Chaotic image encryption method based on multi-cavity attractor features

Abstract

The invention relates to the technical field of image encryption, in particular to a chaotic image encryption method based on multi-cavity attractor features, which comprises the following steps: constructing a TDHNN model by using an activation function and a toroidal surface parameter equation; an MC-TDHNN model based on a TDHNN model is constructed by using a multi-stage step state extension method; and generating a chaos sequence by using the MC-TDHNN model, and performing scrambling operation and diffusion operation on a plaintext image by using the chaos sequence to realize plaintext image encryption. The method solves the problem that the conventional encryption means is not tightly combined with the chaotic system, the inherent randomness of the chaotic system is not fully utilized, and the encryption effect is limited.

Description

Technical Field

[0001] The present invention relates to the technical field of image encryption, and particularly to a chaotic image encryption method based on the characteristics of multi-cavity attractors. Background Art

[0002] As an intuitive and efficient information transmission method, images are becoming increasingly important; chaotic systems have become a research hotspot in the field of image encryption due to their global stability, pseudo-randomness, unpredictability, and initial value sensitivity.

[0003] The existing HNN chaotic image encryption algorithm has been increasingly applied in image encryption research due to its non-linearity and associative memory ability; for example, in the literature "Chaotic image encryption with Hopfield neural network", a fractional-order HNN is used as a key stream generator to develop an image cryptosystem based on FO-HNN; in the literature "Multidirectional Multidouble-Scroll Hopfield Neural Network With Application to Image Encryption", a multi-directional double-scroll HNN is constructed by introducing memristive synapses, and an image encryption scheme based on homogeneous chaotic sequences is designed; however, although these methods all use the pseudo-random sequences generated by system equations as the input of the encryption algorithm and operate on them; the existing improved HNNs (such as DHNN) have problems such as periodicity and insufficient complexity in the generated random sequences due to the use of simple non-linear activation functions;

[0004] In addition, the above methods do not combine conventional encryption means with chaotic systems closely enough, and do not fully utilize the inherent randomness of chaotic systems, resulting in limited encryption effects; therefore, how to more deeply integrate the dynamic characteristics of chaotic systems has become a key research direction for further improving the security and reliability of image encryption. Summary of the Invention

[0005] Aiming at the deficiencies of the existing methods, the present invention constructs a chaotic image encryption model with the characteristics of multi-cavity attractors, introduces the concept of direction index, closely associates the generation of chaotic sequences with the initial value offset characteristics of multi-cavity attractors of chaotic equations, and constructs a unique encryption mechanism.

[0006] The technical solution adopted by the present invention is: a chaotic image encryption method based on the characteristics of multi-cavity attractors includes the following steps:

[0007] Step 1: Construct a TDHNN model by using an activation function and a torus parametric equation;

[0008] As a preferred embodiment of the present invention, the formula of the TDHNN model is:

[0009]

[0010] where μ is the decay factor of the neuron; k is the coupling connection weight; R, r, and m are the parameter variables of the one-dimensional torus parametric equation; x n and y n are the n-th iteration values.

[0011] Step 2: Construct an MC-TDHNN model based on the TDHNN model using the multi-level step state expansion method;

[0012] As a preferred embodiment of the present invention, the multi-level step state expansion method is to add controller variables u n and v n .

[0013] Step 3: Use the MC-TDHNN model to generate a chaotic sequence, and use the chaotic sequence to perform scrambling and diffusion operations on the plaintext image to obtain an encrypted plaintext image;

[0014] As a preferred embodiment of the present invention, the improved scrambling operation includes: determining the number of pixel indices using the control parameters of the multi-level step function, and realizing pixel exchange between the RGB planes and channels.

[0015] As a preferred embodiment of the present invention, the improved scrambling operation specifically includes:

[0016] Step 31: Use the MC-TDHNN model to iteratively generate one-dimensional chaotic sequences U and V; expand the chaotic sequences U and V into an initial direction index matrix U1 and an initial step matrix V1 that are the same size as the plaintext image;

[0017] Step 32: Perform expansion and modulo operations on each data in U1 and V1, and divide them into a direction index matrix E and a step index matrix F at the color plane and pixel levels;

[0018] Step 33: Determine the new pixels to be permuted between the same color layers, and define the horizontal shift weight D x , the vertical shift weight D y direction matrix directions; calculate the coordinates of the new permutation position, and the formula is:

[0019]

[0020] where I x , I y are the coordinates of the position originally prepared for permutation; S is the moving step, and mod is the modulo; M’ 、N ’ is the length and width of the input plaintext;

[0021] Step 34: Determine the new pixels to be permuted between different color layers, and shift the pixels at the corresponding positions between all layers together each time.

[0022] Step 35: Exchange the data at the old and new positions to complete one scrambling operation.

[0023] Step 36: Repeat Steps 31 - 35 to traverse all the input plaintext data until the data exchange is completed at the last position.

[0024] As a preferred embodiment of the present invention, the improved diffusion operation includes: combining the initial value offset characteristic of the MC - TDHNN model with the plane direction and completing the diffusion with the idea of DNA bit - by - bit decomposition.

[0025] As a preferred embodiment of the present invention, the improved diffusion operation specifically includes:

[0026] Step 41: Vectorize the direction index matrix and the step - size index matrix to obtain the corresponding sequences E1 and F1.

[0027] Step 42: Calculate the current iteration controller variable using the previous iteration controller variable of the MC - TDHNN model and the horizontal and vertical shift weights. The formula is:

[0028]

[0029] where a is the perturbation factor of the initial value iteration;

[0030] Step 43: Set the total number of pixels of the plaintext image as the number of iterations. When iterating, determine the initial value of the coordinates (u, v) according to the value of the index sequence E1; generate the final chaotic sequences U ’ and V ’ ;

[0031] Step 44: Obtain the R, G, and B color channel values of the plaintext image; according to the hierarchical order of the RGB three color channels, sequentially use R, G, and B as the inputs of the diffusion algorithm.

[0032] As a preferred embodiment of the present invention, taking the R channel as the first - channel input of the diffusion algorithm, the input order of the channels is R, G, B, and the corresponding formula of the diffusion algorithm is:

[0033]

[0034] where ⊕ represents bit - by - bit exclusive OR.

[0035] As a preferred embodiment of the present invention, a chaotic image encryption system based on multi-cavity attractor features includes: a memory for storing instructions executable by a processor; and a processor for executing the instructions to implement a chaotic image encryption method based on multi-cavity attractor features.

[0036] As a preferred embodiment of the present invention, a computer-readable medium storing computer program code, the computer program code implementing a chaotic image encryption method based on multi-cavity attractor features when executed by a processor.

[0037] Advantages of the present invention:

[0038] 1. The present invention performs multi-level step state expansion on the TDHNN seed mapping to obtain a multi-cavity chaotic mapping, and designs a general scrambling algorithm applicable to the multi-cavity chaotic system by using its high-performance chaotic sequence and the control parameters of the multi-stage step function; as the control parameters of the multi-stage step function increase, the set upper limit of the direction index also increases, solving the problem of single traditional scrambling pixel exchange.

[0039] 2. The present invention closely associates the generation of the chaotic sequence with the initial value offset characteristics of the multi-cavity attractor of the chaotic equation, and proposes a general diffusion algorithm suitable for the multi-cavity chaotic system; different from traditional encryption algorithms that simply introduce the chaotic sequence generated by the chaotic system, this association introduces more complex dynamic behaviors, making the chaotic characteristics in the encryption process richer and more difficult to predict, thus significantly improving the security and complexity of the encryption algorithm. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 is a flowchart of the chaotic image encryption method based on multi-cavity attractor features of the present invention;

[0041] Figure 2 shows the influence of different parameters of the present invention on the morphology of the torus activation function;

[0042] Figure 3 is the 5×5 grid multi-cavity attractor phase diagram of MC-TDHNN of the present invention;

[0043] Figure 4 shows the coexistence of the initial offset enhanced attractor plane of MC-TDHNN of the present invention;

[0044] Figure 5 is the direction index constraint diagram controlled by the multi-stage step function parameters of the present invention;

[0045] Figure 6 shows the 16 plane direction indexes and 2 spatial direction indexes introduced by the present invention;

[0046] Figure 7This is the motion path of the pixels of the present invention in different planar directions;

[0047] Figure 8 This is a schematic diagram of the spatial RGB level scrambling of the present invention;

[0048] Figure 9 This is the correspondence between the initial value of the initial value offset enhanced attractor of the present invention and the plane index;

[0049] Figure 10 This is a schematic diagram of the present invention using DNA bit-by-bit decomposition for data shifting, conversion, and combination. Detailed implementation manners

[0050] The present invention will be further described below in conjunction with the accompanying drawings and embodiments. This figure is a simplified schematic diagram, which only illustrates the basic structure of the present invention in a schematic manner. Therefore, it only shows the components related to the present invention.

[0051] As Figure 1 shown, a chaotic image encryption method based on multi-cavity attractor features includes the following steps:

[0052] Step 1: Construct a TDHNN model by using one dimension of the sin activation function and the torus parameter equation;

[0053] The system equation of the TDHNN model is as follows:

[0054]

[0055] where μ is the decay factor of the neuron; k is the coupling connection weight; R, r, and m are the parameter variables of the one-dimensional torus parameter equation; x n and y n are the nth iteration values.

[0056] Compared with the DHNN that uses simple tanh and sin functions as activation functions, the TDHNN has three adjustable parameters R, r, and m by introducing the ellipsoidal ring function, enabling flexible adjustment of the shape and behavior of the activation function; among the geometric parameters of the torus, R defines the radius of the central ring, and an increase in its value will cause a linear amplification of the overall size of the torus; when the half-width r of the change amplitude of the cross-sectional radius of the torus and the frequency factor m remain unchanged, the change in R directly reflects the overall scale change of the torus in the two-dimensional plane and is manifested as a corresponding adjustment of the function amplitude; when R and m are fixed, an increase in the value of r not only expands the range of change in the cross-sectional radius of the torus but also has a subtle impact on the frequency and amplitude of the waveform; m, as the frequency factor, controls the fluctuation frequency of the torus; when R and r are unchanged, the change in m significantly changes the waveform period of the torus, thereby significantly affecting the frequency of the wave; as Figure 2As shown, the three parameters R, r, and m precisely regulate the morphological characteristics of the ring from three aspects: overall scale, cross-sectional variation, and frequency, providing stronger flexibility and adaptability for the TDHNN.

[0057] Step 2: Use the multi-level step state expansion method to construct a multi-cavity chaotic map MC-TDHNN based on TDHNN, and the expression is:

[0058]

[0059] where, u n and v n are two controller variables; k1, k2 are scaling parameters; S(.) is a multi-level step function, and the mathematical model can be expressed as:

[0060]

[0061] where, M = 1, 2, 3,...; sgn(·) is the sign function, and S(ξ) can be set by controlling the parameter M; set (μ, k, R, r, m, k1, k2) = (0.5, 2, 11, 3, 5, 0.32, 0.15), and the control parameter M is set to 2, then a chaotic attractor with 5×5 multi-cavities can be constructed as Figure 3 shown.

[0062] Set (x0, y0) = (1, 1), (u0, v0) = (-4 / -2 / 0 / 2 / 4, -4 / -2 / 0 / 2 / 4), when (μ, a, R, r, m, k1, k2) = (0.5, 2, 11, 3, 5, 0.3, 0.13), 25 coexisting chaotic attractors are generated in the u-v phase plane; Figure 4 The planar coexistence of the initial offset enhanced attractor of MC-TDHNN is given.

[0063] Step 3: Use the control parameter M of the multi-level step function to determine the number of pixel indices. When M = 2, there are 16 direction indices in the plane and 2 direction indices between color levels, for a total of 18; realize pixel exchange between the RGB plane and channels;

[0064] When M = 1, a 3×3 multi-cavity attractor can be constructed based on MC-TDHNN, corresponding to the direction indices of the first layer around the central pixel; similarly, when M = 2, a 5×5 multi-cavity attractor can be formed, representing the direction indices of the second layer around the central pixel; when M = 3, a 7×7 multi-cavity attractor is generated, corresponding to the direction indices of the third layer around the central pixel, as Figure 5 shown.

[0065] All the direction indexes can be manually constructed to ensure that the central pixel has a unique direction in each index; as M increases, the number of constructible direction indexes also increases, resulting in more random pixel exchanges; in this embodiment, M = 2 is set, and a two-layer direction index is established around the central pixel. Under this constraint, there are a total of 16 different planar direction indexes.

[0066] The specific steps of Step 3 include:

[0067] Step 31: Use the iteration of MC-TDHNN to generate two one-dimensional chaotic sequences U and V; according to the size of the plaintext image, expand these two chaotic sequences into two matrices consistent with the image size, named the initial direction index matrix U1 and the initial step matrix V1 respectively;

[0068] Step 32: Multiply each data in the initial direction index matrix U1 by 2 14 times, and then limit it to the range of 1 to 18 through modulo operation; among them, 1 to 16 are used as the 16 direction indexes of the color plane, and 17 and 18 are used as the 2 direction indexes between pixel levels, for a total of 18 directions, as Figure 6 shown; after the above operations are completed, name the obtained new matrix the direction index matrix E; similarly, after expanding the data in the step matrix V1 and limiting it to the range from 1 to the maximum length and width of the plaintext image through modulo operation, obtain a new matrix named the step index matrix F;

[0069] Step 33: Determine the new pixels to be permuted between the same color layers, and define a 16-row and 2-column direction matrix directions; directions: (-1,1), (0,1), (1,1), (1,0), (1,-1), (0,-1), (-1,-1), (-1,0), (-1,2), (1,2), (2,1), (2,-1), (1,-2), (-1,-2), (-2,-1), (-2,1) respectively correspond to Figure 6 the direction indexes of plane parts 1 to 16, where the first column represents the horizontal shift weight D x and the second column represents the vertical shift weight D y , and the shift weight D x is determined by the direction index matrix E during each iteration; define S as the moving step, and S is determined by the value generated by the step index matrix F during each iteration; the formula for determining the new position is as follows:

[0070]

[0071] where, I x , I y are the original position coordinates to be permuted, I ’x ,I ’ y is the new position coordinate after replacement, M ’ 、N ’ is the length and width of the input plaintext; the remainder function is used to ensure that the coordinates of the new position are always within the valid range of the image matrix. When the coordinates of the original position move out of the valid range of the image for the first time, the remainder function will return this position to the other side of the image boundary, achieving a loop effect similar to that of a greedy snake, thereby simplifying the processing of the boundary.

[0072] like Figure 7 As shown in FIG. 1 , the original pixels of the 5×5 plaintext block move 4 steps to the new position under different direction indexes (1,1) and (1,2), respectively. The red pixels represent the original positions and the blue pixels represent the new positions.

[0073] Step 34: Determine the new pixels to be replaced between different color layers. Since color images only have three different color levels, R, G, and B, there are only two moving directions, up and down. The step length of each move is determined by the modulo 3 of the step length index matrix F. When the moving direction is up and down, the schematic diagram of the level scrambling is as follows: Figure 8 As shown, each movement will shift the pixels at corresponding positions between all layers together, and each 3 steps constitutes a cycle;

[0074] Step 35: swap the data at the new and old positions to complete a scrambling operation;

[0075] Step 36: Repeat steps 31 to 35 to traverse all input plaintext data until the data exchange is completed at the last position;

[0076] The essence of scrambling is to exchange pixels at different positions in the plaintext image. The farther the exchange distance and the more chaotic the order, the better the scrambling effect. Although conventional scrambling methods perform well in this regard, many methods increase the difficulty of implementation in pursuit of complexity, and fail to establish an effective connection with the random characteristics of chaotic systems. In contrast, the present invention cleverly combines the grid distribution of multi-cavity attractors, uses multi-level step functions to control the number of attractor plane layers, and introduces a large number of direction indexes based on the central pixel on this basis, and determines the moving step length by the chaotic sequence, which significantly enhances the randomness of pixel exchange. This method is not only simple to implement, but also has a better scrambling effect.

[0077] like Figure 10 ,Step 4, combine the initial value offset characteristics of the multi-cavity chaotic system with the plane direction, and use the idea of DNA bitwise decomposition to complete the diffusion;

[0078] Step 4 specifically includes:

[0079] Step 41: According to the iteration of the seed mapping TDHNN, scale the generated X-dimensional chaotic sequence to the range of 1 to 16 according to 16 plane direction indices as the direction index sequence E1; then vectorize the step matrix F obtained in the scrambling operation to get the step sequence F1; the index sequence E1 and the step sequence F1 are the one-dimensional values corresponding to the matrices E and F, respectively.

[0080] Step 42: Based on the seed mapping TDHNN, a two-dimensional multi-cavity model MC-TDHNN is constructed. Compared with the original system equation, two additional dimensions u and v are added to this system; according to the dynamic characteristics of the system, the initial value offset of the 5×5 multi-cavity system is regulated. Specifically, the initial values of u and v are selected as 0, ±2, ±4. During this process, 16 attractors corresponding one-to-one with the direction indices are selected. As Figure 9 shown, it is found by observation that there is a quantitative relationship with a multiple of 2 between these attractors and the shift weights of the directions. Therefore, a mathematical relationship can be constructed between the shift weights and the initial values of u and v. The mathematical relationship formula is as follows:

[0081]

[0082] where a = 0.1 is the perturbation factor for the initial value iteration, which is used to correlate the current initial values (u n , v n ) with the next iteration's initial values (u n+1 , v n+1 ), so that each initial value is related to the original initial value;

[0083] Using the direction matrix directions defined by the scrambling operation, determine the horizontal shift weight D x and the vertical shift weight D y ;

[0084] Step 43: Set the total number of pixels of the plaintext image as the number of iterations. For each iteration, determine the initial values of the coordinates (u, v) according to the corresponding values in the index sequence E1; then, apply MC-TDHNN to perform the iterative process; in this way, each iteration is based on the result of the previous iteration, thereby generating a chaotic sequence with high complexity and randomness; after completing all the predetermined number of iterations, the final chaotic sequences U ’ and V ’ are obtained;

[0085] Step 44: According to the hierarchical order of the three RGB color channels, sequentially use RGB as the input of the diffusion algorithm; specifically, the input of the first channel is P, the input of the second channel is PX, and the input of the third channel is PX ’; In this process, the R channel is used as the first-channel input of the diffusion algorithm, such that the input order of the channels is R, G, B. The complete diffusion algorithm is as follows:

[0086]

[0087] Where i represents the position of the element in the sequence, P i represents the i-th element in the first-channel input sequence, PX i represents the i-th element in the second-channel input sequence, PX ’ i represents the i-th element in the third-channel input sequence, U ’ i represents the i-th element in the chaotic sequence U ’ where V ’ i represents the i-th element in the final chaotic sequence V ’ where C i represents the i-th element of the forward diffusion output sequence C, T i+1 represents the (i + 1)-th element of the reverse diffusion output sequence T, F 1(i-1) represents the (i - 1)-th element in the step sequence F1; ⊕ represents bitwise exclusive OR; C and T are the forward diffusion output and reverse diffusion output of the encryption algorithm respectively; BP is a two-bit decomposition operation function that receives three numerical inputs in the range from 0 to 255 and a parameter f that affects the arrangement of bits in each value. The parameter f is taken from the values in the step sequence F1. By taking different multiples of the parameter f and taking the remainder when divided by 3, the BP function moves each group of decomposed bits respectively, achieving a pairwise exchange between the initial position and the moved position; this operation function uses methods such as bit splitting, step cyclic permutation, remainder operation wrapping, and recombined exclusive OR output to implement a method of converting and combining data. Taking f = 4 as an example, its operation process is as Figure 7 shown.

[0088] Take the G channel of the plaintext image as the first-channel input of the diffusion algorithm, with the channel input order being GBR, and perform a loop operation, repeating the diffusion algorithm of the previous step; subsequently, take the B channel of the plaintext image as the first-channel input of the diffusion algorithm, with the channel input order being BRG, perform the loop operation again, and repeat the diffusion algorithm of the previous step; after the above loop processing, the final output will form the encrypted image.

[0089] In sequence-based encryption algorithms, conventional methods usually directly use the pseudo-random sequence generated by a chaotic system that determines the initial value and parameters as the input for diffusion and make extensions based on this. However, this approach is only based on a single case, which limits the diversity and security of the algorithm. The diffusion method of the present invention closely associates the generation of chaotic sequences with the initial value offset characteristics of the multi-cavity attractor of the chaotic equation. Each initial value corresponds to a different chaotic sequence, thus greatly enriching the chaotic characteristics in the encryption process, making it more difficult to predict, and significantly enhancing the security of the algorithm. In addition, the selection of the initial value is also correlated with the direction index of the scrambling part, showing coherence and strong integrity in the process design and being easy to implement.

[0090] Inspired by the ideal embodiments of the present invention described above, through the above description, relevant staff can make various changes and modifications completely within the scope of not deviating from the technical idea of this invention. The technical scope of this invention is not limited to the content in the specification, and its technical scope must be determined according to the scope of the claims.

Claims

1. A chaotic image encryption method based on the characteristics of multi - cavity attractors, characterized in that, It includes the following steps: Step 1: Construct a TDHNN model by using an activation function and a torus parametric equation; Step 2: Construct an MC-TDHNN model based on the TDHNN model by using a multi-level step state expansion method; Step 3: Generate a chaotic sequence by using the MC-TDHNN model, and perform scrambling and diffusion operations on the plaintext image by using the chaotic sequence to obtain an encrypted plaintext image.

2. The chaotic image encryption method based on multi-cavity attractor features according to claim 1, characterized in that The formula of the TDHNN model is: where μ is the decay factor of the neuron; k is the coupling connection weight; R, r, and m are the parameter variables of the one-dimensional torus parametric equation; x n and y n are the n-th iteration values.

3. The chaotic image encryption method based on multi-cavity attractor features according to claim 1, characterized in that The improved scrambling operation includes: determining the number of pixel indexes by using the control parameter of the multi-level step function, and realizing pixel exchange between the RGB planes and channels.

4. The chaotic image encryption method based on multi-cavity attractor features according to claim 3, characterized in that, The improved scrambling operation specifically includes: Step 31: Iteratively generate one-dimensional chaotic sequences U and V by using the MC-TDHNN model; expand the chaotic sequences U and V into an initial direction index matrix U1 and an initial step matrix V1 with the same size as the plaintext image; Step 32: Perform expansion and modulo operations on the data in U1 and V1, and divide them into a direction index matrix E and a step index matrix F at the color plane and pixel levels; Step 33: Determine the new pixels to be replaced between the same color layers, and define the horizontal shift weight D x , the vertical shift weight D y direction matrix directions; calculate the coordinates of the new replacement position, and the formula is: Among them, I x , I y are the position coordinates originally prepared for replacement; S is the moving step length, and mod is the modulo operation; M ’ , N ’ are the length and width of the input plaintext; Step 34: Determine the new pixels to be permuted between different color layers, and shift the pixels at the corresponding positions between all layers together each time; Step 35: Exchange the data at the old and new positions to complete one scrambling operation; Step 36: Repeat steps 31-35, traverse all the input plaintext data until the data exchange is completed at the last position.

5. The chaotic image encryption method based on multi-cavity attractor features according to claim 1, wherein The improved diffusion operation includes: combining the initial value offset characteristic of the MC-TDHNN model with the plane direction, and completing diffusion by means of the idea of DNA bitwise decomposition.

6. The chaotic image encryption method based on multi-cavity attractor features according to claim 5, characterized in that The improved diffusion operation specifically includes: Step 41: Vectorize the direction index matrix and the step index matrix to obtain corresponding sequences E1 and F1; Step 42: Calculate the current iteration controller variable by using the previous iteration controller variable of the MC-TDHNN model and the horizontal and vertical shift weights, and the formula is: where a is the perturbation factor of the initial value iteration; Step 43: Set the total number of pixels of the plaintext image as the number of iterations. When iterating, determine the initial value of the coordinates (u, v) according to the value of the index sequence E1; generate the final chaotic sequences U ’ and V ’ ; Step 44: Obtain the R, G, and B color channel values of the plaintext image and store them in the color matrix PX; according to the hierarchical order of the RGB three color channels, sequentially use R, G, and B as the inputs of the diffusion algorithm.

7. The chaotic image encryption method based on multi-cavity attractor features according to claim 6, wherein Taking the R channel as the first channel input of the diffusion algorithm, the input order of the channels is R, G, B, and the corresponding formula of the diffusion algorithm is: where ⊕ represents bitwise exclusive OR.

8. The chaotic image encryption method based on multi-cavity attractor features according to claim 1, characterized in that The multi-level step state expansion method is to add controller variables u n and v n .

9. A chaotic image encryption system based on the characteristics of multi-cavity attractors, characterized in that It includes: A memory for storing instructions executable by a processor; A processor for executing the instructions to implement the chaotic image encryption method based on the multi-cavity attractor feature as described in any one of claims 1-8.

10. A computer-readable medium storing computer program code, characterized in that, The computer program code implements the chaotic image encryption method based on the multi-cavity attractor feature as described in any one of claims 1-8 when executed by the processor.

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