An image encryption and decryption method based on elastic sampling and model-free data driving

Abstract

The application belongs to the technical field of image processing and information security, and specifically discloses an image encryption and decryption method based on elastic sampling and model-free data driving. The method first establishes a driving model-free chaotic neural network and a response model-free chaotic neural network dynamics model. Then, an offline data sequence is collected, and a data-driven error representation is constructed. An elastic sampling strategy is designed according to the denial of service (DoS) attack characteristics, which actively stops the transmission of control input in the attack interval and restores the transmission of control input after the attack. Based on the data-driven error representation and the elastic sampling strategy, a data-driven closed-loop system model is obtained, and a data-driven attack resistance ratio is defined. Then, a data-driven criterion is established to ensure the synchronization of the driving model-free chaotic neural network and the response model-free chaotic neural network. Finally, the driving model-free chaotic neural network and the response model-free chaotic neural network are used to realize the encryption and decryption of insulator images.

Description

Technical Field

[0001] This invention belongs to the field of image processing and information security technology, specifically relating to an image encryption and decryption method based on elastic sampling and model-free data-driven methods. Background Technology

[0002] In the field of privacy protection, chaotic sequences generated by driving chaotic neural networks are used to scramble and propagate image pixel values. Successful decryption depends on the response chaotic neural network generating the same chaotic sequence, which requires precise synchronization between the driving and response chaotic neural networks. Meanwhile, in networked control systems, sensor measurements and control commands are transmitted via shared communication networks, offering advantages such as reduced wiring, lower installation and maintenance costs, and increased flexibility. However, this reliance on shared communication networks inevitably expands the attack surface, making the control system vulnerable to denial-of-service (DoS) attacks. DoS attacks intermittently interrupt control inputs by blocking the successful transmission of data packets, leading to system performance degradation and loss of closed-loop stability, and even catastrophic failures in safety-critical applications. Therefore, researchers have proposed various synchronization control strategies to ensure stable system operation and maintain control performance in the presence of DoS attacks.

[0003] However, existing synchronization control strategies heavily rely on precise knowledge of the system model. While they have achieved good results in power systems, multi-agent systems, and chaotic systems, they become inapplicable when the system matrix is ​​unknown or only partially known. In practical engineering, chaotic neural networks typically have complex and highly interconnected structures, making it difficult to obtain accurate mathematical models and extremely challenging to accurately identify all system parameters.

[0004] Data-driven techniques have garnered increasing attention for addressing control problems in model-free systems. In recent years, Lyapunov-based direct data-driven control has attracted widespread interest, achieving significant results in areas such as event-triggered control. However, most of these studies are based on model-free systems under ideal communication environments, and the issue of data-driven event-triggered synchronization control using chaotic neural networks has received little attention, especially regarding data-driven security in the presence of DoS attacks.

[0005] Most existing data-driven linear matrix inequality criteria are derived based on the matrix S-lemma, meaning that a model-based linear matrix inequality criterion must first be established before the data-driven criterion can be obtained through the S-lemma. This implies that the collected data sequences cannot be directly incorporated into the construction of the data-driven criterion, and the S-lemma imposes dimensional constraints on the model conditions. Furthermore, these methods are mainly applicable to discrete-time systems; for continuous-time systems, due to the quadratic form of the derivative of the Lyapunov function, existing methods are difficult to apply. Summary of the Invention

[0006] The purpose of this invention is to propose an image encryption and decryption method based on elastic sampling and model-free data-driven methods. This method constructs a data-driven closed-loop system through an elastic sampling strategy and data-driven criteria, and uses a driving model-free chaotic neural network and a response model-free chaotic neural network to encrypt and decrypt insulator images, so as to achieve secure encrypted transmission of images under DoS attack environment and lossless recovery of the original image at the receiving end.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: The image encryption and decryption method based on elastic sampling and model-free data-driven methods includes the following steps: Step 1. Establish dynamic models of the driving model-free chaotic neural network and the response model-free chaotic neural network; Step 2. Collect offline data sequences and construct a data-driven representation of the errors; Then, based on the characteristics of Denial-of-Service (DoS) attacks, an elastic sampling strategy is designed to actively stop the transmission of control input during the attack period and resume the transmission of control input after the attack ends. Based on error-driven data representation and elastic sampling strategy, a data-driven closed-loop system model is obtained. At the same time, the data-driven attack resistance ratio is defined to evaluate the attack resistance capability of the data-driven closed-loop system. Step 3. Establish data-driven criteria to ensure synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network; Step 4. Use a model-free chaotic neural network to generate an encryption key to encrypt the insulator image, thus obtaining an encrypted image; Step 5. Generate a decryption key using a response model-free chaotic neural network, and perform the inverse operation on the encrypted image to obtain the decrypted image.

[0008] Furthermore, based on the aforementioned image encryption and decryption method based on elastic sampling and model-free data-driven methods, this invention also proposes a corresponding image encryption and decryption system based on elastic sampling and model-free data-driven methods, which adopts the following technical solution: An image encryption and decryption system based on elastic sampling and model-free data-driven methods includes the following modules: The chaotic neural network model building module is used to build dynamic models of driving model-free chaotic neural networks and responding model-free chaotic neural networks. The data-driven closed-loop system model building module is used to collect offline data sequences and construct a data-driven representation of errors. Then, based on the characteristics of Denial-of-Service (DoS) attacks, an elastic sampling strategy is designed to actively stop the transmission of control input during the attack period and resume the transmission of control input after the attack ends. Based on error-driven data representation and elastic sampling strategy, a data-driven closed-loop system model is obtained. At the same time, the data-driven attack resistance ratio is defined to evaluate the attack resistance capability of the data-driven closed-loop system. The data-driven criterion establishment module is used to establish data-driven criteria to ensure synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network. The image encryption module is used to generate an encryption key using a model-free chaotic neural network to encrypt the insulator image, thus obtaining an encrypted image. And an image decryption module, which uses a response model-free chaotic neural network to generate a decryption key and performs the inverse operation on the encrypted image to obtain a decrypted image.

[0009] The present invention has the following advantages:

[0010] As described above, this invention proposes an image encryption / decryption method based on elastic sampling and model-free data-driven approaches. Addressing the problem that existing synchronization control strategies heavily rely on precise knowledge of system models, this invention employs an open-loop data acquisition method. A limited-amplitude excitation signal is applied to the system as a control input, inducing system state oscillations. Data is collected from these oscillations, which fully reflect the actual dynamic behavior characteristics of the system. Then, an offline data sequence is acquired using a high-frequency sampler, and a theoretical lower bound for the data sequence length is provided. Based on this, a data information equation is constructed, directly establishing a data-driven representation of the error. This invention eliminates the need to identify any system parameters; it can accurately characterize the system dynamics solely through offline data sequences, breaking through the dependence of traditional methods on precise mathematical models and enabling synchronization control even when the chaotic neural network model is completely unknown. Most existing data-driven methods focus on ideal communication environments. When the communication network suffers a DoS attack that interrupts data transmission, the system performance will severely degrade or even become unstable. To address this issue, this invention proposes a data-driven elastic sampling strategy. The sampling time is jointly determined by the conventional sampling rules and the characteristics of the DoS attack signal. During the attack interval, the transmission of control input is actively stopped, and the transmission of control input is quickly restored after the attack ends. This invention also introduces a data-driven attack resistance ratio to quantitatively evaluate the system's resilience against DoS attacks. Even when the communication network suffers a malicious attack that interrupts data packet transmission, this invention can still guarantee the stability and synchronization performance of the data-driven closed-loop system. Moreover, the data-driven attack resistance ratio can be directly measured from the collected data sequence without knowing the system matrix, thus achieving true model-free elastic control. To address the problems of cumbersome existing data-driven criterion processes and the inability to directly incorporate data sequences into criterion construction, this invention constructs a piecewise Lyapunov function to characterize the hybrid dynamics of a data-driven closed-loop system under the alternating effects of elastic sampling intervals and attack intervals. Combined with the data-driven representation of errors, data-driven criters based on data sequences are directly derived. Therefore, this invention eliminates the need for any model-based intermediate steps; all synchronization criteria are directly obtained from the data sequence, significantly reducing design complexity. In practical applications of insulator image privacy protection, this invention achieves secure encrypted transmission of insulator images under attack environments and enables lossless recovery of the original insulator image at the receiving end. Attached Figure Description

[0011] Figure 1 This is a flowchart of an image encryption / decryption method based on elastic sampling and model-free data-driven methods in an embodiment of the present invention. Figure 2 This is an architecture diagram of an image encryption / decryption method based on elastic sampling and model-free data-driven methods in an embodiment of the present invention; Figure 3The diagrams illustrate sampling methods; (a) represents traditional sampling; (b) represents a DoS attack; and (c) represents resilient sampling. Figure 4 Here, (a) is a chaotic attractor for a three-neuron model-free chaotic neural network; where (a) is (a) chaotic attractor; (b) for Chaotic attractors; Figure 5 This is the data sequence obtained during the data acquisition phase; where (a) is... The sampled values; where (b) is The sampled values; where (c) is The sampled values; where (d) is The sampled values; Figure 6 The images are the original and encrypted / decrypted images of the insulator; where (a) is the original insulator image; (b) is the encrypted image; and (c) is the decrypted image. Figure 7 Here are the histograms of the original insulator image and the encrypted image; where (a) is the histogram of the original insulator image and (b) is the histogram of the encrypted image. Detailed Implementation

[0012] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments: Example 1 This embodiment 1 describes an image encryption and decryption method based on elastic sampling and model-free data-driven methods, such as... Figure 1 As shown, the method includes the following steps: Step 1. Establish dynamic models for the driving model-free chaotic neural network and the response model-free chaotic neural network.

[0013] The dynamic models for driving model-free chaotic neural networks and response model-free chaotic neural networks are as follows: (1) (2) in, , These are the state vectors that drive the model-free chaotic neural network and the state vectors that respond to the model-free chaotic neural network, respectively. This indicates a control input.

[0014] For activation function, , ; , They are respectively , The One portion, ; , It is a monotonically non-decreasing and Lipschitz continuous function, that is, for a constant... and , There exists a constant , making Established.

[0015] , , The matrix is ​​unknown; express A column vector of dimension, express A 3D matrix; The number of neurons in the chaotic neural network. To control the input dimension.

[0016] During the data acquisition phase, control input signals Typically applied in an open-loop manner, the excitation signal can be intentionally designed as a bounded signal, such as an amplitude-limited sine or cosine wave. In this context, both the amplitude and frequency of the control input signal can be used as adjustable parameters. By appropriately selecting these parameters, the system state can be maintained within an acceptable operating range throughout the data acquisition process, thereby avoiding violations of physical constraints or instability. This design flexibility makes the method of this invention highly suitable for practical engineering applications. It is worth emphasizing that in many real-world scenarios, the system state is often constrained by operating boundaries and safety-related limitations.

[0017] Define error ,and ; The dynamic model of the error is then: (3).

[0018] Figure 2 The overall architecture of the method of this invention is illustrated. The system consists of two parts: a driving model-free chaotic neural network and a response model-free chaotic neural network, which interact with each other through a shared communication network. The driving model-free chaotic neural network applies bounded excitation signals (i.e., control input signals) in an open-loop manner to generate chaotic sequences for image encryption; simultaneously, the system collects error derivatives through a high-frequency sampler. ,error Activation function and control input Offline data sequences are used to construct a data-driven representation of the error, which is then used to solve for the security controller gain. In the wireless network, the resilient sampling strategy dynamically adjusts the sampling time according to the denial-of-service attack state. During the attack interval, the transmission of control input is actively paused, and resumed rapidly after the attack ends, ensuring effective updates of the control input signal. The control signal output by the controller is converted into a continuous signal by a zero-order hold, and then applied to the response model-free chaotic neural network by the actuator, driving its state evolution. Under the action of the controller, the response model-free chaotic neural network achieves synchronization with the driving model-free chaotic neural network, generating the same chaotic sequence for image decryption. The entire architecture achieves the unification of data-driven modeling, resilient sampling control, and image encryption / decryption.

[0019] Step 2. Collect offline data sequences and construct an error-driven data representation; then design an elastic sampling strategy based on the characteristics of Denial-of-Service (DoS) attacks, actively stop the transmission of control inputs during the attack interval, and resume the transmission of control inputs after the attack ends; based on the error-driven data representation and the elastic sampling strategy, obtain a data-driven closed-loop system model, and define the data-driven attack resistance ratio to evaluate the attack resistance capability of the data-driven closed-loop system.

[0020] Step 2.1. For cases where the system model is completely unknown, a data-driven representation is constructed using offline collected error derivatives, errors, activation functions, and control input data sequences to obtain a combined expression of the unknown system matrix.

[0021] This invention addresses the scenario where the system model is completely unknown. It does not rely on any prior model information and utilizes only the collected data sequence to complete the controller design and data-driven criterion construction. Since matrices C, A, and B are unknown, designing model-based control laws for such model-less chaotic neural networks is infeasible. To solve this problem, a data-driven approach is used to model the data-driven closed-loop system, specifically comprising the following two parts: first, collecting data sequences; second, constructing a data-driven representation of the error.

[0022] In practical engineering applications, during the data acquisition phase, the system operates in an open-loop state, and all state variables can be directly measured by the sampler. Therefore, data sequences can be acquired without knowing the system matrix.

[0023] Directly using a high-frequency sampler from variables , , and Collect data information to obtain a set of data sequences. , , and Its expression is: ; ; ; ; in, For the first The sampled value of the error derivative at each sampling time. In the first The sampled value of the error at each sampling time. In the first The sampled values ​​of the activation function at each sampling time. For the first Each sampling moment controls the input sample value; ; This indicates the number of data collections, and the matrix... The condition of full rank is met.

[0024] Full rank requirements satisfy This establishes the data sequence in this invention. , , , The theoretical lower bound for length. In practical applications, at least... Only a certain number of samples can guarantee the matrix It has full row rank. This condition is fundamentally important for capturing the basic dynamic characteristics of unknown systems. It should be noted that this requirement is consistent with the continuous excitation condition commonly found in system identification and data-driven control.

[0025] In the data-driven representation construction phase, the collected data sequences are used to characterize the dynamic properties of the model-free chaotic neural network.

[0026] Based on formula (3), the following data information equation is established: .

[0027] The results were: .

[0028] Due to the matrix Full rank matrix , so exist.

[0029] The combined expression of the unknown matrices C, A, and B obtained through matrix operations is: ; This process requires no iterative identification or parameter estimation.

[0030] Therefore, we get: .

[0031] To simplify the following notation, let's denote it as: , , , ; in It is an identity matrix.

[0032] Then we have: , , Substituting this into formula (3), we obtain the data-driven representation of the error: (4).

[0033] It should be noted that the method of this invention does not require explicit identification of matrices C, A, and B. In fact, the designed data information equation, i.e., formula (8), provides a theoretical equivalence between data-based neural network representations and model-based representations. This equivalence is used for analytical purposes, namely: Based on this, the linear matrix inequality conditions in the first and second sets of data-driven criteria are directly applied to the data sequence. , , , This allows the design of the safety controller gain to be solved directly without any system identification steps. This means that the method of this invention falls entirely within the scope of direct data-driven control.

[0034] Step 2.2. Based on the characteristics of non-periodic energy-constrained DoS attacks, design a resilient sampling strategy.

[0035] like Figure 3 As shown in (a), the traditional sampling period is: ; in , Indicates the sampling time. Indicates the sampling period. ; .

[0036] Typically, control inputs are designed as follows: ), , ; in Indicates the controller gain. () indicates the error at the sampling time. The state value.

[0037] like Figure 3As shown in (b), a power-limited DoS attack may clog sampling data packets. Based on the characteristics of the jammer signal, in the... attack cycles Internal DoS attack signal The description is as follows: ; in, For rest area, This is the attack range. ; For the first The start time of each attack cycle, For the first The end of an attack cycle For the first The duration of rest periods within each attack cycle.

[0038] ;in It is a positive integer, and .

[0039] This indicates the start time of the first attack cycle, which begins at t=0. ; This indicates that the next attack cycle will only begin after the attack phase ends.

[0040] definition , .

[0041] when At that time, the DoS attack was in a dormant state, that is... At this point, the sampled data packet can reach the zero-order hold; when At that time, the DoS attack becomes active, that is... At this point, the zero-order hold cannot receive the sampled data packet.

[0042] Ideal data sampling methods have been widely used to achieve synchronization between drive and response systems; however, traditional sampling schemes fail when DoS attacks are present because these attacks occur within a certain range. Internally, the transmission of control information will be interrupted. To mitigate the adverse effects of a DoS attack, a resilient sampling strategy is constructed as follows: (5) in, Indicates the first The first attack cycle The sampling point, the first Initial sampling points for each attack cycle ; For the first The first attack cycle One sampling point.

[0043] and ,Right now ; in, Denotes the supremum of a set. Indicates the sampling period; Indicates the first The first attack cycle One sampling point, Indicates the first The first attack cycle One sampling point.

[0044] The elastic sampling strategy proposed in this invention actively suspends the transmission of control input during the attack interval and quickly resumes the transmission of control input after the attack ends, thus avoiding the injection of erroneous data. Even when the communication network suffers a DoS attack and data transmission is interrupted, it can still ensure the global asymptotic stability synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network.

[0045] Step 2.3. Based on the constructed error-driven data-driven representation and elastic sampling strategy, a data-driven closed-loop system model is obtained, and a data-driven attack resistance ratio is introduced to quantitatively evaluate the system's attack resistance capability, i.e., the system's tolerance to attacks of different intensities.

[0046] According to formula (5): (6) in, ; Indicates the gain of the safety controller. () indicates the error at the sampling time. The state value.

[0047] definition mod([ );in, This indicates the modulo operation.

[0048] like Figure 3 As shown in (c) The value can take two cases. When hour, ;when hour, .

[0049] Substituting formula (6) into formula (4), we obtain the data-driven closed-loop system model as follows: (7) To assess the resilience of data-driven attacks, data-driven attack resistance is compared to... Defined as: ; in For the shortest rest duration, ; For the maximum attack duration, ; ; It represents the infimum of a set.

[0050] The larger the value, the higher the percentage of continuous attack duration that the system can tolerate, meaning the stronger the system's resilience against DoS attacks.

[0051] It should be noted that the method of the present invention includes two distinct stages: an offline data acquisition stage and an online control stage. In the offline data acquisition stage, control input... It is applied in an open-loop manner, without considering the impact of DoS attacks; even if DoS attacks are considered at this stage, as long as the collected data sequence... , , , These data are generated through system iterations and still reflect the actual dynamic evolution of the error. In other words, the method proposed in this invention does not require the data sequences to be acquired under ideal communication conditions; what is truly important is that the acquired data sequences can realistically characterize the underlying dynamic behavior of the model-free chaotic neural network. Therefore, the presence of DoS attacks during the offline data acquisition phase has no essential impact on the subsequent controller design of this invention.

[0052] Step 3. Establish data-driven criteria to ensure synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network.

[0053] Based on Lyapunov stability theory, a piecewise Lyapunov function is constructed to address the alternating effects of sampling and attack intervals in elastic sampling strategies. Combining this with data-driven error representation, matrix theory, inequality estimation methods, and stability theory are used to derive sufficient conditions for linear matrix inequalities that depend solely on the acquired data sequence (i.e., the first set of data-driven criteria), which are directly presented in the form of a data matrix. Simultaneously, a data-driven attack resistance ratio is introduced to quantify the system's tolerance to DoS attacks. Solving the aforementioned linear matrix inequalities yields the security controller gain.

[0054] For ease of analysis, we define: , .

[0055] For a data-driven closed-loop system, consider the following piecewise Lyapunov function: ; in, ,in ; ,in This represents the exponential decay rate of the system during the DoS sleep interval. The larger the value, the faster the energy convergence rate of the system during the period when the control signal is available.

[0056] ; in for matrix, ; for matrix.

[0057] ,in ; .

[0058] Based on the aforementioned piecewise Lyapunov function, the time derivatives of the piecewise Lyapunov function along the trajectory of the data-driven closed-loop system, i.e., formula (7), are calculated in each time interval. Analysis is then performed in the rest interval and the attack interval to characterize the decay and growth patterns of the system energy under attack. Within this framework, an integral inequality method is introduced to handle the quadratic integral term, and the inequality properties are used to handle the sector-bounded constraint of the activation function, incorporating the influence of nonlinear dynamics on system stability into the framework of linear matrix inequalities. After processing, a set of sufficient conditions characterized by linear matrix inequalities is obtained, which constitutes the first set of data-driven criteria. This data-driven criterion ensures that the piecewise Lyapunov function satisfies the corresponding negative definiteness or growth bound conditions in each time interval, thereby ensuring the global asymptotic stability of the data-driven closed-loop system under the action of the elastic sampling strategy.

[0059] The first set of data-driven criteria is as follows: For a given data sequence , scalar , , , , sum matrix .

[0060] If it exists matrix , , diagonal matrix , , , matrix , , , , , , , , This makes linear matrix inequalities (8)-(12) and inequality (13) hold: (8) (9) (10) (11) (12) (13) in, This represents the exponential divergence rate of the system within the DoS attack range. The smaller the value, the slower the energy growth rate of the system during the control signal interruption. , Together, they determine the stability margin of a data-driven closed-loop system under DoS attacks.

[0061] , , ; , , ; , , ; , , ; , , ; , , ... This represents a constant diagonal matrix representing the activation function. represents a diagonal matrix; 0 represents a zero matrix, and * represents the transpose of its symmetric part.

[0062] The data-driven closed-loop system, i.e., formula (7), is globally asymptotically stable. That is, under the data-driven elastic sampling strategy, the driving model-free chaotic neural network and the response model-free chaotic neural network can achieve synchronization.

[0063] By decoupling the first set of data-driven criteria, we obtain another set of sufficient conditions characterized by linear matrix inequalities, which is the second set of data-driven criteria.

[0064] The second set of data-driven criteria is as follows: For a given data sequence , scalar , , , , , , , .

[0065] If it exists matrix , , matrix , diagonal matrix , , , matrix , , , , , This makes inequality (13) and linear matrix inequalities (14)-(18) hold: (14) (15) (16) (17) (18) in, , , ; , , ; , , ; , , ; , , .

[0066] when At that time, the data-driven closed-loop system, i.e., formula (7), is globally asymptotically stable. That is, under the data-driven elastic sampling strategy, the driving model-free chaotic neural network and the response model-free chaotic neural network can achieve synchronization.

[0067] The relationship between the first set of data-driven criteria and the second set of data-driven criteria: The first set of data-driven criteria defines a matrix. ,Bundle The first set of data-driven criteria is used as a known condition for solving the problem, while the second set of data-driven criteria decouples the linear matrix inequalities obtained from the first set of data-driven criteria to solve for the matrix. .

[0068] To prove the validity of the second set of data-driven criteria, that is, to prove that when the linear matrix inequalities (14)-(18) have solutions, the data-driven closed-loop system, i.e., formula (7), is globally asymptotically stable under the elastic sampling strategy, we will establish the equivalence relationship between the second set of data-driven criteria and the first set of data-driven criteria through decoupling, variable substitution, and matrix transformation. The proof process is as follows: remember , , , , , , , , , , , , , , .

[0069] definition , , , , ;in Represents a column vector.

[0070] Use respectively and Multiply equation (8) on the left and right, using... and Multiply equation (9) on the left and right, and use... and Multiply equation (10) on the left and right, and use... and Multiply equation (11) on the left and right, and use... and Multiplying equation (12) by its left and right sides yields equations (14)-(18). Therefore, it can be obtained through... Solving for the results Proof complete.

[0071] To date, several fundamental studies on networked linear systems have been based on Lyapunov data-driven control. It is important to emphasize that the core issue in these works is how to derive data-driven criteria from model-based conditions. Specifically, model-based conditions must be established beforehand, and then data-driven criteria are constructed using the proposed transformation technique. Clearly, applying such data-driven methods involves considerable work. Furthermore, all existing work focuses on linear systems, not nonlinear systems. Compared to these transformation-based methods, the method proposed in this invention directly establishes the data-driven criteria; the security controller gain and data-driven attack resistance ratio are directly synthesized from the acquired data sequence; this direct and practical method not only eliminates the intermediate model-based steps but also adapts to offline data sequences. Therefore, this invention has significant advantages in terms of simplicity and practicality.

[0072] This invention enables the design of security controller gain and the construction of data-driven criteria based solely on offline data sequences, even without knowing the precise mathematical model of the chaotic neural network. Furthermore, the data acquisition phase allows for denial-of-service attacks, does not require an ideal communication environment, and closely approximates actual engineering conditions. Moreover, the proposed data-driven criteria are constructed directly from the offline data sequences of the data matrix, eliminating the need for intermediate model conversion and reducing design complexity.

[0073] Step 4. Use a model-free chaotic neural network to generate an encryption key to encrypt the insulator image, thus obtaining an encrypted image.

[0074] In smart grids, monitoring systems for substations or transmission lines need to transmit images of critical equipment such as insulators and transformers to the dispatch center via communication networks for status monitoring and fault diagnosis. These images carry sensitive equipment structural information, which may lead to information leakage during transmission. Therefore, it is necessary to encrypt the transmission of these images.

[0075] Get the size as Original insulator image, in which and These represent the width and height of the image, respectively.

[0076] The original insulator image is reconstructed into a one-dimensional vector, denoted as . ; in , ... These represent the grayscale values ​​of each pixel in the original insulator image.

[0077] Generating encryption keys using a model-free chaotic neural network ; in , ... These represent the pseudo-random numbers that drive the output of the model-free chaotic neural network.

[0078] Then using the encryption key right Perform an XOR operation to obtain the encrypted image. Its expression is: .

[0079] Under appropriate data acquisition scale and accuracy, obtain data sequences. , , , This data is then used as the second set of data-driven criteria to solve for the safety controller gain. and data-driven anti-attack ratio The resulting safety controller gain This ensures synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network.

[0080] Taking a three-neuron driven model-free chaotic neural network as an example, its system matrix... , , All are unknown.

[0081] The activation function is preset as follows: ; Initial values ​​for the state vectors of a model-free chaotic neural network Initial values ​​of the state vector of a model-free chaotic neural network .

[0082] The chaotic attractor of this three-neuron driving model-free chaotic neural network is as follows: Figure 4 As shown. , , It refers to the state variables that drive the three neurons of a model-free chaotic neural network.

[0083] Data sequence collection , , , Design control input , , as follows: A sine wave with an amplitude of 1 and a frequency of 1 Hz is used as... A sine wave with the same amplitude but a frequency of 4 Hz was used as the... ;for A chirped signal with an amplitude of 0.01 was used, the target time was 1 second, and the frequency was 2Hz.

[0084] , , , The trend of change is as follows Figure 5 As shown. , , These are the three state components of the error; , , The derivatives of the three state components of the error; , , Here is the activation function for the three channels.

[0085] Set the number of data collections ,but: ; ; ; ; At the same time, matrix The term of office has been completed.

[0086] When the data precision is 4 (experimental data are rounded to four decimal places), the data sample is as follows.

[0087] ; ; ; ; See the original insulator image Figure 6 (a) where IR represents the temperature at various locations on the insulator in the photograph, and the color intensity at different locations indicates the temperature distribution across different areas of the insulator. Such images carry sensitive equipment structural information, and traditional text encryption alone is insufficient to provide adequate security. If the image is intercepted during transmission, attackers could use it to reconstruct the power grid topology and identify critical components such as interconnecting circuit breakers and transformer units.

[0088] After XOR operation, the encrypted image is obtained, such as... Figure 6 As shown in (b), compared with the original insulator image, the encrypted image has a more uniform distribution of pixel gray values, and the statistical correlation between adjacent pixels is significantly reduced.

[0089] Figure 7 The histogram comparison between the original insulator image and the encrypted image is shown, where... Figure 7 (a) Histogram corresponding to the original insulator image, Figure 7 (b) The histogram corresponding to the encrypted image shows a clear uniform distribution.

[0090] Step 5. Generate a decryption key using a response model-free chaotic neural network, and perform the inverse operation on the encrypted image to obtain the decrypted image.

[0091] A chaotic sequence identical to the encryption key is generated using a response-free model-free chaotic neural network and used as the decryption key. Then, the decryption key is used to perform the inverse XOR operation on the encrypted image to obtain the decrypted image, thereby recovering the original insulator image.

[0092] Figure 6 (c) shows the decrypted image, which is related to... Figure 6 The original image in (a) is visually highly consistent. This demonstrates that even under the presence of a denial-of-service attack, the response model-free chaotic neural network and the driving model-free chaotic neural network can maintain precise synchronization, thereby ensuring the complete recovery of image information.

[0093] Given: , , , , ; , , , , ; Data-driven attack resistance ; , , , The second set of data-driven criteria; Find: ; ; ; And the safety controller gain It is applied to the synchronization control process of driving model-free chaotic neural networks and responding model-free chaotic neural networks.

[0094] The successful recovery of the decrypted image verifies the effectiveness of the proposed method. The above results demonstrate that, even under DoS attacks, the proposed resilient sampling strategy can reliably achieve image encryption and decryption, and the invention can be applied to privacy protection of insulator images. The invention maintains high effectiveness, robustness, and engineering applicability even under conditions of unknown model and communication interference.

[0095] In the data acquisition phase, this invention applies bounded excitation signals in an open-loop manner, maintaining the system state within a safe operating range by appropriately selecting signal parameters. Without identifying the unknown system model, it directly collects offline data sequences of states, activation functions, and open-loop control inputs, establishing equivalent data information equations to provide a data-based representation of the dynamic behavior of model-free chaotic neural networks. Secondly, this invention proposes a flexible sampling strategy to address signal transmission interruption issues under DoS attacks. Using Lyapunov stability theory, two data-driven criteria are constructed to ensure synchronization between the driving and response models of the model-free chaotic neural network. Unlike existing data-driven methods, this invention does not require establishing model-based conditions before transformation; all data-driven criteria are directly obtained from the data sequence, achieving true model-free data-driven design. Simulation results show that the acquisition scale and accuracy of the data sequence significantly affect the system's anti-attack capability, and the proposed method can adapt to different acquisition conditions, exhibiting good universality.

[0096] Example 2 This embodiment 2 describes an image encryption and decryption system based on elastic sampling and model-free data-driven methods. This system is based on the same inventive concept as the image encryption and decryption method based on elastic sampling and model-free data-driven methods in embodiment 1 above.

[0097] An image encryption and decryption system based on elastic sampling and model-free data-driven methods includes the following modules: The chaotic neural network model building module is used to build dynamic models of driving model-free chaotic neural networks and responding model-free chaotic neural networks. The data-driven closed-loop system model building module is used to collect offline data sequences and construct a data-driven representation of errors. Then, based on the characteristics of Denial-of-Service (DoS) attacks, an elastic sampling strategy is designed to actively stop the transmission of control input during the attack period and resume the transmission of control input after the attack ends. Based on error-driven data representation and elastic sampling strategy, a data-driven closed-loop system model is obtained. At the same time, the data-driven attack resistance ratio is defined to evaluate the attack resistance capability of the data-driven closed-loop system. The data-driven criterion establishment module is used to establish data-driven criteria to ensure synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network. The image encryption module is used to generate an encryption key using a model-free chaotic neural network to encrypt the insulator image, thus obtaining an encrypted image. And an image decryption module, which uses a response model-free chaotic neural network to generate a decryption key and performs the inverse operation on the encrypted image to obtain a decrypted image.

[0098] It should be noted that any content not mentioned in the above-described functional modules of the system described in Embodiment 2 can be referred to the step description of the corresponding method in Embodiment 1 above, and will not be repeated in detail here.

[0099] Of course, the above description is only a preferred embodiment of the present invention. The present invention is not limited to the above-described embodiments. It should be noted that any equivalent substitutions or obvious modifications made by those skilled in the art under the guidance of this specification fall within the scope of this specification and should be protected by the present invention.

Claims

1. An image encryption / decryption method based on elastic sampling and model-free data-driven approach, characterized in that, Includes the following steps: Step 1. Establish dynamic models of the driving model-free chaotic neural network and the response model-free chaotic neural network; Step 2. Collect offline data sequences and construct a data-driven representation of the errors; The specific process of constructing a data-driven representation of the error is as follows: Using a high-frequency sampler from variables , , and Collect data information to obtain a set of data sequences. , , and Its expression is: ； ； ； ； in, For the first The sampled value of the error derivative at each sampling time. In the first The sampled value of the error at each sampling time. In the first The sampled values ​​of the activation function at each sampling time. For the first Each sampling moment controls the input sample value; ; This indicates the number of data collections, and the matrix... The row rank condition is satisfied; For error, To control the input, For activation functions; The number of neurons in the chaotic neural network. To control the input dimension; The following data information equations are established based on the error-based dynamic model: ； in , , The matrix is ​​unknown; The results were: ; Due to the matrix Full rank matrix , so exist; The combined expression of the unknown matrices C, A, and B obtained through matrix operations is: ； Therefore, we get: ; To simplify the following notation, let's denote it as: , , , ; in It is the identity matrix; Then we have: , , Substituting this into the error dynamics model, we obtain the data-driven representation of the error: (4) Then, based on the characteristics of Denial-of-Service (DoS) attacks, an elastic sampling strategy is designed to actively stop the transmission of control input during the attack period and resume the transmission of control input after the attack ends. The specific process of constructing a resilient sampling strategy is as follows: In the attack cycles Internal DoS attack signal The description is as follows: ； in, For rest area, This is the attack range. ; For the first The start time of each attack cycle, For the first The end of an attack cycle For the first The duration of rest periods within an attack cycle; definition , ; when At that time, the DoS attack was in a dormant state, that is... At this point, the sampled data packet can reach the zero-order hold; when At that time, the DoS attack becomes active, that is... At this point, the zero-order hold cannot receive the sampled data packet; The elastic sampling strategy is constructed as follows: (5) in, Indicates the first The first attack cycle The sampling point, the first Initial sampling points for each attack cycle ; Indicates the first The first attack cycle One sampling point; and ,Right now ; in, Denotes the supremum of a set. Indicates the sampling period; Indicates the first The first attack cycle One sampling point, Indicates the first The first attack cycle One sampling point; Based on error-driven data representation and elastic sampling strategy, a data-driven closed-loop system model is obtained. At the same time, the data-driven attack resistance ratio is defined to evaluate the attack resistance capability of the data-driven closed-loop system. Step 3. Establish data-driven criteria to ensure synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network; Step 4. Use a model-free chaotic neural network to generate an encryption key to encrypt the insulator image, thus obtaining an encrypted image; Step 5. Generate a decryption key using a response model-free chaotic neural network, and perform the inverse operation on the encrypted image to obtain the decrypted image.

2. The image encryption and decryption method based on elastic sampling and model-free data-driven processing according to claim 1, characterized in that, Step 1 specifically involves: The dynamic models for driving model-free chaotic neural networks and response model-free chaotic neural networks are as follows: (1) (2) in, , These are the state vectors that drive the model-free chaotic neural network and the state vectors that respond to the model-free chaotic neural network, respectively. Indicates control input; For activation functions; , , The matrix is ​​unknown; The number of neurons in the chaotic neural network. To control the input dimension; Define error ,and ; The dynamic model of the error is then: (3)。 3. The image encryption and decryption method based on elastic sampling and model-free data-driven processing according to claim 2, characterized in that, In step 2, according to formula (5), we get: ) (6) in, ; Indicates the gain of the safety controller. () indicates the error at the sampling time. State value; definition mod([ );in, This represents the modulo operation; when hour, ;when hour, ; Substituting formula (6) into formula (4), we obtain the data-driven closed-loop system model as follows: (7)。 4. The image encryption and decryption method based on elastic sampling and model-free data-driven processing according to claim 3, characterized in that, In step 2, data-driven anti-attack ratio Defined as: ; in , , ; It represents the infimum of a set.

5. The image encryption and decryption method based on elastic sampling and model-free data-driven processing according to claim 4, characterized in that, In step 3, the data-driven criteria include two sets; definition: , ; The first set of data-driven criteria is as follows: For a given data sequence , scalar , , , , sum matrix ; If it exists matrix , , diagonal matrix , , , matrix , , , , , , , , This makes linear matrix inequalities (8)-(12) and inequality (13) hold: (8) (9) (10) (11) (12) (13) in, , ， ； ， ， ； ， ， ； ， ， ； ， ， ； , , ... This represents a constant diagonal matrix representing the activation function. represents a diagonal matrix; 0 represents a zero matrix; * represents the transpose of its symmetric part; Then the data-driven closed-loop system, i.e., formula (7), is globally asymptotically stable. That is, under the data-driven elastic sampling strategy, the driving model-free chaotic neural network and the response model-free chaotic neural network can achieve synchronization. The second set of data-driven criteria is as follows: For a given data sequence , scalar , , , , , , , ; If it exists matrix , , matrix , diagonal matrix , , , matrix , , , , , This makes inequality (13) and linear matrix inequalities (14)-(18) hold: (14) (15) (16) (17) (18) in, , ， ； ， ， ； ， ， ； ， ， ； ， ， ； when At that time, the data-driven closed-loop system, i.e., formula (7), is globally asymptotically stable. That is, under the data-driven elastic sampling strategy, the driving model-free chaotic neural network and the response model-free chaotic neural network can achieve synchronization.

6. The image encryption and decryption method based on elastic sampling and model-free data-driven processing according to claim 5, characterized in that, Step 4 specifically involves: Get the size as Original insulator image, in which and These are the width and height of the image, respectively; The original insulator image is reconstructed into a one-dimensional vector, denoted as . ; in , ... These represent the grayscale values ​​of each pixel in the original insulator image; Generating encryption keys using a model-free chaotic neural network ; in , ... These represent the pseudo-random numbers that drive the output of the model-free chaotic neural network; Then using the encryption key right Perform an XOR operation to obtain the encrypted image. Its expression is: .

7. The image encryption and decryption method based on elastic sampling and model-free data-driven processing according to claim 6, characterized in that, Step 5 specifically involves: A chaotic sequence identical to the encryption key is generated using a response-free model-free chaotic neural network and used as the decryption key. Then, the decryption key is used to perform the inverse XOR operation on the encrypted image to obtain the decrypted image.

8. An image encryption and decryption system based on elastic sampling and model-free data driving for implementing the image encryption and decryption method based on elastic sampling and model-free data driving as described in claim 1, characterized in that, An image encryption and decryption system based on elastic sampling and model-free data-driven methods includes the following modules: The chaotic neural network model building module is used to build dynamic models of driving model-free chaotic neural networks and responding model-free chaotic neural networks. The data-driven closed-loop system model building module is used to collect offline data sequences and construct a data-driven representation of errors. Then, based on the characteristics of Denial-of-Service (DoS) attacks, an elastic sampling strategy is designed to actively stop the transmission of control input during the attack period and resume the transmission of control input after the attack ends. Based on error-driven data representation and elastic sampling strategy, a data-driven closed-loop system model is obtained. At the same time, the data-driven attack resistance ratio is defined to evaluate the attack resistance capability of the data-driven closed-loop system. The data-driven criterion establishment module is used to establish data-driven criteria to ensure synchronization between the driving model-free chaotic neural network and the response model-free chaotic neural network. The image encryption module is used to generate an encryption key using a model-free chaotic neural network to encrypt the insulator image, thus obtaining an encrypted image. And an image decryption module, which uses a response model-free chaotic neural network to generate a decryption key and performs the inverse operation on the encrypted image to obtain a decrypted image.

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