image, wherein, (a) is Lena original image, (b) is Cameraman original image, (c) is Baboon original image, (d) is the histogram of Lena original image, (e) is the histogram of Cameraman original image, (f) is the histogram of Baboon original image, (g) is the histogram of Lena ciphertext image, (h) is Cameraman ciphertext image The histogram of , (i) is the histogram of the Baboon ciphertext image.

Figure 8 is the value of adjacent pixels in each direction in the Lena original image and the ciphertext image, where (a) is in the horizontal direction in the Lena original image, (b) is in the vertical direction in the Lena original image, and (c) is In the diagonal direction in the original Lena image, (d) is in the horizontal direction in the Lena ciphertext image, (e) is in the vertical direction in the Lena ciphertext image, (f) is in the diagonal direction in the Lena ciphertext image .

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to 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, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the protection scope of the present invention.

As shown in Figure 1, a Joseph traversal and hyperchaotic system image encryption method based on pixel value association is characterized in that its steps are as follows:

Step 1: Input the original image with a size of Height×Width into the SHA-384 hash function to obtain a 384-bit binary sequence H; divide the 384-bit binary sequence H into 48 parts, and perform operations in groups of eight to obtain a piecewise linear map and Chen initial values for hyperchaotic systems.

The present invention uses the SHA-384 function to generate the key required for encryption. The SHA-384 function is a kind of Hash security hash function, which can convert the original image of any length, that is, the plaintext image, into a binary sequence with a length of 384 bits. The SHA-384 function is irreversible, so it is impossible to deduce any plaintext information through the key. There are two chaotic systems used in the present invention, which are one-dimensional piecewise linear mapping and four-dimensional Chen hyperchaotic mapping. There are six initial values of these chaotic systems, respectively u ₁ , p, x ₁ , y ₁ , z ₁ , w ₁ . The method for calculating the initial value of the piecewise linear map and the Chen hyperchaotic system in the step 1 is: divide the 384-bit binary sequence H into 48 parts to obtain the 8-bit binary sequence h ₁ , h ₂ , ..., h ₄₈ , use The calculation method of the piecewise linear map and the initial value of the Chen hyperchaotic system is:

Among them, u ₁ and p are the initial values of the piecewise linear map, x ₁ , y ₁ , z ₁ , and w ₁ are the initial values of the four-dimensional Chen hyperchaotic system, Indicates that XOR operation is performed on corresponding position elements in two 8-bit binary sequences, u′ ₁ , x′ ₁ , y′ ₁ , z′ ₁ , and w′ ₁ are the given initial values.

Step 2: Bring the initial value of the piecewise linear map into the piecewise linear map for iteration to obtain a pseudo-random sequence U of length Height*Width, and bring the initial value of the Chen hyperchaotic system into the Chen hyperchaotic system for iteration to obtain four A pseudorandom sequence x, y, z, w of length Height*Width/4.

The chaotic system is of great application value in cryptography. It is usually used as a pseudo-random number generator because of its sensitive initial value and strong track ergodicity. The pseudo-random index value sequence used in the scrambling process in the present invention is generated by piecewise linear mapping. The iterative equation of the piecewise linear mapping is:

Among them, p is the parameter of the piecewise linear mapping, and the value range is (0,0.5); u(i+1)∈[0,1) represents the pseudo-random number generated by the ith iteration of the piecewise linear mapping, i= 1,2,...,Height+Width-1; u(i)∈[0,1) is the pseudo-random number or initial value u ₁ of the previous iteration used in the i-th iteration. The piecewise linear mapping uses the initial value u ₁ to iterate Height+Width-1 times, and the initial value u ₁ and the obtained Height+Width-1 pseudo-random numbers form a chaotic sequence u with a length of Height+Width. In the chaotic sequence u The value range of the element is [0,1).

In the Joseph scrambling method associated with pixel values, the index value must be an integer greater than 0, so the pseudo-random chaotic sequence u generated by the piecewise linear mapping is reconstructed, and the chaotic sequence u is reconstructed into a pseudo-random sequence U:

Among them, u(:) is the element in the chaotic sequence u, the position of the element in the pseudo-random sequence U corresponds to the position of the element in u(:), and mod(,) represents the remainder function.

The first Height values of the sequence U are used as the initial values of the row scrambling in the Joseph scrambling method associated with the pixel values, and the last Width values are used as the initial values of the Joseph scrambling method column scrambling associated with the pixel values. For example, when u(1)=0.23046875 and p=0.37890625 of the piecewise linear map, the improved Joseph traversal method is used to scramble the original images of Lena and Cameraman, and the result is shown in Figure 3. It can be seen from Figure 3 that the scrambled Lena image and Cameraman image are obviously different from the original Lena image and Cameraman image.

The chaotic system used in the pixel replacement process in the present invention is a Chen hyperchaotic system. Compared with the low-dimensional chaotic system, the high-dimensional chaotic system has more initial values and system parameters, more complex orbits, and a safer system. The dynamic equation of the Chen hyperchaotic system is:

in, and respectively represent the reciprocal of the dynamic parameters x', y', z', w', a, b, c, d, k are the parameters of Chen's hyperchaotic system respectively, when a=36, b=3, c=28, When d=16 and -0.7≤k≤0.7, the Chen hyperchaotic system is in a hyperchaotic state. When x(1)=0.312, y(1)=0.3828125, z(1)=0.4765625, w(1)=0.45703125, k=0.2, use the Runge-Kutta method to iterate the Chen hyperchaotic system, Chen hyperchaos The phase orbit diagram of the system is shown in Figure 2. It can be seen from Figure 2 that Chen's chaotic system has complex orbits and strong ergodicity, and can be used for information encryption.

The method of obtaining the pseudo-random sequence x, y, z, w is as follows: bring the initial values x ₁ , y ₁ , z ₁ , w ₁ into the four-dimensional Chen hyperchaotic system, and iterate the Chen hyperchaotic system Height×Width/4 +999 times, the value of the first 999 iterations is discarded to eliminate the transient effect, and four pseudo-random sequences x, y, z, w of length Height×Width/4 are obtained.

Step 3: Use the first Height pseudo-random numbers of the pseudo-random sequence U as index values, and use the Joseph scrambling method associated with the pixel values to perform row scrambling operations on the original image to obtain the ciphertext image C ₁ .

It is said that the famous Jewish historian Josephus had the following story: After the Romans occupied Chotapat, 39 Jews hid in a cave with Josephus and his friends, and the 39 Jews decided that they would rather die than be caught by the enemy , so a suicide method was decided, 41 people lined up in a circle, the first person started to count, every time the third person was counted, the person had to commit suicide, and then the next person counted again, until everyone committed suicide until death. However Josephus and his friends did not want to comply. The question is, given the total number and step size, where to stand in the first place to avoid being executed? Josephus asked his friend to pretend to obey first, and he placed his friend and himself in the 16th and 31st positions, escaping the death game. For the Joseph problem, it can be quantified into a Joseph function S′=F(S,k). In this function, the S sequence represents the permutation set of elements {s ₁ ,s ₂ ,s ₃ ,…,s _N }, and k represents the step size . Taking the sequence S={1, 2, 3, 4, 5, 6, 7}, k=3 as an example, the solution of the Joseph function is {3, 6, 2, 7, 5, 1, 4}, the solution The elements 3, 6, 2, 7, 5, 1, and 4 in the representative sequence are selected in turn. The Joseph problem is a classic computational problem, also known as the Joseph ring. Later generations expanded the Joseph problem, adding the starting point and the direction of the cycle, which greatly enriched the connotation of the Joseph problem.

In order to make the Joseph traversal associated with the plaintext, the present invention reforms the Joseph problem, and provides a Joseph traversal method associated with the pixel value, which expands the Joseph function and applies it to the pixel scrambling process to obtain an association with the pixel value. The Joseph scrambling method. The extended Joseph function is: S′=F(S,InitialValue); where, F(·,·) is the Joseph function, S is the sequence to be scrambled and S={s ₁ ,s ₂ ,s ₃ ,…, s _N }, S′ is the scrambled sequence, and InitialValue is the initial step size.

The implementation method of the extended Joseph function is as follows: the step size during the initial traversal is InitialValue, and the step size during the nth cycle is S' _n-1 +1; where 2≤n≤N, N is the sequence S The number of elements; the specific implementation method is:

Step 1: traverse the Joseph ring composed of sequence S with the step size as InitialValue, and obtain S′ ₁ ;

Step 2: traverse the Joseph ring with a step size of S′ ₁ +1 to obtain S′ ₂ ;

Step 3: Cycle Step 2: Traversing the Joseph ring with a step size of S′ _n-1 +1 to obtain S′ _n ;

Step 4: traverse the Joseph ring with a step size of S′ _N-1 +1 to obtain S′ _N ;

Step 5: The sequence composed of S' ₁ , S' ₂ , ..., S' _n , ..., S' _N is the scrambled sequence S'.

Assuming a given sequence S={1, 0, 255, 128, 100, 80, 60}, InitialValue=3, the traversal steps of the extended Joseph function F(S, InitialValue) are:

Step 1: traverse the Joseph ring with a step size of 3, and obtain S′ ₁ =255;

Step 2: traverse the Joseph ring with a step size of S′ ₁ +1=256, and obtain S′ ₂ =60;

Step 3: traverse the Joseph ring with a step size of S′ ₂ +1=61, and obtain S′ ₃ =1;

Step 4: traverse the Joseph ring with a step size of S′ ₃ +1=2, and obtain S′ ₄ =128;

Step 5: traverse the Joseph ring with a step size of S′ ₄ +1=129, and obtain S′ ₅ =0;

Step 6: traverse the Joseph ring with a step size of S′ ₅ +1=1, and obtain S′ ₆ =100;

Step 7: Traversing the Joseph ring with a step size of S′ ₆ +1=101 to obtain S′ ₇ =80.

Therefore, when S={1, 0, 255, 128, 100, 80, 60}, InitialValue=3, the solution of the function F(S, InitialValue) is the sequence S'={255, 60, 1, 128, 0 , 100, 80}. The modified Joseph function is used in the pixel scrambling process, where InitialValue is the key of the scrambling process. The scrambling process is reversible, that is, the original sequence S can be restored by knowing the encrypted pixel sequence S' and the initial step size InitialValue. The restoration process of the scrambling method can be described as S=F ⁻¹ (S′, InitialValue).

The implementation method of the Joseph scrambling method associated with the pixel value in the step 3 is as follows: the element of the pseudo-random sequence U(j) is used as the initial step size InitialValue, and the element of the jth row of the original image is brought into the expansion as the element of the sequence S The obtained sequence S′ is the jth row element of the scrambled ciphertext image C ₁ ; where j=1,2,...,Height. The specific method is:

S1: Take the elements of the pseudo-random sequence U(j) as the initial step size, and the elements of the jth row of the original image as the sequence S into the extended Joseph function, and traverse the Joseph ring to obtain S′ _j1 , where j=1,2, ......, Height;

S2: Use S′ _j1 +1 as the step size to traverse the Joseph ring to get S′ _j2 ;

S3: Loop step S2: use S' _j(q) +1 as the step size to traverse the Joseph ring to obtain S'_j(q+1); where, q=1,2,...,Width-1;

S4: Use S′ _j(Width-1) +1 as the step size to traverse the Joseph ring to obtain S′ _j(width) ;

S5: S' _j1 , S' _j2 ..., S' _j(q+1) , S' _j(width) The sequence sequence S' _j is the sequence after the jth line of the original image is scrambled, that is, S' _j1 , S′ _j2 . . . , S′ _j(q) , S′ _j(width) are the elements of the jth row of the ciphertext image C ₁ .

Methods using Joseph scrambling associated with pixel values include row scrambling operations and column scrambling operations. Because the length of the pseudo-random index value sequence used for scrambling is small, the pseudo-random index value sequence used for scrambling will be generated by the piecewise linear map.

Step 4: Divide the ciphertext image C ₁ into 4 equal parts to obtain ciphertext blocks, use the pseudo-random sequence x, y, z, w to perform pixel replacement on the pixels in the 4 ciphertext blocks and reassemble the replaced ciphertext blocks , to get the ciphertext image C ₂ .

Pixel scrambling scrambles the position of the image and destroys the correlation between adjacent pixels, but the pixels have not undergone any calculations and cannot effectively resist cryptographic attacks. Further pixel replacement can completely confuse the plaintext image and ciphertext relationship between images. The present invention uses XOR and cross operations to perform replacement operations on pixels. Crossover operation is a commonly used operator in genetic algorithm, and the schematic diagram of crossover operation is shown in Figure 4(a). That is, assuming that there are two pixels A and B to be encrypted, in order to make the interleaving operation reversible, a control word C is added here, and A, B, and C are all converted into eight-bit binary numbers: when a certain value in the control word C When the bit value is 1, the binary characters at the positions corresponding to the control bits in pixels A and B are exchanged; when a certain bit value in the control word C is 0, the binary characters at the positions corresponding to the control bits in pixels A and B are not changed. operate.

As shown in Figure 4(b), the method for pixel replacement in step 4 is:

Step S1: Reconstruct the four pseudo-random sequences x, y, z, and w to obtain four pseudo-random matrices X, Y, Z, and W whose values range from [0,255].

The reconstruction method of the pseudo-random matrix is:

Among them, x(:), y(:), z(:), w(:) are elements in the sequence x, y, z, w respectively, is the rounding down symbol, mod(·,·) is the remainder function, and the reshape function reshape(a1,b1,c1) means to rearrange the array a1 into an array of size b1×c1 in the order of column priority.

Step S2: Divide the ciphertext image C ₁ whose size is Height×Width into ciphertext blocks Block ₁ , Block ₂ , Block ₃ , and Block ₄ whose size is Height×Width/4, and use pseudorandom matrices X, Y, Z , W and the ciphertext blocks Block ₁ , Block ₂ , Block ₃ , and Block ₄ are respectively bitwise XORed to obtain four matrices Block′ ₁ , Block′ ₂ , Block′ ₃ , and Block′ ₄ .

The matrix Block' ₁ = bitxor (Block ₁ , X), Block' ₂ = bitxor (Block ₂ , Y), Block' ₃ = bitxor (Block ₃ , Z), Block' ₁ = bitxor (Block ₄ , W) , where, bitxor(·,·) is a binary bit XOR operation, that is, convert the elements at the corresponding positions of the ciphertext block and the pseudo-random matrix into 8-bit binary numbers, and then perform binary bit XOR calculations on the binary numbers at the corresponding positions, and then The conversion of 8-bit binary numbers into decimal numbers is the elements of the corresponding positions of the matrix Block′ ₁ , Block′ ₂ , Block′ ₃ , and Block′ ₄ .

Step S3: Calculate the control matrix ControlMatrix ₁ and the control matrix ControlMatrix ₂ respectively by using the pseudo-random matrices X, Y, Z, and W.

The control word matrix ControlMatrix ₁ =bitxor(X,Y), and the control word matrix ControlMatrix ₂ =bitxor(Z,W). That is, two pseudo-random matrices are used to perform binary bit XOR operation to obtain a control word matrix.

Step S4: using the control word matrix ControlMatrix ₁ , matrix Block' ₁ and matrix Block' ₄ are interleaved to obtain crossed matrix Block" ₁ and matrix Block"₄; using ControlMatrix ₂ as the control word matrix, matrix Block' ₂ Perform crossover operation with matrix Block′ ₃ to obtain crossover matrix Block″ ₂ and matrix Block″ ₃ .

The method of using the control word matrix to carry out the interleaving operation is as follows: the elements at the corresponding positions in the two matrices to be intersected are respectively converted into eight-bit binary numbers A and binary numbers B, and the elements at the corresponding positions in the control word matrix are identified Convert it into an eight-bit binary number to obtain the control word C. If the value of a certain control bit in the control word C is 1, the binary characters in the binary number A and the binary number B corresponding to the control bit are exchanged; when the control word C When the value of a certain control bit of is 0, the binary characters corresponding to the control bit in binary number A and binary number B do not perform any operation.

Step S5: Recompose the crossed matrices Block″ ₁ , Block″ ₂ , Block″ ₃ and Block″ ₄ into a ciphertext image C ₂ .

Step 5: Use the last Width pseudo-random numbers of the pseudo-random sequence U as index values, and use the Joseph scrambling method associated with pixel values to perform a column scrambling operation on the ciphertext image C ₂ to obtain the ciphertext image C ₃ .

The implementation method of the Joseph scrambling method associated with the pixel value in the step 5 is: the element of the pseudo-random sequence U(Height+q) is used as the initial step size _InitialValue , and the qth column element of the ciphertext image C2 is used as the sequence S The elements of are brought into the extended Joseph function, and the obtained sequence S' is the jth row element of the scrambled ciphertext image C ₁ ; where, q=1,2,...,Width. The specific implementation method is the same as step three.

Step 6: Use two key pixels Pixel ₁ and Pixel ₂ to perform a pixel diffusion operation on the pixels in the ciphertext image C ₃ to obtain the encrypted ciphertext image C.

In order to make the pixels in the ciphertext interact with each other, the pixels are further diffused. First, the image matrix of the ciphertext image C ₃ is converted into a one-dimensional sequence, and then two operators are used to diffuse the pixels backward and forward to ensure that each pixel in the image is affected by other pixels. The method for performing pixel diffusion in step 6 is:

Convert the ciphertext image C ₃ into a one-dimensional pixel sequence PixelSequence whose length is Height*Width, and use the key pixel Pixel ₁ to diffuse the one-dimensional pixel sequence PixelSequence backward to obtain the one-dimensional pixel sequence PixelSequence':

Use the key pixel Pixel ₂ to diffuse backward to obtain a one-dimensional pixel sequence PixelSequence'forward diffusion to obtain a one-dimensional pixel sequence PixelSequence":

Reconstruct the one-dimensional pixel sequence "PixelSequence" whose length is Height*Width into a two-dimensional matrix whose size is Height×Width to obtain the encrypted ciphertext image C. The decryption process of the pixel diffusion operation is the inverse process of the encryption process, and will not be repeated here .

The decryption method of the present invention is the reverse process of the encryption method, so the present invention will not repeat it.

When the encrypted initial keys are u′ ₁ =0, x′ ₁ =0, y′ ₁ =0, z′ ₁ =0, w′ ₁ =0, Pixel ₁ =127, Pixel ₁ =255, use The original image and ciphertext image encrypted by the present invention are shown in Figure 5, and the simulation experiment was completed on the MATLABR2018a platform. Through observation, the encrypted ciphertext image of the present invention has completely lost the characteristics expressed by the original image. Because the encryption algorithm of the present invention is lossless, the decrypted image of the ciphertext image obtained by using the present invention is exactly the same as the original image, and will not be listed here. In the following, the security of the encrypted ciphertext image using the present invention will be quantitatively analyzed to prove the security of the encryption algorithm.

The invention uses the binary sequence generated by the SHA-384 algorithm and two pixels as the initial key, and its key space is sufficient to resist exhaustion attack. The two chaotic systems used in the present invention are very sensitive to the initial value of the system. When the initial value of the chaotic system changes slightly, the decrypted image will change greatly. Figure 6 lists the original image of Lena with a size of 256×256, the ciphertext image, and the decrypted image when the key changes slightly. Through comparison, it can be seen that when the key changes slightly, the ciphertext image cannot be detected. correct decryption. The present invention has extremely strong key sensitivity. Mean square error (MSE) and peak signal-to-noise ratio (PSNR) can be used to measure the difference between the original image and the decrypted image when the key changes slightly. The calculation method of MSE is shown in formula (11), and the calculation method of PSNR is shown in formula As shown in (12):

In formulas (11) and (12), P ₁ (i, j) represents the pixel in the i-th row and j-th column of the original image, and P ₂ (i, j) represents the i-th pixel of the decrypted image when the key changes slightly. The pixels in row and column j, Height and Width are the height and width of the image respectively. In general, when MSE ≥ 30, it means that the difference between the two images is significant. The smaller the value of PSNR, the greater the difference between the two images. Taking the Lena image as an example, when the key changes slightly, the values of MSE and PSNR between the original image and the decrypted image are shown in Table 1. It can be seen from the comparison that the key sensitivity of the present invention is very strong.

Table 1 The values of MSE and PSNR between the original image and the decrypted image when the key changes slightly

A histogram can reflect the statistics of the values of all pixels in an image. In Figure 7, (a)-(c) is the original image, (d)-(f) is the histogram of the original image, and (g)-(i) is the histogram of the ciphertext image. In the histogram of the original image, the distribution of pixel values is relatively concentrated and has certain statistical characteristics, so it has no resistance to exhaustive attacks. However, the distribution of pixels in the ciphertext image is uniform and scattered, the distribution of pixels is broken, and no longer has statistical characteristics. Attackers cannot use statistical characteristics to exhaustively enumerate the original information of the image. Therefore, the present invention can well resist statistical analysis. attack.

The distribution law of the pixel histogram can be measured by the histogram variance. Use hist _i (i=0,1,...,255) to represent the histogram of the image, and the calculation formula of the histogram variance is as described in formula (13):

The histogram variance indicates the uniformity of pixel value distribution in the image histogram, the smaller the variance of the histogram, the more uniform the pixel value distribution. The variance of the images listed in Figure 7 is shown in Table 2. It can be seen from the comparison that the present invention greatly changes the variance of the image and has a good ability to break the histogram distribution of the original image.

Table 2 Variance statistics of the histogram

Information entropy is a concept proposed by Shannon to quantify the amount of information. Shannon uses information entropy to quantify the uncertainty of information. The calculation method of information entropy is as formula (14):

In the formula, s represents the total number of possible events in the source, and p(i) represents the probability of each event i in the source. Information entropy H(s) can be used to measure the randomness of an image. When using formula (14) to calculate the information entropy of an image, hist _i (i=0,1,...,255) represents the histogram of the image. An ideal random image has a probability of 1/256 for each value in a pixel, so ideally the information entropy of a random image is 8. When the information entropy of the image is close to 8, it means that the randomness of the image is very strong. Table 3 lists the information entropy of the original image and the ciphertext image using the encryption method of the present invention. By comparison, it can be seen that the information entropy of the ciphertext image encrypted using the present invention is close to 8, so the randomness of the ciphertext image is very strong.

Table 3 Information entropy of original image and ciphertext image

(a)-(c) in Figure 8 is the statistics of the values of adjacent pixels in the horizontal direction, vertical direction and diagonal direction of the Lena original image, and (d)-(f) is the horizontal direction and vertical direction of the Lena ciphertext image and the statistics of the values of adjacent pixels in the diagonal direction. The values between adjacent pixels in most areas of the original image are very close, and the correlation between the values of adjacent pixels in the image is very strong. The ciphertext image breaks the strong correlation between pixels, which is of great significance for resisting statistical analysis attacks. The calculation method of the correlation coefficient between adjacent pixels is shown in formula (15):

Among them, _xi represents the value of the selected pixel, y _i represents the value of the adjacent pixel to _xi , N represents the total number of selected pixels, and E(x) is the average value of the selected pixels , E(y) is the average value of the pixels adjacent to the selected pixel, D(x) represents the variance of the selected pixel, D(y) is the variance of the pixels adjacent to the selected pixel, cov(x,y) represents the covariance between x and y, and r _xy represents the covariance between x and y.

Randomly select 10,000 pairs of pixels, and make statistics on the correlation coefficients of the original image and the ciphertext image in the horizontal direction, vertical direction and diagonal direction. The statistical results are shown in Table 4. The statistical results in Table 4 show that the randomly selected pixels in the original image have a strong correlation, while the correlation coefficient between pixels in the ciphertext image is close to 0. The encryption method proposed by the present invention can better disrupt the correlation between pixels, so it can better resist statistical analysis attacks.

Table 4 Correlation coefficients of original image and ciphertext image in each direction

Differential attack refers to making small changes in the plaintext, and then comparing and analyzing the changes in the ciphertext. Number of Pixel Change Rate (NPCR) and Normalized Average Change Intensity (UACI) two indicators can reflect the size of the algorithm's ability to resist differential attacks. The calculation method of NPCR is shown in formula (16), and the calculation method of UACI is shown in formula ( 17) as shown:

In formula (12), sign() is a sign function, and its calculation method is shown in formula (18):

The ideal value of NPCR is 100%, and the ideal value of UACI is 33.4635%. After a slight change in the plaintext, the closer the values of NPCR and UACI are to the ideal value, the stronger the ability of the encryption algorithm to resist differential attacks. Table 5 lists the values of NPCR and UACI of their ciphertext images when the plaintext images change by 1 bit. Through comparison, it can be seen that the present invention has good anti-differential attack capability.

Table 5 Values of NPCR and UACI after a 1-bit change in the plaintext image

The present invention improves the Joseph problem, and proposes a Joseph scrambling method associated with pixel values. This scrambling method associates the Joseph problem, the chaotic system and the pixel value, increases the sensitivity to plaintext, and The number of iterations of the chaotic system in the scrambling process is reduced. The present invention uses a high-dimensional chaotic system to perform replacement operations on pixels, which increases the randomness of the encryption system and improves the security of the system at the same time. In the present invention, all pixels in the ciphertext image are affected by other pixels by using pixel diffusion operation, thereby improving the connection between pixels. Simulation results and safety analysis show that the invention is safe and can be used for image encryption.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the scope of the present invention. within the scope of protection.

Claims (10)

1. A Joseph traversal and hyperchaotic system image encryption method based on pixel value association is characterized by comprising the following steps:

the method comprises the following steps: inputting an original image with Height multiplied by Width into an SHA-384 hash function to obtain a 384-bit binary sequence H; equally dividing the 384-bit binary sequence H into 48 parts, and performing operation on eight groups to obtain piecewise linear mapping and an initial value of the Chen hyperchaotic system;

step two: substituting the initial value of the piecewise linear mapping into the piecewise linear mapping for iteration to obtain a pseudorandom sequence U with the length of Height and Width, and substituting the initial value of the Chen hyperchaotic system into the Chen hyperchaotic system for iteration to obtain four pseudorandom sequences x, y, z and w with the length of Height and Width/4;

step three: scrambling the original image by using the first Height pseudo random numbers of the pseudo random sequence U as index values and using a Josephson scrambling method associated with pixel values to obtain a ciphertext image C₁；

Step four: the ciphertext image C₁Equally dividing into 4 parts to obtain ciphertext blocks, respectively performing pixel replacement on pixels in the 4 ciphertext blocks by using pseudorandom sequences x, y, z and w, and recombining the replaced ciphertext blocks to obtain a ciphertext image C₂；

Step five: using the last Width pseudo random numbers of the pseudo random sequence U as index values, the ciphertext image C is scrambled using the Josephson scrambling method associated with the pixel values₂Performing a column scrambling operation to obtain a ciphertext image C₃；

Step six: using two key pixels Pixel₁And Pixel₂For ciphertext image C₃And performing pixel diffusion operation on the pixels in the encrypted ciphertext image C to obtain the encrypted ciphertext image C.

2. The method for image encryption based on Joseph traversal and hyperchaotic system with associated pixel values as claimed in claim 1, wherein the method for calculating the initial values of piecewise linear mapping and Chen hyperchaotic system in the first step is: equally dividing the 384-bit binary sequence H into 48 parts to obtain 8-bit binary sequence H₁，h₂，...，h₄₈The method for calculating the initial value of the Chen hyperchaotic system by using the piecewise linear mapping comprises the following steps:

wherein u is₁And p is the initial value of the piecewise-linear mapping, x₁、y₁、z₁、w₁Is an initial value of a four-dimensional Chen hyperchaotic system,represents that corresponding position elements in two 8-bit binary sequences are subjected to exclusive OR operation u'₁，x′₁，y′₁，z′₁，w′₁Is given an initial value.

3. The method for encrypting the image based on the Josephson traversal and hyperchaotic system with associated pixel values as claimed in claim 1 or 2, wherein the iterative equation of the piecewise linear mapping is:

wherein, p is a parameter of the piecewise linear mapping, and the value range is (0, 0.5); u (f +1) ∈ [0,1) represents a pseudo-random number generated by the ith iteration of piecewise linear mapping, i is 1,2, … …, Height + Width-1; u (f) e [0,1) is the pseudo-random number or initial value u of the previous iteration used for the ith iteration_i；

The power equation of the Chen hyperchaotic system is as follows:

wherein,andthe parameters respectively represent the reciprocal of kinetic parameters c ', y' z ', w', a, b, c, d and k are parameters of the Chen hyperchaotic system, and when a is 36, b is 3, c is 28, d is 16 and-0.7 is less than or equal to k is less than or equal to 0.7, the Chen hyperchaotic system is in a hyperchaotic state.

4. The pixel value association based Josephson traversal and hyper-chaotic system image encryption of claim 3The method is characterized in that the realization method of the pseudo-random sequence U comprises the following steps: piecewise linear mapping using an initial value u₁Iterating the Height + Width-1 times, and adding the initial value u₁And the obtained Height + Width-1 pseudo random numbers form a chaos sequence u with the length of Height + Width; reconstructing the chaotic sequence U into a pseudo-random sequence U:

wherein u (:) is an element in the chaotic sequence u, and mod (,) represents a complementation function;

the method for obtaining the pseudorandom sequences x, y, z and w comprises the following steps: will start value x₁、y₁、z₁、w₁And (3) bringing the four-dimensional Chen hyperchaotic system into, iterating the Chen hyperchaotic system for Height multiplied by Width/4+999 times, and omitting the value of the former 999 iterations to obtain four pseudorandom sequences x, y, z and w with the length of the Height multiplied by Width/4.

5. The method for encrypting the Josephson traversal and hyperchaotic system images based on pixel value association according to claim 1, wherein the Josephson scrambling method associated with pixel values in step three is implemented by: taking the element of a pseudorandom sequence U (,) as an initial step InitialValue, taking the element of the jth line of an original image as the element of a sequence S to be brought into an expanded Joseph function, and obtaining a sequence S' which is a scrambled ciphertext image C₁Row j elements of (1); wherein j is 1,2, … …, Height;

the implementation method of the Josephson scrambling method related to the pixel values in the step five is as follows: taking the element of the pseudorandom sequence U (Height + q) as the initial step InitialValue and the ciphertext image C₂The q-th row of elements of (1) as elements of the sequence S are substituted into an extended Joseph function, and the obtained sequence S' is a scrambled ciphertext image C₁Row j elements of (1); wherein q is 1,2, … …, Width.

6. The pixel value association-based josephson traversal and hyperchaotic system image encryption method of claim 1, wherein the extended josephson function is: s ═ F (S, InitialValue);

wherein, F (,) is a Joseph function, S is a sequence to be scrambled, S' is a scrambled sequence, and InitialValue is an initial step size;

the implementation method of the extended Joseph function is as follows: the step size for the initial pass is InitialValue, and the step size for the nth cycle is S'_{n_1}+ 1; wherein N is more than or equal to 2 and less than or equal to N, and N is the number of elements in the sequence S; the specific implementation method comprises the following steps:

step 1: obtaining S 'by using Joseph ring formed by traversal sequence S with InitialValue step length'₁；

Step 2: is S 'in step length'₁+1 traverse the Joseph ring to obtain S'₂；

And step 3: and (3) circulating the step 2: is S 'in step length'_{n_1}+1 traverse the Joseph ring to obtain S'_n；

And 4, step 4: is S 'in step length'_{N_1}+1 traverse the Joseph ring to obtain S'_N；

And 5: s'₁、S′₂、……、S′_n、……、S′_NThe composed sequence is the scrambled sequence S'.

7. The method for encrypting the Josephson traversal and hyperchaotic system image based on pixel value association according to claim 1, wherein the method for pixel replacement in the fourth step is:

step S1: reconstructing the four pseudo-random sequences x, y, z and w to obtain four pseudo-random matrixes X, Y, Z, W with value intervals of [0,255 ];

step S2: the ciphertext image C with the size of Height multiplied by Width₁Equally dividing the Block into ciphertext blocks with the size of Height multiplied by Width/4_l、Block₂、Block₃、Block₄Respectively connected with the ciphertext blocks by using a pseudo-random matrix X, Y, Z, W_l、Block₂、Block₃、Block₄Performing a bitwise XOR operation to obtain fourMatrix Block'₁，Block′₂，Block′₃，Block′₄；

Step S3: respectively calculating control word matrix ControlMatrix by using pseudo-random matrix X, Y, Z, W₁And control word matrix control matrix₂；

Step S4: using a control word matrix control matrix_lMatrix Block'₁And matrix Block'₄Performing cross operation to obtain a crossed matrix Block ″)₁And the matrix Block ″)₄(ii) a Using ControlMatrix₂As a control word matrix, the matrix Block'₂And a matrix Block; performing cross operation to obtain a crossed matrix Block ″)₂And the matrix Block ″)₃；

Step S5: the matrix Block after crossing ″', is₁、Block″₂、Block″₃And Block ″)₄Reconstituting ciphertext image C₂。

8. The method for encrypting the Josephson traversal and hyperchaotic system image based on pixel value association according to claim 7, wherein the reconstruction method of the pseudo random matrix is as follows:

wherein, x (: y (: z) and w (:) are elements in the sequence x, y, z and w respectively,to round the symbol down, mod (·,) is a complementation function, the recombination function reshape (a1, b1, c1) represents the rearrangement of array a1 into an array of size b1 × c1 in column-first order;

the matrix Block'₁＝bitxor(Block_l，X)，Block′₂＝bitxor(Block₂，Y)，Block′₃＝bitxor(Block₃，Z)，Block′₄＝bitxor(Block₄W), wherein, bitxor (·,) is a binary bit XOR operation; the control word matrix ControlMatrix_lControl word matrix control matrix (X, Y) ═ bitxor₂＝bitxor(Z，W)。

9. The method for encrypting the Josephson traversal and hyperchaotic system images based on pixel value association according to claim 7, wherein the method for performing the crossover operation by using the control word matrix is: converting elements at corresponding positions in two matrixes to be crossed into eight-bit binary numbers A and B respectively, converting elements at corresponding positions in a control word matrix into eight-bit binary numbers respectively to obtain a control word C, and exchanging binary numbers at positions corresponding to control bits in the binary numbers A and B if the value of a certain control bit in the control word C is 1; when the value of a control bit in the control word C is 0, the binary character in the binary number a and the binary number B at the position corresponding to the control bit does not perform any operation.

10. The method for encrypting the Josephson traversal and hyperchaotic system image based on pixel value association according to claim 1, 5 or 7, wherein the method for pixel diffusion in the sixth step is:

the ciphertext image C₃Converting into one-dimensional Pixel sequence Pixel sequence with length of Height Width, and using key Pixel Pixel₁Back-diffusing the one-dimensional pixel sequence PixelSequence to obtain a one-dimensional pixel sequence PixelSequence':

pixel Using Key Pixel₂Forward diffusion of the backward diffusion resulting one-dimensional pixel sequence PixelSequence' results in a one-dimensional pixel sequence PixelSequence ":

reconstructing a one-dimensional pixel sequence PixelSequence with the length of Height × Width into a two-dimensional matrix with the size of Height × Width to obtain an encrypted ciphertext image C; wherein bitxor (·, ·) is a binary bit xor operation.