An image encryption method based on a two-dimensional hyperchaotic system with constant positive Lyapunov exponents

Abstract

The present invention relates to an image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent, belonging to the fields of chaos theory and secure communications. The method comprises: using any one-dimensional chaotic map as a first seed map and any composite function with a growth rate greater than a preset threshold as a second seed map, introducing the Lyapunov exponent, constructing and iterating a two-dimensional hyperchaotic system based on a constant positive Lyapunov exponent to obtain a chaotic sequence, and constructing a scrambling matrix and a diffusion matrix based on the chaotic sequence; obtaining an image matrix of an image to be encrypted and segmenting it; and performing scrambling-diffusion-scrambling operations on all image sub-blocks obtained by segmentation based on the scrambling matrix and the diffusion matrix, and then combining them to obtain an encrypted image matrix, thereby achieving image encryption. The present invention aims to solve the problem of insufficient multi-dimensional correlation destruction in existing image encryption methods, and can effectively expand the dimension of the chaotic system and enhance the system's complexity, randomness, and other dynamic behaviors.

Description

Technical Field

[0001] The invention relates to an image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent, and belongs to the fields of chaos theory and secure communication. Background Art

[0002] With the rapid growth of network bandwidth, image and video data have become the core carriers of internet information exchange. Chaotic systems, due to their initial value sensitivity and randomness, are widely used in image encryption algorithms. However, existing chaotic mapping designs rely heavily on empirical parameter tuning, lacking rigorous mathematical proof of their chaotic behavior. This can lead to the system degenerating into periodic motion within certain parameter domains. This theoretical flaw creates the risk of inflated encryption keyspaces, allowing attackers to exploit parameter vulnerabilities for selective cracking, threatening the security of practical applications.

[0003] Current color image encryption schemes suffer from insufficient multidimensional correlation destruction: mainstream methods only repeat the grayscale encryption process for the R, G, and B channels, ignoring inter-channel correlations. This allows attackers to infer the global key through a single-channel crack. Although Chai et al. introduced a DNA crossover operation to establish channel correlation, this did not completely destroy intra-channel pixel correlations. Tang et al.'s scheme based on three-dimensional zigzag extraction achieved cross-channel pixel reorganization but failed to effectively eliminate intra-plane correlations. Zhang enhanced correlation through multi-plane zigzag sequence reconstruction, but the intra-plane pixel position correlation remained above the security threshold. Furthermore, existing block encryption strategies (such as non-overlapping blocks) are restricted by fixed boundaries, exposing inter-block pixel correlation features and creating a breakthrough for statistical attacks. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide an image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent, aiming to solve the problem that the existing image encryption method has insufficient multi-dimensional correlation destruction.

[0005] The technical solution of the present invention is: an image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent. First, through rigorous mathematical derivation, it is proved that the proposed two-dimensional hyperchaotic system framework has hyperchaotic characteristics in the entire parameter domain, breaking through the limitation of traditional mapping relying on experience to tune parameters; secondly, the proposed two-dimensional hyperchaotic system framework is a universal model that can generate a large number of chaotic systems by replacing the seed mapping; finally, the chaotic sequence is combined with the Latin square matrix to design a cross-plane dynamic scrambling model, and a nonlinear coupling mechanism is constructed between the three channels R, G, and B (such as cross-plane pixel replacement and orthogonal Latin square matrix fusion), which completely destroys the pixel correlation within and between channels, and further introduces an overlapping block strategy to eliminate the inter-block correlation residue caused by fixed boundaries. This method is highly sensitive to small changes in the plaintext image and can effectively resist various attack methods such as brute force cracking attacks and differential attacks. It has a large key space and high security, and has good application prospects in the fields of image security communication and information protection. The specific steps are as follows:

[0006] S1: taking any one-dimensional chaotic map as a first seed map, and any composite function with a growth rate greater than a preset threshold as a second seed map, designing an objective function for the first seed map, solving the objective function, and obtaining a solution result for the first seed map;

[0007] S2: constructing a two-dimensional hyperchaotic system based on the first seed mapping, the second seed mapping, and the solution results of the first seed mapping;

[0008] S3: Introducing the Lyapunov exponent and constructing a two-dimensional hyperchaotic system based on the constant positivity of the Lyapunov exponent;

[0009] S4: Obtain an image matrix composed of the R, G, and B channels and each pixel of the image to be encrypted, and divide the image matrix according to the order of the preset image sub-block matrix and the size of the overlapping block to obtain a plurality of image sub-blocks;

[0010] S5: iterating the two-dimensional hyperchaotic system based on the constant positive Lyapunov exponent to obtain a chaotic sequence, and constructing a scrambling matrix and a diffusion matrix based on the chaotic sequence;

[0011] S6: Based on the scrambling matrix and the diffusion matrix, perform scrambling-diffusion-scrambling operations on all image sub-blocks obtained by segmentation one by one, combine all image sub-blocks that have undergone the scrambling-diffusion-scrambling operations to obtain an encrypted image matrix, and implement image encryption.

[0012] Optionally, the objective function of the first seed mapping is to find the minimum value of the seed mapping, which is expressed as:

[0013]

[0014] Where, is the objective function of the first seed mapping, is the first seed mapping;

[0015] The second seed mapping Any growth rate The composite function is greater than the preset threshold 1, and the expression is .

[0016] Optionally, the mathematical model for constructing the two-dimensional hyperchaotic system is:

[0017]

[0018] Where, It is The iteration value at the moment, It is The iteration value at the moment, It is Iterate value at a moment.

[0019] Optionally, the S3 is specifically:

[0020] The expression of Lyapunov exponent is introduced as:

[0021]

[0022] in, is the Lyapunov exponent of the chaotic system, , Representative Matrix The sth eigenvalue of the matrix The Jacobian matrix of the chaotic system from observation time 0 to N-1 The product is obtained, the expression is:

[0023]

[0024]

[0025] Where, is the Jacobian matrix of the chaotic system at time i, The observation state at this moment;

[0026] because and ,get 1, thus obtaining ,Therefore, the construction of a two-dimensional hyperchaotic system based on the constant positivity of the Lyapunov exponent is completed.

[0027] Optionally, the two-dimensional hyperchaotic system based on the constant positive Lyapunov exponent is iterated to obtain the chaotic sequence:

[0028] The SHA-256 hash algorithm is used to generate a hash value for the encrypted image, and the generated 256-bit hash value is divided into 32 8-bit decimal values ​​K={K1,K2…K 32}, then initialize the initial value and control parameters of the chaotic system based on K, record this state as the state S of the chaotic system, and iterate the two-dimensional hyperchaotic system based on the Lyapunov exponent constant positive based on the state S to obtain the chaotic sequence S X , S Y , where the expression of state S is:

[0029]

[0030] Where, is the initial value of the chaotic system, is the control parameter.

[0031] Optionally, the scrambling matrix is ​​constructed as follows:

[0032] The chaotic sequence S X The randomness of and the orthogonal properties of the Latin square matrix are combined, and the first, second and third scrambling matrices are determined according to the image sub-block size S1×S1, which are L1, L2, and L3 respectively:

[0033]

[0034] Where, , is the modulo function.

[0035] Optionally, the diffusion matrix is ​​constructed as follows:

[0036] For the chaotic sequence S X , S Y Perform XOR operation to obtain the chaotic sequence S XY :

[0037]

[0038] Where, is the exclusive OR operation;

[0039] The chaotic sequence S X , S Y , S XY The elements in are reconstructed into the first, second and third diffusion matrices after integer definition, which are L4, L5 and L6 respectively:

[0040]

[0041]

[0042] In the formula, the diffusion matrix range is [0,255], and the reshape function reconstructs the one-dimensional sequence into The matrix of .

[0043] Optionally, the scrambling-diffusion-scrambling operation is specifically to first perform a first scrambling operation on the image sub-block to be processed, then perform a diffusion operation on the image sub-block after the first scrambling operation, and finally perform a second scrambling operation on the image sub-block after the diffusion operation, wherein the first scrambling operation is specifically cross-plane scrambling:

[0044] By comparing the element values ​​of the first scrambling matrix L1 and the second scrambling matrix L2, the pixel values ​​of the R channel and the G channel of the image sub-block are exchanged;

[0045] By comparing the element values ​​of the second scrambling matrix L2 and the third scrambling matrix L3, the pixel values ​​of the G channel and the B channel of the image sub-block are exchanged;

[0046] By comparing the element values ​​of the third scrambling matrix L3 and the first scrambling matrix L1, the pixel values ​​of the B channel and the R channel of the image sub-block are exchanged;

[0047] The diffusion operation is specifically plane internal diffusion:

[0048] Performing an exclusive OR operation on the first, second, and third diffusion matrices L4, L5, and L6 and the R, G, and B channels of the image sub-block respectively to complete the change of pixel information of the image sub-block, wherein the exclusive OR operation is a bitwise operation;

[0049] The second scrambling operation is specifically a plane internal scrambling:

[0050] In the R channel, the elements of the corresponding positions in the first scrambling matrix L1 and the second scrambling matrix L2 are combined as coordinates, and the R channel of the original image is rearranged to generate a new R channel image;

[0051] In the G channel, the elements of the corresponding positions in the second scrambling matrix L2 and the third scrambling matrix L3 are combined as coordinates, and the G channel of the original image is rearranged to generate a new G channel image;

[0052] In the B channel, the elements at corresponding positions in the third scrambling matrix L3 and the first scrambling matrix L1 are combined as coordinates, and the B channel of the original image is rearranged to generate a new B channel image.

[0053] Optionally, the performing the scrambling-diffusion-scrambling operations one by one is specifically:

[0054] During the movement of the image sub-block, the entire image to be processed is traversed in a left-to-right and top-to-bottom order. When the image sub-block approaches the boundary of the image to be processed, its position is dynamically adjusted so that the boundaries of the scrambling matrix and the diffusion matrix are aligned with the boundary of the image to be processed, ensuring that all pixels can fully participate in the scrambling-diffusion-scrambling operation.

[0055] The beneficial effects of the present invention are:

[0056] (1) The method for constructing a two-dimensional hyperchaotic system with a constantly positive Lyapunov exponent provided by the present invention can not only generate a large number of two-dimensional hyperchaotic maps by replacing different seed mappings, but also generate a hyperchaotic system with high complexity.

[0057] (2) This paper conducts a theoretical analysis of the construction framework of a two-dimensional hyperchaotic system based on the Lyapunov exponent, and proves that the system generated by the framework always has hyperchaotic characteristics.

[0058] (3) The two-dimensional hyperchaotic mapping generated by the method for constructing a two-dimensional hyperchaotic system based on Lyapunov exponents provided by the present invention consistently exhibits positive Lyapunov exponents throughout the entire parameter space, and exhibits uniform phase space distribution and high complexity. NIST testing has shown that the pseudorandom sequences generated by the novel mapping quantization method of the present invention exhibit excellent randomness.

[0059] (4) The present invention combines chaotic sequences with Latin square matrices to construct a cross-plane dynamic scrambling model and introduces a nonlinear coupling mechanism between the R, G, and B channels. By fusing cross-plane pixel permutation with an orthogonal Latin square matrix, the correlation between pixels is completely destroyed, improving the ability to resist statistical analysis attacks. In addition, the overlapping block strategy is adopted to further eliminate the inter-block correlation residue caused by fixed boundaries, thereby enhancing the diffusion characteristics and ensuring the effectiveness of the encryption method. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a structural schematic diagram of the present invention;

[0061] Figure 2 is the plane attractor phase diagram of the present invention; wherein, Figure 2 (a) is a phase diagram of 2D-LPI mapping according to an embodiment of the present invention, Figure 2 (b)- Figure 2 (d) Phase diagrams of existing two-dimensional mappings 2D-CLSS, 2D Hyperchaotic, and 2D-SIMM, respectively;

[0062] Figure 3 is the Lyapunov exponent spectrum of the present invention; wherein, Figure 3(a) is the Lyapunov exponent map of the 2D-LPI mapping according to an embodiment of the present invention, Figure 3 (b)- Figure 3 (d) Lyapunov exponent maps of existing two-dimensional mappings 2D-CLSS, 2D Hyperchaotic, and 2D-SIMM, respectively;

[0063] Figure 4 is a schematic diagram of the sample entropy and permutation entropy complexity of the present invention; wherein, Figure 4 (a) is a schematic diagram of sample entropy complexity of the 2D-LPI mapping according to an embodiment of the present invention and the existing two-dimensional mappings 2D-CLSS, 2D Hyperchaotic, and 2D-SIMM; Figure 4 (b) is a schematic diagram of the permutation entropy complexity of the 2D-LPI mapping according to an embodiment of the present invention and the existing two-dimensional mappings 2D-CLSS, 2D Hyperchaotic, and 2D-SIMM;

[0064] Figure 5 are the encryption and decryption results of the present invention, wherein, Figure 5 (a) is the plaintext image of image 4.2.03, Figure 5 (b) is the ciphertext image of image 4.2.03, Figure 5 (c) is the decrypted image of image 4.2.03, Figure 5 (d) is the plaintext image of image 4.2.06, Figure 5 (e) is the ciphertext image of image 4.2.06, Figure 5 (f) is the decrypted image of image 4.2.06, Figure 5 (g) is the plaintext image of image 4.2.07, Figure 5 (h) is the ciphertext image of image 4.2.07, Figure 5 (i) is the decrypted image of image 4.2.07;

[0065] Figure 6 This is an example diagram of the key sensitivity test of the present invention, wherein: Figure 6 (a) is the correct key, Figure 6 (b) For key a+10 -15 disturbance, Figure 6 (c) is the key b+10 -15 disturbance;

[0066] Figure 7 is the histogram comparison of the present invention, wherein, Figure 7 (a) is the plaintext image of image 4.2.03, Figure 7 (b) is the R channel histogram of the 4.2.03 plaintext image. Figure 7 (c) is the G channel histogram of the plaintext image 4.2.03. Figure 7(d) is the B channel histogram of the plaintext image 4.2.03. Figure 7 (e) is the ciphertext image of image 4.2.03, Figure 7 (f) is the R channel histogram of the 4.2.03 ciphertext image. Figure 7 (g) is the G channel histogram of the 4.2.03 ciphertext image. Figure 7 (h) is the B channel histogram of the 4.2.03 ciphertext image. DETAILED DESCRIPTION

[0067] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0068] Example 1: Figure 1 As shown in FIG, a structural diagram of an image encryption method based on a two-dimensional hyperchaotic system with a constant Lyapunov exponent is shown. The method includes two stages. The first stage is the construction of a two-dimensional hyperchaotic system based on a constant Lyapunov exponent, corresponding to S1-S3. The second stage is the image encryption based on a two-dimensional hyperchaotic system with a constant Lyapunov exponent, corresponding to S4-S6. Each stage will be further explained below.

[0069] S1: taking any one-dimensional chaotic map as a first seed map, and any composite function with a growth rate greater than a preset threshold as a second seed map, designing an objective function for the first seed map, solving the objective function, and obtaining a solution result for the first seed map;

[0070] S2: constructing a two-dimensional hyperchaotic system based on the first seed mapping, the second seed mapping, and the solution results of the first seed mapping;

[0071] S3: Introducing the Lyapunov exponent and constructing a two-dimensional hyperchaotic system based on the constant positivity of the Lyapunov exponent;

[0072] Specifically, the above is the first stage. In this embodiment, any existing one-dimensional chaotic map is combined with any composite function with a growth rate greater than 1, and transformed using modular operations and the objective function. This reduces the adverse effects of finite precision on the chaotic system, expands the parameter range of the chaotic system, improves the uniformity of the chaotic system, and ultimately enhances the security of the image encryption algorithm based on the chaotic system. Specifically, the chaotic system is constructed using the following formula:

[0073]

[0074] Where, is an arbitrary one-dimensional chaotic seed map, is any composite function with a growth rate greater than 1, To solve the domain The minimum value of the objective function, It is The iteration value at the moment, It is The iteration value at the moment, It is The iteration value at the moment, It is Iterate value at a moment.

[0075] Furthermore, the Lyapunov exponent is introduced, and the expression is:

[0076]

[0077] in, is the Lyapunov exponent of the chaotic system, , Representative Matrix The sth eigenvalue of the matrix The Jacobian matrix of the chaotic system from observation time 0 to N-1 The product is obtained, the expression is:

[0078]

[0079]

[0080] Where, is the Jacobian matrix of the chaotic system at time i, The observation state at this moment;

[0081] The following will further demonstrate that the system constructed in this embodiment always has hyperchaotic characteristics. Let:

[0082]

[0083] therefore:

[0084]

[0085]

[0086] thus, It can be described as:

[0087]

[0088]

[0089]

[0090]

[0091] By mathematical induction we can get:

[0092]

[0093] make:

[0094]

[0095]

[0096] Therefore, when N is an odd number:

[0097]

[0098] When N is an even number:

[0099]

[0100] The above inferences provide a theoretical analysis of the proposed chaotic system. The analysis results show that the number of positive Lyapunov exponents of the system is 2, which fully confirms that it has hyperchaotic characteristics.

[0101] Optionally, the first seed mapping can be any existing one-dimensional chaotic mapping, including but not limited to Logistic mapping, Tent mapping, Sine mapping, Chebyshev mapping and Gauss mapping; the second seed mapping satisfies When , it can be any composite function, including but not limited to a composite of polynomial functions, trigonometric functions, inverse trigonometric functions and the like; in this embodiment, a composite function consisting of a Logistic map and a polynomial-inverse trigonometric function is selected as a seed mapping to generate a new type of two-hyperchaotic mapping (2D-LPI) based on the Logistic map and the polynomial-inverse trigonometric composite function.

[0102] The following is a specific structure of the 2D-LPI mapping in this embodiment:

[0103] first, For the first seed mapping, we choose the Logistic mapping. A polynomial-inverse trigonometric composite function is selected for the second seed mapping, where the system equation and objective function of the logistic mapping are:

[0104]

[0105] in, is the control parameter, is the minimum value of the Sine map;

[0106] The iterative expression of the polynomial-inverse trigonometric composite function is:

[0107]

[0108] in, is a parameter;

[0109] Then, the above seed mapping and objective function value are coupled to the chaotic system expression, and the coupled expression is as follows:

[0110]

[0111] In this embodiment, the control parameters , initial value ;

[0112] Finally, to demonstrate the effectiveness of the present invention, a comparative analysis of chaotic systems of the same dimension as the present invention is provided. The three systems are a two-dimensional cross-mode hyperchaotic map (2D-CLSS) based on logic and sinusoidal mapping, a cross-two-dimensional hyperchaotic map (2D Hyperchaotic), and a two-dimensional modulation map (2D-SIMM) based on a sinusoidal feedback Sine map and infinite collapse iterative chaotic mapping. This example uses multiple performance indicators for comparative analysis, with the 2D-LPI performance demonstrating greater complexity. Its performance evaluation includes attractor phase diagrams, Lyapunov exponent spectrum (LE), sample entropy (SE), and permutation entropy (PE). Detailed information on the three existing chaotic maps is shown in Table 1.

[0113] Table 1 Three existing two-dimensional chaotic maps

[0114]

[0115] Further, Figure 2 (a)-(d) are the xy-plane attractor phase diagrams of 2D-LPI, 2D-CLSS, 2D Hyperchaotic, and 2D-SIMM, respectively, of the embodiments of the present invention, where the control parameters of the 2D-LPI mapping are =80, b=10, control parameters of 2D-CLSS mapping =0.5, control parameters of 2D Hyperchaotic mapping =2, b=1, control parameters of 2D-SIMM mapping =2, b=π, c=0.5; obviously, the trajectory of the present invention occupies the entire parameter space and is more uniform, indicating that 2D-LPI has richer dynamic behavior.

[0116] Further, Figure 3(a)-(d) Comparison of Lyapunov exponents (LE) of 2D-LPI, 2D-CLSS, 2D Hyperchaotic, and 2D-SIMM, respectively (normalized to different mapping parameter ranges). ); Among them, the control parameters of 2D-LPI , control parameters of 2D-CLSS , control parameters of 2D hyperchaotic , control parameters of 2D-SIMM ,Depend on Figure 3 (a) Analysis shows that within the entire parameter space, the 2D-LPI always maintains a hyperchaotic state and does not experience hyperchaotic degradation. In addition, the value of LE is positively correlated with the control parameter. When the control parameter is increased, a larger value is obtained, and the system exhibits more complex dynamic behavior. Figure 4 To compare the complexity of 2D-LPI with 2D-CLSS, 2D Hyperchaotic, and 2D-SIMM, this example used sample entropy (SE) and permutation entropy (PE) to measure data complexity. The results show that 2D-LPI consistently maintains a higher complexity. Therefore, 2D-LPI exhibits more complex dynamic behavior.

[0117] In order to verify the randomness of the sequence generated by the proposed two-dimensional hyperchaotic system for image encryption, the randomness test suite NIST SP800-22 is used to test the chaotic sequence. This test suite includes fifteen sub-tests, and the uniformity test for each test is 𝑃_𝑣𝑎l𝑢𝑒> When the pass rate (PR) is within the confidence interval, it is considered to have passed the randomness test. The confidence interval of the pass rate is defined as:

[0118]

[0119] in, , is the significance level, and m is the number of test sequence groups. When m=100, When , it is required that 𝑃_𝑣𝑎l𝑢𝑒>0.01 and PR∈[0.96,1].

[0120] Furthermore, the 2D-LPI is iterated in state S Then, the chaotic sequence x n and y n The middle section is located The sequence is converted into a binary sequence to obtain a pseudo-random number by the following formula and ,Finally, the obtained pseudo-random number is subjected to NIST SP800-22 test.

[0121]

[0122]

[0123] in, is the floor function, is the modulo function.

[0124] Furthermore, in this test, = 0.01, m = 100, and each sequence bit length is 1,000,000 bits. The test results are shown in Table 2, where the data are the averages of each test. Table 2 clearly shows that the 𝑃_𝑣𝑎l𝑢𝑒 and PR for each test are greater than 0.01 and 0.96, respectively, indicating that the random numbers generated based on the 2D-LPI chaotic system pass the test, further demonstrating the applicability of this chaotic system in the field of cryptography.

[0125] Table 2 NIST SP800-22 test results

[0126]

[0127] Based on the first stage, the second stage is to perform image encryption based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent. This method fully utilizes the advantages of chaotic systems such as extreme sensitivity to initial values ​​and unpredictability, while effectively avoiding the disadvantages of commonly used chaotic systems such as sensitivity to finite precision effects, narrow chaotic parameter range, and poor uniformity. This improves the security of image encryption algorithms built based on chaotic systems, enhances the randomness of encrypted images, and improves the image encryption system's ability to resist known plaintext attacks, chosen plaintext attacks, known ciphertext attacks, and chosen ciphertext attacks. The following are the specific steps of the second stage:

[0128] S4: Obtain an image matrix composed of the R, G, and B channels and each pixel of the image to be encrypted, and divide the image matrix according to the order of the preset image sub-block matrix and the size of the overlapping block to obtain a plurality of image sub-blocks;

[0129] S5: iterating the two-dimensional hyperchaotic system based on the constant positive Lyapunov exponent to obtain a chaotic sequence, and constructing a scrambling matrix and a diffusion matrix based on the chaotic sequence;

[0130] S6: Based on the scrambling matrix and the diffusion matrix, perform scrambling-diffusion-scrambling operations on all image sub-blocks obtained by segmentation one by one, combine all image sub-blocks that have undergone the scrambling-diffusion-scrambling operations to obtain an encrypted image matrix, and implement image encryption.

[0131] Specifically, in this embodiment, the width of the image to be encrypted is defined as W, the height is defined as H, the image sub-block size is S1×S1, and the overlap size along the x-axis and y-axis is t x , t y ,in , the number of blocks along the x-axis and y-axis are N respectively x , N y :

[0132]

[0133] Among them, ceil is the upward rounding function, Is a prime number and its value is greater than the number of channels;

[0134] Furthermore, the SHA-256 hash algorithm is used to generate a hash value for the encrypted image, and the generated 256-bit hash value is divided into 32 8-bit decimal values ​​K={K1, K2…K 32}, and then initialize the initial value of the chaotic system based on K and control parameters , this state is recorded as the state S of the chaotic system, and based on the state S, the two-dimensional hyperchaotic system based on the constant positive Lyapunov exponent is iterated to obtain the chaotic sequence S X , S Y , where the expression of state S is:

[0135]

[0136] Where, is the initial value of the chaotic system, is the control parameter.

[0137] Specifically, this embodiment utilizes the SHA-256 hash algorithm, a hash function with strong collision resistance (it's difficult for different inputs to produce the same output). This makes it difficult for an attacker to reverse engineer the complete key, even if they know partial key information, thereby enhancing the system's anti-attack capabilities. Furthermore, this embodiment converts the original image input into the initial value of the chaotic system, ensuring that the same key always maps to the same initial state, guaranteeing reversibility of encryption and decryption. Furthermore, different plaintext images correspond to different initial chaotic states, preventing the generation of regularities when the same key is used to encrypt different images.

[0138] Specifically, the segmentation is as follows: multiple image sub-blocks are obtained based on the original image segmentation, where the matrix order of each sub-block is selected as a prime number, and the matrix order is greater than the number of channels. This design ensures the mathematical specificity of the sub-blocks and is suitable for multi-channel image processing. During the sliding process, there is a certain overlap between adjacent sub-blocks, that is, the current sub-block and the next sub-block share a part of the pixels. The role is to enhance the global scrambling-diffusion-scrambling process, so that the pixel transformation is not limited to a single sub-block, but can also be diffused across the entire image. This mechanism ensures the effective propagation of pixel information between different sub-blocks, thereby enhancing the encryption effect, improving security, and reducing the risk of possible cropping attacks.

[0139] Furthermore, the initial value { , , , } Bring the 2D-LPI system into iterative (S1×S1+3000) times. To avoid transient errors, discard the first 3000 results and finally obtain the chaotic sequence A with a length of S1×S1 X , A Y , A XY , where A X , A Y Store the chaotic sequences of x and y dimensions respectively, A XY By chaotic sequence A X , A Y Obtained through XOR calculation.

[0140] Further, a scrambling matrix is ​​determined to scramble the image sub-blocks;

[0141] First, intercept the chaotic sequence A X The first S1 times, sort in descending order to get the index value SA X Next, the Latin square matrices L1, L2, and L3 are calculated using the following formula:

[0142]

[0143] in, , A X The jth element in A X The ordering of the entire sequence, L1, L2, L3 is mutually orthogonal, and the element values ​​in the matrix are all in [1, S1];

[0144] Then, use L1, L2, and L3 to randomly scramble the pixels in the image sub-block across the plane:

[0145]

[0146]

[0147]

[0148] in, , the number of blocks along the x and y axes for the three channels R, G, and B are , , and Respectively represent the first Pixels on the block The values, R1, G1, B1, represent the R channel, G channel, and B channel after cross-plane scrambling.

[0149] Furthermore, according to the chaotic sequence A X , A Y , A XY Determine the diffusion matrix and diffuse the image sub-blocks;

[0150] First, the chaotic sequence A X , A Y , A XY The element value is defined as an integer, and the value range is :

[0151]

[0152] Then, the one-dimensional chaotic sequence A X , A Y , A XY Reconstruct to match the spatial structure of the image data:

[0153]

[0154] Among them, the reshape function converts the one-dimensional sequence into , L4, L5, L6 are the first, second and third diffusion matrices;

[0155] Finally, the diffusion matrix is ​​used to change the pixel information of the image sub-block. The calculation process mainly uses the exclusive OR operation, which belongs to the category of bit operations. It can enhance the dependence between each plane of the image and the diffusion matrix, thereby enhancing the encryption strength:

[0156]

[0157] Among them, R2, G2, and B2 represent the R channel, G channel, and B channel after diffusion.

[0158] Furthermore, scrambling is performed within the three channels using scrambling matrices L1, L2, and L3. Due to the orthogonal nature of L1, L2, and L3, the index table formed by any two scrambling matrices does not contain duplicate values, thus ensuring that each pixel is fully rearranged during the scrambling process. This method effectively improves the uniformity and unpredictability of pixel scrambling, providing higher security for image encryption:

[0159]

[0160] Among them, R3, G3, B3 represent the R channel, G channel and B channel after the second scrambling.

[0161] This embodiment utilizes a cross-combination of multiple scrambling matrices to enhance the globality and unpredictability of pixel scrambling. Furthermore, this method boasts high computational efficiency and can achieve complex and uniform pixel rearrangement without compromising the structural integrity of the image, making it difficult to recover the original information from the encrypted image through statistical analysis.

[0162] Furthermore, a scrambling-diffusion-scrambling operation is performed on each segmented image sub-block. Specifically, as the image sub-block moves, it traverses the entire image from left to right and from top to bottom. When the image sub-block approaches the boundary of the image, its position is dynamically adjusted to align the boundaries of the scrambling matrix and the diffusion matrix with the boundary of the image, ensuring that all pixels can fully participate in the scrambling-diffusion-scrambling operation. This method not only enhances the generalization capability of the encryption scheme, enabling encryption of images of various sizes, but also ensures global consistency of the encryption process, avoiding information loss in edge areas.

[0163] In this embodiment, the following formula describes or Adjustment method of B plane:

[0164]

[0165] Among them, end is the end position of the matrix, Indicates an offset from the last column of the matrix , and then move i units to the right. Indicates a forward shift from the last column of the matrix , and then move j units to the right.

[0166] Specifically, the encryption method in this embodiment is symmetric encryption, and the decrypted image can be obtained by performing the reverse process of encryption. The present invention is further illustrated by specific experiments below.

[0167] In this embodiment, Matlab 2019a is used to conduct simulation experiments and analyze the results. The experiment selects three 512×512 color images from the University of Southern California image database, namely 4.2.03, 4.2.06, and 4.2.07. Figure 5 (a)-(c) are the original image, encrypted image and decrypted image of 4.2.03 respectively; (d)-(f) are the original image, encrypted image and decrypted image of 4.2.06 respectively; (g)-(i) are the original image, encrypted image and decrypted image of 4.2.07 respectively; Figure 5 As shown in the figure, the ciphertext image loses its visual meaning and the image information cannot be directly obtained. At the same time, the decrypted image can completely restore the plaintext image information, demonstrating a good encryption and decryption effect.

[0168] In order to effectively resist brute force attacks, relevant research points out that the key space should be greater than 2 100 The key of the present invention is generated by SHA-256 algorithm, and the key space size is 2 256 , much larger than 2 100 Can effectively resist brute force attacks.

[0169] To test the sensitivity of the key, Figure 6 As shown, (b)-(c) are respectively the use of the correct key { , , , }Applied 10 -15 Decryption results of the wrong key after perturbation. The results show that the key structure has good sensitivity and the image cannot be successfully decrypted even if the wrong key differs slightly from the correct key.

[0170] Figure 7 The pixel value distribution of different channels in the plaintext image and the ciphertext image is shown in Figure 2. It can be seen that the pixel values ​​of the plaintext image are unevenly distributed, while those of the ciphertext image are evenly distributed. A uniform pixel value distribution helps to hide information in the image.

[0171] In order to quantitatively analyze the distribution of different pixel values ​​in the ciphertext image, we use The ciphertext image is analyzed by the test. The following formula is Calculation formula, the smaller the calculation result, the more uniform the pixel value distribution. Under the condition of significance level α=0.05, when the calculation result is less than the critical value 293.2478, the image can be considered to have passed test.

[0172]

[0173] in, is the frequency of occurrence of gray value i, is the expected frequency.

[0174] As shown in Table 3, each channel of the experimental image has passed The test shows that the pixel values ​​of the ciphertext image are evenly distributed and have good security.

[0175] Table 3 Test results

[0176]

[0177] Furthermore, the correlation analysis of adjacent pixels is an important indicator for evaluating the performance of image encryption schemes. In this example, 10,000 pairs of adjacent pixels were randomly selected in the horizontal, vertical, and diagonal directions of the plaintext image and the ciphertext image, and the correlation coefficient of the adjacent pixels was calculated using the following formula:

[0178]

[0179] in, is the correlation coefficient, x and y are the grayscale values ​​of the selected adjacent pixels, is the expected value, is the standard deviation.

[0180] The results are shown in Table 4. The correlation coefficients of each channel of the ciphertext image are close to 0, which effectively reduces the risk of being cracked due to redundant information.

[0181] Table 4 Correlation coefficient test results of adjacent pixels

[0182]

[0183] Furthermore, Shannon entropy is an effective method to measure the randomness and uncertainty of information, which is defined as follows: 8 For an image with gray levels, the theoretical value of information entropy is 8.

[0184]

[0185] in, Represented as pixel values ​​in the image The probability of occurrence, is the information entropy of the entire image.

[0186] The calculation results of Shannon entropy of each channel of the experimental image are shown in Table 5, which shows that the Shannon entropy of the ciphertext image is close to the ideal value of 8, effectively hiding the information contained in the image and reducing the risk of being cracked.

[0187] Table 5 Information entropy test results

[0188]

[0189] Furthermore, in order to analyze the ability of the proposed encryption scheme to resist differential attacks, two indicators, namely the pixel change rate (NPCR) and the unified change intensity (UACI), are introduced. For a 512×512 image, when the confidence level is 0.05, the significance threshold The confidence interval of UACI is 99.5893%. is (33.3730%, 33.5541%), when NPCR is greater than UACI is located in When , the encryption scheme is considered to have passed the NPCR and UACI tests and is able to resist differential attacks.

[0190]

[0191]

[0192]

[0193] Where, and Images , In the image, the pixels are The pixel value at It is a binary discriminant matrix used to quantify the difference between two images (usually the ciphertext image before and after a slight modification of the plaintext) at the pixel position. The difference between is the number of pixels. The pixel information distribution of images Q1 and Q2 differs by only 1 bit.

[0194] Table 6 shows the NPCR and UACI test results for the ciphertext image obtained by modifying the pixel information at the (1, 1) position of the R component of the experimental image by 1 bit. As can be seen, even with only a 1-bit pixel change in one channel, all channels of the experimental image pass the test, effectively resisting differential attacks.

[0195] Table 6 NPCR and UACI test results

[0196]

[0197] The above describes the specific embodiments of the present invention in detail with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Various changes can be made within the knowledge of ordinary technicians in this field without departing from the scope of the present invention.

Claims

1. An image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent, characterized in that: The following steps are involved: S1: taking any one-dimensional chaotic map as a first seed map, and any composite function with a growth rate greater than a preset threshold as a second seed map, designing an objective function for the first seed map, solving the objective function, and obtaining a solution result for the first seed map; S2: constructing a two-dimensional hyperchaotic system based on the first seed mapping, the second seed mapping, and the solution results of the first seed mapping; S3: Introducing the Lyapunov exponent and constructing a two-dimensional hyperchaotic system based on the constant positivity of the Lyapunov exponent; S4: Obtain an image matrix composed of the R, G, and B channels and each pixel of the image to be encrypted, and divide the image matrix according to the order of the preset image sub-block matrix and the size of the overlapping block to obtain a plurality of image sub-blocks; S5: iterating the two-dimensional hyperchaotic system based on the constant positive Lyapunov exponent to obtain a chaotic sequence, and constructing a scrambling matrix and a diffusion matrix based on the chaotic sequence; S6: Based on the scrambling matrix and the diffusion matrix, perform a scrambling-diffusion-scrambling operation on all the image sub-blocks obtained by segmentation one by one, combine all the image sub-blocks that have undergone the scrambling-diffusion-scrambling operation to obtain an encrypted image matrix, thereby achieving image encryption; The objective function of the first seed mapping is to find the minimum value of the seed mapping, which is expressed as: ； Where, is the objective function of the first seed mapping, is the first seed mapping; The second seed mapping Any growth rate The composite function is greater than the preset threshold 1, and the expression is ; The mathematical model for constructing the two-dimensional hyperchaotic system is: ； Where, It is The iteration value at the moment, It is The iteration value at the moment, It is Moment iteration value; The S3 is specifically: The expression of Lyapunov exponent is introduced as: ； in, is the Lyapunov exponent of the chaotic system, , Representative Matrix The sth eigenvalue of the matrix The Jacobian matrix of the chaotic system from observation time 0 to N-1 The product is obtained, the expression is: ； ； Where, is the Jacobian matrix of the chaotic system at time i, The observation state at this moment; because and ,get 1, thus obtaining ,Therefore, the construction of a two-dimensional hyperchaotic system based on the constant positivity of the Lyapunov exponent is completed.

2. The image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent according to claim 1, characterized in that: Iterating the two-dimensional hyperchaotic system based on the constant positive Lyapunov exponent, the chaotic sequence is obtained as follows: The SHA-256 hash algorithm is used to generate a hash value for the encrypted image, and the generated 256-bit hash value is divided into 32 8-bit decimal values ​​K={K1,K2…K 32 }, then initialize the initial value and control parameters of the chaotic system based on K, record this state as the state S of the chaotic system, and iterate the two-dimensional hyperchaotic system based on the Lyapunov exponent constant positive based on the state S to obtain the chaotic sequence S X , S Y , where the expression of state S is: ； Where, is the initial value of the chaotic system, is the control parameter.

3. The image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent according to claim 1, characterized in that: The construction of the scrambling matrix is ​​specifically as follows: The chaotic sequence S X The randomness of and the orthogonal properties of the Latin square matrix are combined, and the first, second and third scrambling matrices are determined according to the image sub-block size S1×S1, which are L1, L2, and L3 respectively: ； Where, , is the modulo function.

4. The image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent according to claim 1, characterized in that: The construction of the diffusion matrix is ​​specifically as follows: For the chaotic sequence S X , S Y Perform XOR operation to obtain the chaotic sequence S XY : ； Where, is the exclusive OR operation; The chaotic sequence S X , S Y , S XY The elements in are reconstructed into the first, second and third diffusion matrices after integer definition, which are L4, L5 and L6 respectively: ； ； In the formula, the diffusion matrix range is [0,255], and the reshape function reconstructs the one-dimensional sequence into The matrix of .

5. The image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent according to claim 1, characterized in that: The scrambling-diffusion-scrambling operation is specifically to first perform a first scrambling operation on the image sub-block to be processed, then perform a diffusion operation on the image sub-block after the first scrambling operation, and finally perform a second scrambling operation on the image sub-block after the diffusion operation. The first scrambling operation is specifically cross-plane scrambling: By comparing the element values ​​of the first scrambling matrix L1 and the second scrambling matrix L2, the pixel values ​​of the R channel and the G channel of the image sub-block are exchanged; By comparing the element values ​​of the second scrambling matrix L2 and the third scrambling matrix L3, the pixel values ​​of the G channel and the B channel of the image sub-block are exchanged; By comparing the element values ​​of the third scrambling matrix L3 and the first scrambling matrix L1, the pixel values ​​of the B channel and the R channel of the image sub-block are exchanged; The diffusion operation is specifically plane internal diffusion: Performing an exclusive OR operation on the first, second, and third diffusion matrices L4, L5, and L6 and the R, G, and B channels of the image sub-block respectively to complete the change of pixel information of the image sub-block, wherein the exclusive OR operation is a bitwise operation; The second scrambling operation is specifically a plane internal scrambling: In the R channel, the elements of the corresponding positions in the first scrambling matrix L1 and the second scrambling matrix L2 are combined as coordinates, and the R channel of the original image is rearranged to generate a new R channel image; In the G channel, the elements at corresponding positions in the second scrambling matrix L2 and the third scrambling matrix L3 are combined as coordinates, and the G channel of the original image is rearranged to generate a new G channel image; In the B channel, the elements at corresponding positions in the third scrambling matrix L3 and the first scrambling matrix L1 are combined as coordinates, and the B channel of the original image is rearranged to generate a new B channel image.

6. The image encryption method based on a two-dimensional hyperchaotic system with a constant positive Lyapunov exponent according to claim 1, characterized in that: The specific steps of performing scrambling-diffusion-scrambling operations one by one are as follows: During the movement of the image sub-block, the entire image to be processed is traversed in a left-to-right and top-to-bottom order. When the image sub-block approaches the boundary of the image to be processed, its position is dynamically adjusted so that the boundaries of the scrambling matrix and the diffusion matrix are aligned with the boundary of the image to be processed, ensuring that all pixels can fully participate in the scrambling-diffusion-scrambling operation.

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