CN112422266B

CN112422266B - Hyperchaotic encryption method based on Joseph traversal and bit plane reconstruction

Info

Publication number: CN112422266B
Application number: CN202011169538.7A
Authority: CN
Inventors: 赵进超; 张勋才; 刘洁; 牛莹; 吴涛; 王时达
Original assignee: Zhengzhou University of Light Industry
Current assignee: Zhengzhou University of Light Industry
Priority date: 2020-10-28
Filing date: 2020-10-28
Publication date: 2022-07-15
Anticipated expiration: 2040-10-28
Also published as: CN112422266A

Abstract

The invention provides a hyperchaotic encryption method based on Joseph traversal and bit plane reconstruction, which comprises the following steps: converting the grayscale image into an image matrix; inputting image matrixSHA‑256Taking the algorithm as a key, and obtaining an initialization parameter according to the key; introducing the initialization parameters into a hyper-chaos Lorenz system to iterate and respectively generate four chaotic sequences, processing one chaotic sequence to be used as a variable step length sequence, and scrambling the pixel position of a one-dimensional sequence of an image matrix by using an improved Joseph traversal method and rearranging the one-dimensional sequence into a pixel scrambling matrix; dividing the pixel chaotic matrix into eight bit planes, and performing bit plane reconstruction on the pixel chaotic matrix by using the other three chaotic sequences and a bit plane reconstruction method to generate a bit plane reconstruction image matrix; and diffusing the bit plane reconstructed image matrix to generate a final encrypted image. The invention can encrypt different types of images into uniformly distributed encrypted images, has very sensitive key sensitivity and can resist various attacks.

Description

Hyperchaotic encryption method based on Joseph traversal and bit plane reconstruction

Technical Field

The invention relates to the technical field of image encryption, in particular to a hyperchaotic encryption method based on Joseph traversal and bit plane reconstruction.

Background

With the rapid development of modern communication technology, more and more image information is transmitted on a social network, and many digital images carry private information, so that the method becomes a research hotspot for protecting the private information. Image encryption is an effective technique for securing digital images. Due to inherent characteristics of strong correlation, high redundancy and the like between adjacent pixels of a digital image, some conventional encryption methods such as Data Encryption Standard (DES) or Advanced Encryption Standard (AES) are not suitable for encrypting the digital image because of the disadvantage of low efficiency when the conventional encryption algorithm is used for encrypting the digital image.

The advantages of the chaotic system, such as pseudo-randomness, initial value sensitivity, ergodicity, unpredictability and the like, are paid attention to by people, so that the chaotic system is widely applied to an image encryption scheme. Chaotic systems can be divided into two broad categories, high-dimensional (HD) and one-dimensional (1D) chaotic systems. Because the one-dimensional chaotic system has a simple structure, some image encryption algorithms based on the low-dimensional chaotic system are easy to attack, and the algorithms often have the defects of smaller key space and the like. Compared with a one-dimensional chaotic encryption system, the encryption algorithm of the high-dimensional chaotic system has the advantages of large key space, high dynamic system behavior and high ergodicity and is safer than the one-dimensional chaotic encryption system. Therefore, various image encryption methods based on the hyperchaotic system have been proposed. In the document [ y.wang, Wong Kwok-Wo, f.liao X, t.xiang, R Chen G a char-based image encryption algorithm with variable control parameters Chaos solution Fract,41 (2009), (4), pp.1773-1783], high sum suggests an image encryption algorithm based on hyper-chaotic pixel level permutation, which, although having a large key space, is not resistant to encrypted text attacks. Also, in the current image scrambling method, many algorithms obtain index vectors by ordering the generated chaotic sequence, and then perform a scrambling operation on pixels of an image using the index vectors. The security of the above scrambling method depends on the index vector, and an attacker can obtain the index vector by analyzing the relationship between the ciphertext image and the plaintext image, which may result in the invalidation of the scrambling operation.

Disclosure of Invention

Aiming at the technical problems of poor safety and unstable scrambling operation of the conventional digital image encryption method, the invention provides a hyperchaotic encryption method based on Joseph traversal and bit plane reconstruction.

In order to achieve the purpose, the technical scheme of the invention is realized as follows: a hyperchaotic encryption method based on Joseph traversal and bit plane reconstruction comprises the following steps:

step one, converting a gray image P with the size of M multiplied by N into an image matrix P' with the size of M multiplied by N;

step two, generation of an initial value: inputting the image matrix P' into an SHA-256 algorithm to obtain 256-bit hash values as keys, and obtaining an initialization parameter x of the hyperchaotic Lorenz system according to the keys₀、y₀、z₀And w₀；

Step three, scrambling operation: the image matrix P' is converted into a one-dimensional sequence U, and the initialization parameters x are converted into₀、y₀、z₀And w₀Introducing the chaos sequence X and the chaos sequence Y, Z, W into a hyper-chaos Lorenz system as initial values to perform iteration respectively, processing the chaos sequence X to obtain a variable step length sequence Z1, performing pixel position scrambling on a one-dimensional sequence U by using an improved Joseph traversal method and the variable step length sequence Z1 to generate a pixel scrambling sequence U ', and rearranging the pixel scrambling sequence U ' into a pixel scrambling matrix P ' with the size of M multiplied by N;

step four, bit plane reconstruction: the pixel chaotic matrix P 'is divided into eight bit planes, and the bit plane reconstruction is carried out on the pixel chaotic matrix P' by utilizing the chaotic sequence Y, Z, W and a bit plane reconstruction method to generate a bit plane reconstruction image matrix P₅；

Step five, diffusion operation: reconstructing a bit-plane into an image matrix P₅And in order to convert the one-dimensional sequence R, diffusing the one-dimensional sequence R to obtain a diffusion sequence R ', rearranging the diffusion sequence R' into a matrix with the size of M multiplied by N, and generating a final encrypted image.

The dynamic formula of the hyperchaotic Lorenz system in the step two is as follows:

wherein the content of the first and second substances,

and

are the inverses of the state variables x, y, z and w, respectively, and the hyper-chaotic Lorenz system is in a hyper-chaotic state when a is 10, b is 8/3, c is 28 and-1.52 is r is ≦ 0.06.

Initializing parameter x in the second step₀、y₀、z₀And w₀The generation method comprises the following steps: taking 256 bit hash value as key K, dividing the key K into 32 groups by one group of every 8 bits, converting each group into decimal value of K₁，k₂，k₃...k₃₂And then:

wherein H₁、H₂、H₃And H₄In order to calculate the intermediate variables of the process,

is to perform a bitwise xor operation, mod (,) is a modulo function, and bin2dec () is a function that converts a number from a binary representation to a decimal representation.

The generation method of the variable step length sequence Z1 comprises the following steps: will initialize the parameter x₀、y₀、z₀And w₀The initial value is substituted into a hyper-chaos Lorenz system, M multiplied by N +1000 times is iterated, and the first 1000 times of iteration are omitted to generate a chaos sequence X ═ X₁，x₂，x₃...x_M×NAnd processing the chaotic sequence X:

z_i＝floor(mod(x_i×2³²，8))；

wherein floor () is a floor function, x_iIs the i-th element, z, of the chaotic sequence X_iIs the ith element of the variable step size sequence Z1.

The method for scrambling the pixel position of the one-dimensional sequence U by using the improved Joseph traversal method and the variable step length sequence Z1 in the third step is as follows: bringing the variable step length sequence Z1 into an improved Joseph traversal method, generating a Joseph sequence D as an index vector O after Joseph variable step length traversal, scrambling the one-dimensional sequence U by using the index vector O, and generating a pixel scrambling sequence U'; i.e. the starting position is s ═ z₁Removing the position index in the one-dimensional sequence U as

Is placed in the sequence U ', i.e. U'₁＝u_j1(ii) a Recirculating movement j₂Element, remove position index u_j2Of (a), i.e. u'₂＝u_j2(ii) a This operation is repeated until the last element, which finally results in the joseph sequence U '═ U'₁，u′₂，u′₃...，u′_n}。

The josephson sequence in the improved josephson traversal method is:

D＝JS(n，s，j，Z1)；

where JS () is a josephson function, N ═ M × N represents the total number of pixels in the image matrix P', s represents the start position, and j ═ j { (j) }₁，j₂，j₃...j_i...j_M×NDenotes a shift number sequence, Z1 ═ Z₁，z₂，z₃...z_i...z_M×NRepresenting a variable step length sequence generated by the hyperchaotic Lorenz system;

let s equal z₁Number of shifts j₁＝z₁When i is not less than 1, j_i＝z_i+j_i-1Generate, generateJosephson sequence D.

The method for reconstructing the bit plane in the fourth step comprises the following steps:

step 1: will initialize the parameter x₀、y₀、z₀And w₀The chaos sequence Y is substituted into a hyper-chaos Lorenz system, 8 XN +1000 times and 8 XM XN +1000 times are iterated, the previous 1000 times are omitted, and the chaos sequence Y is generated respectively as { Y ═ Y [ ((Y) } Y [)₁，y₂，y₃...y_8×NZ, chaotic sequence Z ═ Z₁₁，z₁₂，z₁₃...z_1(8×N)W-chaotic sequence W ═ W₁，w₂，w₃...w_8×M×NRespectively returning the index values of the chaotic sequence Y, Z, W according to descending order, and respectively generating index vectors Y ', Z ' and W ';

step 2: decomposing a pixel-scrambling matrix P' into A₁，A₂，A₃，A₄，A₅，A₆，A₇，A₈Eight bit planes, bit plane A₈～A₁Two-dimensional bit matrix P arranged into M multiplied by 8N from top to bottom in sequence₁Using the index vector Y' to the two-dimensional bit matrix P₁Generates a line scrambling matrix P by scrambling each line of the line₂；

And 3, step 3: scrambling the rows by a matrix P₂Divided into eight M × N bit planes, rearranged into M × 8N two-dimensional bit matrix P in left-to-right order₃Using the index vector Z' to pair a two-dimensional bit matrix P₃Generates a row scrambling matrix P by scrambling each row of the matrix₄；

And 4, step 4: scrambling the rows by a matrix P₄Arranging the two sequences into a one-dimensional sequence T of 8 xMxN from left to right and from top to bottom, scrambling the one-dimensional sequence T by using an index vector W', and generating a line scrambling vector T₁(ii) a And scrambling the rows by the vector T₁Rearranging generated bit plane A'₁～A′₈Bit plane A'₁～A′₈Fusing the two-dimensional matrix to obtain a bit plane reconstruction image matrix P₅。

The diffusion method in the fifth step comprises the following steps: reconstructing a bit plane into a pictureImage matrix P5 is converted into a one-dimensional sequence R ═ R₁，r₂，r₃，...，r_i，...r_M×NAnd the diffusion method comprises the following steps:

wherein r'₁、r′₂、r′_i-2、r′_i-1、r′_iAre all elements of diffusion sequence R'.

Compared with the prior image encryption method, the invention has the following beneficial effects: according to the method, a classical confusion structure is adopted, a pseudorandom sequence is generated by a hyper-chaotic Lorenz system, a variable-step Joseph scrambling algorithm is provided by utilizing the Joseph problem in a scrambling stage, pixels of a plaintext image are traversed through the Joseph problem, and the step length increased each time when the pixels are traversed is controlled by the chaotic sequence, so that the pixels of a ciphertext image after scrambling are more random, and the correlation between adjacent pixels can be effectively broken; then, performing bit plane decomposition on the image, sequencing the generated chaotic sequence by using the chaotic sequence to obtain an index vector, scrambling bit positions between bit planes by using the index vector, and reconstructing the bit planes; different from scrambling the pixels of the image by directly utilizing the index vector, the scrambling of the bit positions between the bit planes can not only make the bits between the bit planes more disordered, but also change the pixel value, so that an attacker is difficult to obtain the index vector by analyzing the relationship between the ciphertext image and the plaintext image, and the safety of the algorithm is improved; and finally, performing diffusion operation on the encrypted image by using ciphertext feedback, and generating a key by using an SHA-256 algorithm, thereby further improving the key space. The invention uses classical scrambling and diffusion structure, and utilizes the principle of Josephson problem to move the encrypted image pixel to different positions to realize the confusion characteristic of the image; the bit plane reconstruction method adopts a pseudo-random sequence generated by hyperchaos to confuse and reconstruct different bit planes of the encrypted image; and finally, diffusing the encrypted image by using ciphertext feedback operation. Simulation experiment analysis shows that the method can encrypt different types of images into uniformly distributed encrypted images; the entropy of the encrypted image information can reach more than 7.99, and the statistical attack is effectively resisted; has very sensitive key sensitivity and can resist various attacks.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the embodiments or the prior art descriptions will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and other drawings can be obtained by those skilled in the art without creative efforts.

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

Fig. 2 is a phase diagram of a hyper-chaotic Lorenz system with parameters a-10, b-8/3, c-28, and r-1, where (a) is x_n-y_nA phase diagram; (b) is x_n-z_nA phase diagram; (c) is x_n-w_nA phase diagram; (d) is y_b-z_nA phase diagram; (e) is y_b-w_nA phase diagram; (f) is z_n-z_nAnd (4) phase diagrams.

FIG. 3 is an example of the Josephson traversal process and scrambling of sequences of the present invention.

FIG. 4 is a flow chart of bit plane reconstruction according to the present invention.

Fig. 5 is an encryption effect diagram of the present invention, in which (a) is an Elaine plaintext image, (b) is an Elaine ciphertext image, (c) is a Hill plaintext image, (d) is a Hill ciphertext image, (e) is a Lena plaintext image, and (f) is a Lena ciphertext image.

FIG. 6 is a decrypted image after the parameter change and the decrypted image of the present invention, wherein (a) is a Lena ciphertext image, (b) is a Lena decrypted image, and (c) is x₀Change 10^-10Post-decryption of the image, (d) is y₀Change 10^-10Post-decryption of the image, (e) is z₀Change 10^-10Post-decryption of the image, (f) is w₀Change 10^-10Post-decryption of images

Fig. 7 is histograms of a plaintext image and a ciphertext image according to the present invention, in which (a) is a histogram of a Lena plaintext image, (b) is a histogram of a Lena ciphertext image, (c) is a histogram of a Boat plaintext image, (d) is a histogram of a Boat ciphertext image, (e) is a histogram of a Face plaintext image, and (f) is a histogram of a Face ciphertext image.

Fig. 8 shows the ciphertext image and the decrypted image after the noise attack, where (a) is the 0.01 scale attack ciphertext image, (b) is the 0.05 scale attack ciphertext image, (c) is the 0.1 scale attack ciphertext image, (d) is the 0.01 scale attack decrypted image, (e) is the 0.05 scale attack decrypted image, and (f) is the 0.1 scale attack decrypted image.

Fig. 9 shows the ciphertext image and decrypted image after clipping in accordance with the present invention, where (a) the ciphertext image is clipped at 1/64, (b) the decrypted image is clipped at 1/64, (c) the ciphertext image is clipped at 1/16, (d) the decrypted image is clipped at 1/16, (e) the ciphertext image is clipped at 1/4, and (f) the decrypted image is clipped at 1/4.

Detailed Description

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art based on the embodiments of the present invention without inventive step, are within the scope of the present invention.

As shown in fig. 1, a hyperchaotic encryption method based on joseph traversal and bit plane reconstruction sets the size of an input plaintext image, i.e., a gray image P, to be mxn, and the image encryption method of the present invention is composed of four parts, namely, key generation, scrambling scheme, bit plane reconstruction and diffusion algorithm, and specifically includes the steps of:

step one, converting the gray image P with the size of M multiplied by N into an image matrix P' with the size of M multiplied by N.

Step two, generating an initial value: inputting the image matrix P' into an SHA-256 algorithm to obtain 256-bit hash values as a key, and obtaining an initialization parameter x of the hyperchaotic Lorenz system according to the key₀、y₀、z₀And w₀。

Due to the disadvantages of an over-simple structure, a small key space, a few parameters and the like of the low-dimensional chaotic system, the hyper-chaotic system can generate a more complex chaotic sequence. The hyperchaotic has a plurality of forward Lyapunov exponents, has a higher key space, is convenient to scramble and diffuse in a larger space, reduces the correlation between adjacent pixels and improves the overall security of the ciphertext image.

The dynamic formula of the hyper-chaotic Lorenz system is as follows:

wherein the content of the first and second substances,

and

are the inverses of the state variables x, y, z and w, respectively, and when a is 10, b is 8/3, c is 28 and-1.52 r ≦ -0.06, equation (1) is in a hyper-chaotic state. When r is-1, the four Lyapunov indices in formula (1) are: lambda [ alpha ]₁＝0.3381、λ₂＝0.1586、λ₃＝0、λ₄-15.1752. FIG. 2 is a phase diagram of the hyper-chaotic Lorenz system, showing 6 strange attractors of the hyper-chaotic Lorenz system, wherein x_n-z_nThe phase diagram is shaped like a butterfly and is a well-known butterfly attractor, which also indicates that the generated sequence has pseudo-randomness.

Firstly inputting a gray image P into an SHA-256 algorithm to generate 256-bit Hash values as a key K, then dividing each 8 bits into one group, dividing the key K into 32 groups, converting each group into decimal values with K being respectively₁，k₂，k₃...k₃₂(ii) a Obtaining the intermediate variable H by the formula (2)₁、H₂、H₃And H₄Wherein, in the step (A),

a bit exclusive or operation is performed. Using equation (3) for the intermediateVariable H₁、H₂、H₃And H₄Calculating to obtain an initial parameter x₀，y₀，z₀，w₀As the initial value of the four-dimensional hyper-chaotic Lorenz system.

Where mod (,) is a modulo function and bin2dec () is a function that converts a number from a binary representation to a decimal representation.

Step three, scrambling operation: the image matrix P' is converted into a one-dimensional sequence U, and the initialization parameters x are converted into₀、y₀、z₀And w₀And (3) carrying out iteration by using a hyperchaotic Lorenz system to respectively generate a chaotic sequence X and a chaotic sequence Y, Z, W, processing the chaotic sequence X to obtain a variable step length sequence Z1, scrambling the pixel position of the one-dimensional sequence U by using an improved Joseph traversal method and a variable step length sequence Zl to generate a pixel scrambling sequence U ', and rearranging the pixel scrambling sequence U ' into a pixel scrambling matrix P ' with the size of M multiplied by N.

Josephson problems occur in the roman-jewish war, and in mathematics, the josephson sequence can be expressed as:

D＝JS(n，s，j) (4)

wherein n represents the total number of elements, s represents the start position, j represents the shift number, JS () is a josephson function, D ═ D { (D)₁，d₂，d₃...d_nIs the joseph sequence, d₁，d₂，d₃...d_nAre elements of the Joseph sequence, respectively. In the following, a josephson sequence generation process of n-8, s-4, j-4 is given to better explain the josephson problem, setting the start position to 4, then removing the element numbered 4 and placing it in josephson sequence D, i.e. D ₁4; move 4 elements in a cycle, at this timeThe element number k is 8, the element is deleted and the number 8 is placed in the sequence D, i.e. D ₂8. This operation is repeated until the last element, and finally the josephson sequence D ═ {4, 8, 5, 2, 1, 3, 6, 7} is obtained. The nature of sequence traversability and reversibility produced by the josephson problem is very similar to the idea of digital image scrambling encryption. Thus, image permutation algorithms can be designed using josephson sequences.

The Joseph sequence can be used for effectively scrambling a plaintext image, correlation between adjacent pixels is broken, scrambling safety is not high after pixels are traversed through the Joseph sequence, and an attacker can break through the initial position and the traversal step of the Joseph traversal through analyzing the scrambled sequence through statistical characteristics because the initial position and the traversal step are fixed, so that scrambling operation is disabled. Therefore, the chaos sequence is used for improving the starting position and each traversal step length of Josephson traversal, and ciphertext-only attack can be effectively resisted. The specific scrambling method is as follows:

iterating the hyper-chaos Lorenz system M multiplied by N +1000 times and cutting off the first 1000 times to generate a chaos sequence X ═ X₁，x₂，x₃...x_M×NAnd processing the chaotic sequence X according to a formula (5) to obtain a variable step length sequence Z1:

z_i＝floor(mod(x_i×2³²，8)) (5)

wherein floor () is a floor function, x_iIs the ith element of the sequence X, z_iIs the ith element of the variable step size sequence Z1.

The formula of the josephson sequence is improved, and the formula (6) shows the mathematical expression of the improved josephson sequence:

D＝JS(n，s，j，Z1) (6)

where N is M × N, s is a start position, and j is { j { (j) } represents the total number of pixels in the image matrix P₁，j₂，j₃...j_i...j_M×NRepresents a shift number sequence, Z1 ═ Z₁，z₂，z₃...z_i...z_M×NAnd represents a variable step length sequence generated by the Lorenz chaotic system. Let s be z₁Number of shift j₁＝z₁When i is not less than 1, j_i＝z_i+j_i-1Thereby generating josephson sequence D.

Converting the image matrix P' into a one-dimensional sequence U, wherein the starting position is s-z₁＝j₁The removal position is indexed by

Is placed in the sequence U ', i.e. U'₁＝u_j1(ii) a Recirculating movement j₂An element, remove position index of

Of elements i.e.

This operation is repeated until the last element, resulting in the josephson sequence U '═ { U'₁，u′₂，u′₃...，u′_n}. To explain the scrambling process in detail, the present invention provides a 1 × 8 one-dimensional sequence U ═ {178, 123, 157, 160, 147, 56, 24, 89} and a variable step length sequence Z ═ {4, 1, -3, 1, 1, 0, -1, 0}, and as shown in fig. 3, the process of generating the joseph scrambling sequence generates an index vector O ═ {4, 1, 3, 7, 6, 8, 2, 5} after being traversed by the joseph variable step length. The sequence U is scrambled by the index vector O, and a scrambled sequence U' is generated {160, 178, 157, 24, 56, 89, 123, 147 }.

Step four, bit plane reconstruction: the pixel chaotic matrix P 'is divided into eight bit planes, and the bit plane reconstruction is carried out on the pixel chaotic matrix P' by utilizing the chaotic sequence Y, Z, W and a bit plane reconstruction method to generate a bit plane reconstruction image matrix P₅。

Because the image has the problem of local uneven bit distribution in the bit plane, the encryption effect may be less than ideal without processing. Therefore, in order to enhance the encryption effect and reduce the algorithm overhead, attention should be paid to changing the distribution of the image bit planes. For the problems, the invention provides a bit plane reconstruction method based on a chaotic sequence, and the bits of the encrypted image can be uniformly distributed on the bit planes by changing the distribution of the bits among the bit planes. The method of bit plane reconstruction will be described in detail below:

step 1: will initialize the parameter x₀、y₀、z₀And w₀The chaos sequence Y is generated by iterating 8 XN +1000, 8 XN +1000 and 8 XM XN +1000 times and cutting off the first 1000 times by substituting the hyper-chaos Lorenz system into { Y ═ Y-₁，y₂，y₃...y_8×NZ, chaotic sequence Z ═ Z₁₁，z₁₂，z₁₃...z_1(8×N)W-chaotic sequence W ═ W₁，w₂，w₃...w_8×M×NRespectively returning the index values of the chaotic sequence Y, Z, W in a descending order to respectively generate index vectors Y ', Z ' and W ';

step 2: decomposing a pixel-scrambling matrix P' into A₁，A₂，A₃，A₄，A₅，A₆，A₇，A₈Eight bit planes, bit plane A₈～A₁Two-dimensional bit matrix P arranged into M multiplied by 8N from top to bottom in sequence₁Using the index vector Y' to pair a two-dimensional bit matrix P₁Generates a row scrambling matrix P by scrambling each row of the matrix₂；

And step 3: scrambling the rows by a matrix P₂Divided into eight M × N bit planes, rearranged into M × 8N two-dimensional bit matrix P in left-to-right order₃Using the index vector Y' to pair a two-dimensional bit matrix P₃Generates a row scrambling matrix P by scrambling each column of the input signal₄；

And 4, step 4: scrambling the rows by a matrix P₄Arranging the two sequences into a one-dimensional sequence T of 8 xMxN from left to right and from top to bottom, scrambling the one-dimensional sequence T by using an index vector W', and generating a line scrambling vector T₁(ii) a And scrambling the rows by a vector T₁Rearranging generated bit plane A'₁～A′₈Combining bitset () function to convert bit plane A'₁～A′₈And fusing the two-dimensional matrixes to obtain a bit plane reconstruction image matrix P5. Use of b in matlabThe itget () function can get any bit plane of the plaintext image, for example: bitget (P ', 1) represents the acquisition of the lowest bitplane, A, of the pixel scrambling matrix P'₁Bitget (P ', 8) denotes the acquisition of the most significant bit plane of the pixel scrambling matrix P', namely A₈. Bit plane A 'can be mapped using bitset () function'₁～A′₈Fused into a two-dimensional matrix, e.g. A'₁₂＝bitset(A′₁，2，A′₂And is A'₁，A′₂Two bit planes fuse, A'₁₂₃＝bitset(A′₁₂，2，A′₃And is A'₁，A′₂，A′₃The three bit planes are fused.

To better illustrate the scrambling process, a 2 × 2 image matrix bit-plane reconstruction flow chart as shown in fig. 4 is given below.

Step five, diffusion operation: reconstructing the bit-plane into an image matrix P₅In order to convert the one-dimensional sequence R, the one-dimensional sequence R is diffused to obtain a diffusion sequence R ', the diffusion sequence R' is rearranged into a matrix with the size of M multiplied by N, and a final encrypted image P is generated₂。

In order to enhance the security of the encryption algorithm, the information of the plaintext pixel points is hidden in as many ciphertext pixel points as possible, and the plaintext image is subjected to diffusion operation. The invention uses the previous two pixel values to change the current pixel value, so that a small amount of change of the plaintext image can be diffused to the whole ciphertext image, and the bit plane is used for reconstructing the image matrix P₅Conversion to one-dimensional sequence R ═ R₁，r₂，r₃，...，r_i，...r_M×NAnd the diffusion method comprises the following steps:

wherein r'₁、r′₂、r′_i-2、r′_i-1、r′_iAre all elements of diffusion sequence R'. The diffusion in the decryption process is opposite to the forward operation.

The decryption process of the present invention is the reverse process of the encryption process, and is not described herein again.

In the invention, a computer with a configuration environment of Windows 10, 8.00GB RAM, Intel (R) core (TM) i7-4510 CPU @2.00GHz is used for carrying out experimental simulation on a matlab2019a platform. The test images are from the USC-SIPI image database. All test images are difficult-to-process pattern images, as can be seen in fig. 5(b), 5(d), and 5 (f). The encrypted ciphertext image cannot be identified, and in order to verify the safety and analyze the performance of the encrypted ciphertext image, safety experimental analysis is performed on the encrypted ciphertext image, wherein the safety experimental analysis comprises histogram analysis, correlation coefficient analysis, key space analysis, key sensitivity analysis, differential attack analysis and the like.

The key is the most important component in the encryption scheme, and the larger the key space is, the stronger the resistance to brute force attack is. When the key space is greater than 2¹⁰⁰The key has a sufficient level of security to resist brute force attacks. The invention uses SHA-256 algorithm to generate 256 bit binary hash value generated key, so its key space size is 2¹²⁸And an initial value x₀，y₀，z₀，w₀Has a calculation accuracy of 10^-10So the total key space is 3.4028 × 10⁸²Thus, the key space of the present invention is large enough to resist all types of brute force attacks.

Differential attacks are a common and effective method of attack. The differential attack is to study the influence of the difference between plaintext images on ciphertext images of the plaintext images and aim to establish a relationship between the plaintext images and the ciphertext images corresponding to the plaintext images. The diffusion property indicates that small changes in the plaintext image may propagate throughout the ciphertext image. If the encryption algorithm of the image has a diffusion characteristic, the image can resist differential attack more effectively. Pixel rate of change (NPCR) and pixel mean change intensity (UACI) are two methods of testing whether an encryption algorithm can resist differential attacks. The NPCR calculates the number of different pixels, while the UACI calculates the average rate of change of pixels between two images. Suppose P₁And P₂Are two ciphertext images, and their plaintext image only has one bit difference, then their NPCR and UACI valuesThe calculation is shown below:

if P₁(i，j)≠P₂(i，j)，D _(i，j)1 is ═ 1; otherwise D _(i，j)1. In recent years, the publication [ Y.Wu, J.P.Noonan, S.Agaian NPCR And UACI random tests for image encryption Cyber J.Multidisci.J.Sci.Technol.J.Select.areas Telecommun. (JSAT) (2011), pp.31-38]The strict critical NPCR and UACI values are presented. The simulation was performed using 6 256 × 256 grayscale images in the USC-SIPI image database as test images, and NPCR and UACI values for different image encryption methods are shown in tables 2 and 3, respectively. It can be seen that the present invention can pass all NPCR and UACI tests, while other encryption methods cannot pass some tests. This indicates that the proposed encryption algorithm exhibits high performance in resisting differential attacks. From "5.1.09" to "5.1.14" in table 1, the NPCR results, bold numbers, of the various image encryption methods are test results that failed the NPCR test. ZBC is from the literature [ Y.Zhou, L.Bao, C.L.P.Chen A new 1D cosmetic system for image encryption Signal Process, 97(2014), pp.172-182]WZNS is available from the literature [ Y.Wu, Y.Zhou, J.P.Noonan, S.Agaian Design of image equalizer using squares Inf.Sci. (Ny),264 (2014), pp.317-339]CKW is from the literature [ C.Cao, K.Sun, W.Liu A novel bit-level image encryption algorithm on 2D-licm hyperconductive map Signal Process, 143(2018), pp.122-133]. Table 2 from "5.1.09" to "5.1.14", UACI results for various image encryption methods, bold numbers are test results that failed the UACI test.

TABLE 1 NPCR results for various image encryption methods

Algorithm	ZBC	WZNS	CKW	The invention
					5.1.09	99.5575	99.6506	99.6140	99.5789
5.1.10	99.5544	99.6063	99.5880	99.5941
					5.1.11	99.8123	99.6490	99.6033	99.5926
5.1.12	99.6109	99.6170	99.5651	99.6475
					5.1.13	99.7421	99.5605	99.5789	99.5804
5.1.14	99.6933	99.6216	99.6765	99.6140

Table 2 UACI results for various image encryption methods

Algorithm	ZBC	WZNS	CKW	The invention
					5.1.09	33.7574	33.4387	33.4032	33.4017
5.1.10	33.1739	33.4701	33.3557	33.4746
					5.1.11	33.3198	33.4150	33.4696	33.4991
5.1.12	33.6656	33.5082	33.4634	33.4113
					5.1.13	34.3306	33.4939	33.3046	33.5042
5.1.14	33.1888	33.7240	33.4796	33.4621

To ensure the security of the encryption algorithm, the encryption algorithm should have a high sensitivity to the input key. When an incorrect key is input to decrypt a ciphertext image, the output is an image in which no information can be recognized. The invention uses the original key to encrypt the Lena image and to the initial value x₀、y₀、z₀、w₀Respectively change 10^-10The ciphertext image is decrypted using the modified key. The Lena ciphertext image is shown in fig. 6(a), and the decrypted image of the correct decryption key is shown in fig. 6 (b). The decrypted image using the wrong decryption key is shown in fig. 6(c), 6(d), and 6(e) and fig. 6 (f). Obviously, a slightly changed decryption key cannot decrypt the cryptographic image. Such asThe sensitivity test of the key decryption shown in table 3 shows that the key of the present invention has higher sensitivity.

TABLE 3 sensitivity analysis of key decryption after different parameter changes

The pixel distribution histogram can intuitively reveal the distribution rule of pixel values in a plaintext image, the pixel values of the plaintext image are generally unevenly distributed, and an attacker can obtain effective information in the image by utilizing statistical characteristic attack; the more evenly the pixel distribution in the ciphertext image, the more difficult it is for an attacker to obtain valuable information, and the more desirable the encryption algorithm is. A good encryption system can make the encrypted image have a histogram with a uniform pixel distribution, which can resist any statistical attack. As shown in fig. 7(a), 7(c), and 7(e), the pixel value distribution of the plaintext image is not uniform; however, fig. 7(b), 7(d) and 7(f) of the corresponding ciphertext images show that the distribution of pixel values of the ciphertext images corresponding to the plaintext images exhibits a flat and uniform characteristic. Therefore, the invention can effectively resist the attack of the statistical property.

The adjacent pixels of the plaintext image have high correlation in the horizontal, vertical, and diagonal directions. The ideal encryption algorithm can ensure that the correlation coefficient of the pixel in the ciphertext image has low enough correlation, and can resist statistical attack. The correlation coefficient calculation formula is:

where x and y are pixel values, cov (x, y) is covariance, D (x) is variance, E (x) is mean, ρ_xyIs the correlation coefficient. For analyzing and comparing plaintext image and ciphertextCorrelation of adjacent pixels in an image, 5000 pairs of adjacent pixels are randomly selected from a plaintext image and a ciphertext image by using 6 256 x 256 grayscale images in a USC-SIPI image database as a test image, and a correlation distribution graph of two adjacent pixels is displayed in three directions.

TABLE 4 Adjacent Pixel correlation analysis of plaintext image and ciphertext image from "5.1.09" to "5.1.14

As shown in table 4, the distribution of adjacent pixels in the plaintext image is highly concentrated, which means that the correlation of adjacent pixels in the plaintext image is high. However, the distribution of adjacent pixels in the ciphertext image is random, which means that the correlation of adjacent pixels of the encrypted image is low.

The entropy is the average information with redundant parts removed, and is the most important feature of randomness. The calculation formula of the information entropy is as follows:

where n is the gray level of the image and m is_iIs the ith gray value, P (m), on the image_i) Is the gray value m_iThe probability of occurrence. For a grayscale image, the theoretical value of the information entropy is 8. The 6 images with 256 x 256 gray levels in the USC-SIPI image database are used as test images, the test results are shown in the table 5, and the entropy value of the encrypted ciphertext image is very close to the ideal value.

Table 5 information entropies of plaintext images and ciphertext images from "5.1.09" to "5.1.14

USC-SIPI	Entropy of plaintext image information	Information entropy of ciphertext image
			5.1.09	6.7093	7.9971
5.1.10	7.3318	7.9971
			5.1.11	6.4523	7.9970
5.1.12	6.7057	7.9973
			5.1.13	1.5483	7.9957
5.1.14	7.3424	7.9973

When digital images are transmitted in a network, portions of the data may be lost or disturbed by noise. Therefore, the encryption algorithm should have strong data loss and noise defense capabilities. In order to test the capability of the present invention against noise and data loss attacks, a Lena image with a size of 256 × 256 is used as a test image, 1%, 5%, and 10% salt and pepper noise are added to a ciphertext image of the Lena image after an encryption algorithm, and a decryption image corresponding to the ciphertext image is shown in fig. 8. As is apparent from fig. 8, when the ciphertext image is attacked by salt-and-pepper noise, the decrypted image can still recognize the information of the decrypted image by using the encryption algorithm of the present invention. As shown in fig. 9, since the ciphertext images of Lena are cropped 1/64, 1/16, and 1/4 and then decrypted, the image information can still be recognized, and thus the present invention can resist the analysis of cropping attacks.

The invention mainly provides an image encryption method based on a hyperchaotic system by utilizing a Josephson problem principle and bit plane reconstruction, Josephson scrambling is derived from the Josephson problem, and better scrambling effect is obtained by utilizing improved Josephson traversal to scramble an image; the bit plane reconstruction method reconstructs the bit plane of the plaintext image from bits, so that the bits of the bit plane are randomly distributed, a pixel distribution histogram of the ciphertext image obtained by combining a diffusion algorithm is more uniform, and differential attack can be effectively resisted. Simulation results show that the method can encrypt different types of images into unrecognizable ciphertext images; the security of the key is analyzed in the aspects of key sensitivity, capability of defending differential attack, correlation of adjacent pixels, information entropy, histogram and the like, and the excellent effect is achieved and the good security performance is achieved.

The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims

1. A hyperchaotic encryption method based on Joseph traversal and bit plane reconstruction is characterized by comprising the following steps:

step two, generation of an initial value: inputting the image matrix P' into an SHA-256 algorithm to obtain 256-bit hash values as a key, and obtaining an initialization parameter x of the hyperchaotic Lorenz system according to the key₀、y₀、z₀And w₀；

Step three, scrambling operation: the image matrix P' is converted into a one-dimensional sequence U, and the initialization parameters x are converted into₀、y₀、z₀And w₀Introducing the initial value into a hyper-chaos Lorenz system to iterate and respectively generate a chaos sequence X and a chaos sequence Y, Z, W, processing the chaos sequence X to obtain a variable step length sequence Z1, scrambling the pixel position of a one-dimensional sequence U by utilizing an improved Joseph traversal method and the variable step length sequence Z1 to generate a pixel scrambling sequence U ', and rearranging the pixel scrambling sequence U ' into a pixel scrambling matrix P ' with the size of M multiplied by N;

the josephson sequence in the improved josephson traversal method is:

D＝JS(n，s，j，Z1)；

where JS () is a josephson function, N ═ M × N represents the total number of pixels in the image matrix P ', s represents the start position, j ═ { j ×, N represents the total number of pixels in the image matrix P', and j ═ j represents the start position₁，j₂，j₃...j_i...j_M×NRepresents a shift number sequence, Z1 ═ Z₁，z₂，z₃...z_i...z_M×NThe variable step length sequence generated by the hyperchaotic Lorenz system is represented;

let s equal z₁Number of shifts j₁＝z₁When i is not less than 1, j_i＝z_i+j_i-1Generating joseph sequence D;

step 1: will initialize the parameter x₀、y₀、z₀And w₀The chaos sequence Y is generated by iterating 8 XN +1000, 8 XN +1000 and 8 XM XN +1000 times and cutting off the first 1000 times by substituting the hyper-chaos Lorenz system into { Y ═ Y-₁，y₂，y₃...y_8×NZ, chaotic sequence Z ═ Z₁₁，z₁₂，z₁₃...z_1(8×N)Chaos and chaosSequence W ═ W₁，w₂，w₃...w_8×M×NRespectively returning the index values of the chaotic sequence Y, Z, W according to descending order, and respectively generating index vectors Y ', Z ' and W ';

step 2: decomposing a pixel disorder matrix P' into A₁，A₂，A₃，A₄，A₅，A₆，A₇，A₈Eight bit planes, bit plane A₈～A₁Two-dimensional bit matrix P arranged into M multiplied by 8N from top to bottom₁Using the index vector Y' to pair a two-dimensional bit matrix P₁Generates a line scrambling matrix P by scrambling each line of the line₂；

And step 3: scrambling the rows by a matrix P₂Divided into eight M × N bit planes, rearranged into M × 8N two-dimensional bit matrix P in left-to-right order₃Using the index vector Z' to pair a two-dimensional bit matrix P₃Generates a row scrambling matrix P by scrambling each column of the input signal₄；

And 4, step 4: scrambling the rows by a matrix P₄Arranging the two sequences into a one-dimensional sequence T of 8 XMxN from left to right and from top to bottom, scrambling the one-dimensional sequence T by using an index vector W', and generating a line scrambling vector T₁(ii) a And scrambling the rows by the vector T₁Rearranging generated bit plane A'₁～A′₈Bit plane A'₁～A′₈Fusing the two-dimensional matrix to obtain a bit plane reconstruction image matrix P₅；

Step five, diffusion operation: reconstructing the bit-plane into an image matrix P₅And in order to convert the one-dimensional sequence R, diffusing the one-dimensional sequence R to obtain a diffusion sequence R ', rearranging the diffusion sequence R' into an M multiplied by N matrix, and generating a final encrypted image.

2. The hyperchaotic encryption method based on Joseph traversal and bit-plane reconstruction as claimed in claim 1, wherein the dynamic formula of the hyperchaotic Lorenz system in the second step is:

wherein, the first and the second end of the pipe are connected with each other,

and

3. The hyperchaotic encryption method based on Joseph traversal and bit-plane reconstruction as claimed in claim 1 or 2, characterized in that in step two, parameter x is initialized₀、y₀、z₀And w₀The generation method comprises the following steps: using 256 bit hash value as key K, dividing the key K into 32 groups by 8 bit, converting each group into decimal value K₁，k₂，k₃...k₃₂Then:

4. The method according to claim 1The hyperchaotic encryption method of Joseph traversal and bit plane reconstruction is characterized in that the generation method of the variable step length sequence Z1 is as follows: will initialize the parameter x₀、y₀、z₀And w₀Introducing the initial value into a hyper-chaotic Lorenz system, iterating M multiplied by N +1000 times and omitting the first 1000 times to generate a chaotic sequence X ═ { X ═ X₁，x₂，x₃...x_M×NAnd (4) processing the chaotic sequence X:

z_i＝floor(mod(x_i×2³²，8))；

wherein floor () is a floor function, x_iIs the i-th element, z, of the chaotic sequence X_iIs the ith element of the variable step sequence Z1.

5. The hyperchaotic encryption method based on Joseph traversal and bit-plane reconstruction as claimed in claim 4, wherein the method of pixel position scrambling of one-dimensional sequence U with modified Joseph traversal method and variable step length sequence Z1 in the third step is: bringing the variable step length sequence Z1 into an improved Joseph traversal method, generating a Joseph sequence D as an index vector O after Joseph variable step length traversal, scrambling the one-dimensional sequence U by using the index vector O, and generating a pixel scrambling sequence U'; i.e. the starting position is s ═ z₁Removing the position index in the one-dimensional sequence U as

Is placed in the sequence U', i.e.

Recirculating movement j₂An element, the removal position index is

Of elements i.e.

This operation is repeated until the last element, finallyTo joseph sequence U '═ U'₁，u′₂，u′₃...，u′_n}。

6. The hyperchaotic encryption method based on Joseph traversal and bit-plane reconstruction as claimed in claim 1, 4 or 5, characterized in that the method of diffusion in step five is: reconstructing the bit-plane into an image matrix P₅Conversion to one-dimensional sequence R ═ R₁，r₂，r₃，...，r_i，...r_M×NAnd the diffusion method comprises the following steps: