Method for replay attack-oriented multi-agent system differential privacy consistency control

Abstract

The application belongs to the technical field of distributed control of multi-agent systems, and discloses a multi-agent system differential privacy consensus control method for replay attacks. The method effectively overcomes the shortcomings of existing methods in privacy protection and can provide stronger privacy protection capability. The core idea is that agents do not transmit real data but transmit perturbed data after adding privacy protection noise, which makes it impossible for malicious attackers to obtain real data, thereby achieving the effect of privacy protection. In addition, to resist replay attacks, the method of the application couples the perturbed data to be transmitted with a key before transmitting the perturbed data, so as to achieve the purpose of replay attack detection. In addition, through the design of a time-varying step length based on a stochastic approximation method, the method of the application realizes the privacy protection effect and the mean square asymptotic consistency while ensuring that the noise variance is allowed to increase.

Description

Technical Field

[0001] This invention belongs to the field of distributed control technology for multi-agent systems, specifically relating to a differential privacy consistency control method for multi-agent systems resistant to replay attacks. Background Technology

[0002] Distributed consensus control in multi-agent systems has wide applications in many fields, including distributed Kalman filtering, drone swarm coordination, intelligent traffic scheduling, and medical information sharing. Consistency control aims to design network protocols that enable all agents to asymptotically reach consensus. Individual agents achieve global consensus by exchanging neighbor information, but this information exchange carries privacy risks; attackers may steal communication data. Through reverse engineering, attackers may even obtain the agent's local input, internal state, and final convergence value. Therefore, how to achieve mean-square asymptotic consensus while protecting the sensitive information of each agent from being inferred has become a pressing problem.

[0003] To address the privacy requirements in distributed control, existing technologies have proposed various methods, including homomorphic encryption, state decomposition, and differential privacy. Homomorphic encryption uses encryption techniques to process sensitive information, enabling the system to perform computations directly on encrypted data without decryption. However, it typically introduces heavy communication and computational burdens, making it unsuitable for scenarios with limited computational and network resources. State decomposition works by abstracting the nodes of an agent into two or more virtual child nodes. One child node remains private and does not participate in inter-node communication, while the other child nodes actively interact with neighboring nodes. However, this method is only suitable for specific scenarios with simple system architectures, and its effectiveness is limited in other application scenarios. As the system scales up, this method becomes unsuitable. Differential privacy effectively overcomes the shortcomings of the above two methods. Its core principle is to inject random noise into the transmitted data to mask the real information, thereby ensuring consensus performance while effectively preventing eavesdroppers from obtaining real private information. Moreover, differential privacy requires no additional overhead and is relatively simple to compute, making it suitable for privacy protection in multi-agent systems in open network environments.

[0004] However, existing differential privacy methods suffer from noise form limitations, typically requiring the added noise variance to decay exponentially to zero or remain constant. However, differential privacy methods with exponentially decaying variance usually only provide privacy protection for the initial state, easily leading to the exposure of subsequent states and failing to achieve global privacy protection. Adding noise with constant variance is easily detected by malicious attackers, thus jeopardizing privacy protection. Therefore, adding decaying or constant noise may directly expose the agent's state trajectory over time, failing to provide global privacy protection. Furthermore, many existing methods are also subject to attacker limitations, assuming that attackers cannot access the communication information of all the agent's neighbors, which limits their applicability to more general attack models.

[0005] Meanwhile, open and dynamic network environments are highly vulnerable to various network attacks, among which replay attacks have become a common and serious threat due to their simplicity and high stealth. A replay attack refers to an attacker maliciously intercepting and recording legitimate data packets transmitted between agents, and then repeatedly injecting them into the network at a later time. Unlike data tampering attacks that aim to corrupt data, replay attacks utilize correct but outdated information, making them difficult to completely defend against using traditional encryption methods.

[0006] Therefore, achieving efficient, robust, and universally applicable differential privacy mean-square asymptotic consistency control in the face of replay attacks remains a challenge. Summary of the Invention

[0007] The purpose of this invention is to propose a differential privacy consistency control method for multi-agent systems resistant to replay attacks, in order to solve the problems of high communication and computational overhead, noise form limitations, and attacker limitations in existing methods, and to provide a more general, robust, and adjustable mean square consistency control scheme for multi-agent systems under differential privacy protection.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] A differential privacy consistency control method for multi-agent systems facing replay attacks includes the following steps:

[0010] Step 1. Establish a mathematical model of the multi-agent system;

[0011] Step 2. Add randomly growing noise to the real state information that needs to be transmitted, so that the multi-agent system transmits the disturbed data with added noise during the communication process;

[0012] Step 3. Based on the mathematical model of the multi-agent system established in Step 1, design a distributed controller;

[0013] Step 4. Design an encrypted pairing algorithm to defend against replay attacks;

[0014] Step 5. Using the distributed controller designed in Step 3 and the encrypted pairing algorithm designed in Step 4, implement consensus control of the multi-agent system against replay attacks.

[0015] Furthermore, based on the aforementioned differential privacy consistency control method for multi-agent systems facing replay attacks, this invention also proposes a computer device, which includes a memory and one or more processors.

[0016] The memory stores executable code, and when the processor executes the executable code, it implements the steps of the differential privacy consistency control method for multi-agent systems against replay attacks described above.

[0017] Furthermore, based on the aforementioned differential privacy consistency control method for multi-agent systems against replay attacks, this invention also proposes a computer-readable storage medium storing a program thereon; when executed by a processor, this program is used to implement the steps of the aforementioned differential privacy consistency control method for multi-agent systems against replay attacks.

[0018] The present invention has the following advantages:

[0019] As described above, this invention relates to a differential privacy consistency control method for multi-agent systems resistant to replay attacks. This method uses randomly growing noise with variance increasing over time as privacy noise, avoiding the risk of agent state information being exposed when the noise variance decays to zero or when the noise variance remains constant. This provides more durable and robust global privacy protection than existing methods, and has stronger privacy protection capabilities, making it particularly suitable for application scenarios that require long-term privacy protection.

[0020] The method of this invention has better universality, overcoming the strict limitations of existing differential privacy consensus algorithms on the form of noise. It can guarantee the convergence of the system even with increasing noise variance and calculates the convergence rate, making it more flexible and applicable in practical applications. In terms of privacy protection for infinite time series, the method of this invention outperforms methods that require noise attenuation. The method of this invention also has robustness against communication defects; its noise mechanism can, to some extent, combat communication defects in distributed computing, enhancing the robustness of the system.

[0021] Furthermore, the method of the present invention provides a countermeasure for scenarios where replay attacks exist during system data transmission. It couples a key that changes over time with the transmitted perturbation data. After coupling, communication is achieved through network transmission. The presence of replay attacks can be detected by comparing the keys, so that corresponding countermeasures can be made based on the detection.

[0022] In addition, the method of the present invention has flexible parameter design and controllability. It provides guiding principles for design step size and noise parameters, enabling users to flexibly adjust parameters according to actual application needs to balance convergence accuracy and privacy protection level, achieve customized performance optimization, and achieve a convergence target with preset accuracy while maintaining differential privacy over an infinite time range. Attached Figure Description

[0023] Figure 1 This is a flowchart of a differential privacy consistency control method for a multi-agent system resistant to replay attacks, as described in an embodiment of the present invention.

[0024] Figure 2 This is a communication topology diagram of the differential privacy consistency control method for multi-agent systems facing replay attacks in an embodiment of the present invention.

[0025] Figure 3 This is a graph showing the variation of privacy protection accuracy with noise parameter γ in an embodiment of the present invention.

[0026] Figure 4 This is a graph showing the change in privacy protection level as a function of noise parameter γ in an embodiment of the present invention.

[0027] Figure 5 This is a schematic diagram of a multi-agent system in an embodiment of the present invention.

[0028] Figure 6 This is a schematic diagram of the actual state evolution obtained using the control method of the present invention.

[0029] Figure 7 This is a schematic diagram of the evolution of the disturbance state obtained by the control method of the present invention.

[0030] Figure 8 This is a schematic diagram comparing the real state and the disturbance state of agent 1 obtained by the control method of the present invention.

[0031] Figure 9 This is a schematic diagram of the control inputs of each intelligent agent obtained using the control method of the present invention.

[0032] Figure 10 This is a schematic diagram of the actual state evolution obtained under the noise attenuation mechanism.

[0033] Figure 11 This is a schematic diagram of the perturbation state evolution obtained under the noise attenuation mechanism.

[0034] Figure 12 This is a schematic diagram comparing the real state and the perturbed state of agent 1 obtained under the noise attenuation mechanism. Detailed Implementation

[0035] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0036] Example 1

[0037] To address the shortcomings of existing methods, this invention provides a control method for achieving differential privacy mean-square asymptotic consistency on undirected network graphs in the face of replay attacks. This method effectively overcomes the privacy protection deficiencies of existing methods and provides stronger privacy protection capabilities. Its core idea is that agents do not transmit real data, but instead transmit perturbed data with added privacy-preserving noise (randomly increased noise). This prevents malicious attackers from obtaining the real data, thus achieving privacy protection. Furthermore, to resist replay attacks, this invention couples the perturbed data to be transmitted with a key before transmitting the perturbed data, achieving replay attack detection. Additionally, this invention achieves both privacy protection and mean-square asymptotic consistency by using a time-varying step size design based on a random approximation method, ensuring that the noise variance can be increased.

[0038] The differential privacy consistency control method for multi-agent systems against replay attacks proposed in this invention will be described in detail below. Figure 1 As shown, the differential privacy consistency control method for multi-agent systems facing replay attacks includes the following steps:

[0039] Step 1. Establish a mathematical model of the multi-agent system.

[0040] In this embodiment, step 1 specifically includes:

[0041] A collection of intelligent agents Represented as: .

[0042] edge set Represented as: .

[0043] in, Represents intelligent agents Received from intelligent agent Information.

[0044] intelligent agent Neighbor set Represented as: .

[0045] Adjacency Matrix Represented as: . Adjacency matrix The element, when hour ,otherwise .

[0046] Degree matrix Represented in diagonal matrix form: ,in For nodes The sum of the weights of all edges in the connected component. (The degree matrix is ​​then used.) The maximum and minimum eigenvalues ​​are denoted as follows: and .

[0047] Laplace matrix Defined as: .

[0048] and These are the Laplace matrices. The maximum and minimum non-zero eigenvalues.

[0049] Given an undirected connected graph, consider using... In a multi-agent system network consisting of [number] agents, each agent is considered a node, and the dynamic equation for each agent is:

[0050] .

[0051] in, Represents intelligent agents exist The state at any given moment, Represents intelligent agents exist The state at any given moment. Represents intelligent agents exist Time-based control input.

[0052] intelligent agent The initial state is denoted as , , , All are intelligent agents The true state information.

[0053] In this embodiment, potential attackers are assumed to be eavesdroppers. Specifically, an eavesdropper is an external eavesdropper familiar with the network topology and capable of monitoring the communication channels of the agents. It is assumed that the eavesdropper can continuously monitor the external observable state set of the agents. Compared to other attack forms discussed in multi-agent system security research, such as honest but curious nodes and malicious information injectors, external eavesdroppers are a potentially greater threat because they can infer the true state trajectory of the agents based on observed communication information between them, especially when sensitive information is involved.

[0054] The method of this invention also constructs a replay attack model. The attacker intercepts legitimate network communication data and repeatedly sends it to neighboring agents, recording the total attack duration as [missing information]. Due to the limited performance of storage hardware, attackers typically intercept and store a limited number of signals, thus limiting the duration of the attack. It is finite. The core of a replay attack lies in altering the current signal to a signal from a historical moment, that is, injecting a real signal rather than an artificially synthesized signal. This characteristic gives the attacker a certain degree of disguise ability, making it difficult to detect.

[0055] Step 2. Add randomly growing noise to the real state information that needs to be transmitted, so that the multi-agent system transmits the disturbed data with added noise during the communication process.

[0056] To protect the system's privacy, noise is added when exchanging information with neighbors. i.e., intelligent agent Do not transmit real-state information when communicating with neighbors. Instead, it transmits perturbed data with random noise added. By injecting noise, eavesdroppers cannot steal real data, thus achieving the effect of privacy protection.

[0057] In this embodiment, step 2 specifically includes:

[0058] Information on the actual state Add random noise intelligent agent Transmitting disturbed data with added noise when communicating with neighbors :

[0059] .

[0060] Among them, intelligent agents exist Random noise added at different times Follows a Laplace distribution: Added random noise They are independent of each other.

[0061] This indicates that the mean is 0 and the variance is 0. Laplace distribution, This represents a time-varying, increasing noise parameter. Unlike existing technologies where the noise variance must decay to zero or remain constant, the method of this invention effectively avoids the direct exposure of state trajectories, thus providing stronger privacy protection.

[0062] Step 3. Based on the mathematical model of the multi-agent system established in Step 1, design a distributed controller.

[0063] In this embodiment, the distributed controller is specifically designed as follows:

[0064] ;

[0065] in, It represents the time-varying step size and satisfies the random approximation condition. Represents intelligent agents exist Perturbation data with added noise transmitted in real time.

[0066] The designed distributed controller will be used for subsequent convergence analysis and system state iteration.

[0067] Step 4. Design an encrypted pairing algorithm to defend against replay attacks.

[0068] In this embodiment, step 4 specifically includes:

[0069] A replay attack model is constructed in which an attacker, after intercepting legitimate network communication data of a multi-agent system, repeatedly sends the captured data to the target node.

[0070] The attack time period is defined as: .

[0071] in, to These represent the start times of the first to the sth attacks, respectively. to These represent the time lengths of the first to the sth attacks, respectively.

[0072] Total duration of replaying the attack for: Due to limitations in data storage capacity and device battery life, attackers cannot perform long-term monitoring and repeated replay operations; therefore, the total duration of a replay attack is limited. It is finite.

[0073] To defend against replay attacks, the method of this invention employs an encrypted pairing algorithm to ensure data security. Specifically, before transmitting the disrupted state, a dedicated pairing key is first generated, and then the disrupted state to be transmitted is coupled with this key. After coupling, communication is achieved through the network.

[0074] intelligent agent Send to neighboring intelligent agents The data encryption pairing algorithm is designed as follows:

[0075] .

[0076] in, Represents intelligent agents Send to neighboring intelligent agents Data, It is a non-singular matrix. Represents intelligent agents A key used to encrypt data.

[0077] intelligent agent Received from neighboring intelligent agents The data decryption algorithm is designed as follows:

[0078] .

[0079] in, Represents intelligent agents From neighboring intelligent agents The data received at the location Represents intelligent agents The key used to decrypt the data.

[0080] From neighboring intelligent agents Data obtained from Afterwards, the intelligent agent Data based on decryption algorithm Decryption is performed by comparing the keys. and To detect replay attacks.

[0081] when When a replay attack is detected, it is determined that a replay attack has been detected; when At that time, it was determined that no replay attack was detected.

[0082] If a replay attack is detected, then ;otherwise, .

[0083] The control input is rewritten as:

[0084] .

[0085] Among them, when a replay attack is detected When no replay attack was detected .

[0086] Figure 2 The communication process between intelligent agents is demonstrated, in which Represents intelligent agents exist The state at any given moment, Represents intelligent agents exist Time-based control input, Represents intelligent agents exist Random noise added at any time. (Intelligent agent) Only disturbed data with added noise is transmitted, not the actual state message. Therefore, the data obtained by the eavesdropper is disturbed data. And unable to obtain intelligent agents Real data This effectively protects the privacy data of intelligent agents. Furthermore, the method of this invention also detects the existence of replay attacks through encrypted pairing, thereby further protecting multi-agent systems.

[0087] In step 4 of this embodiment, after completing the design of the encryption pairing algorithm, a convergence analysis is also performed on the multi-agent system controlled by the distributed controller:

[0088] Assumption 1. Assume time-varying step size. Satisfy the random approximation conditions:

[0089] , , .

[0090] in, Indicates time-varying step size The upper bound.

[0091] Assumption 2. Assume time-varying step size. and noise parameters satisfy:

[0092] hour, , , .Right now Approaching hour, Approaching , , Approaching 0.

[0093] Assumption 3. Assumption Diagram It is an undirected connected graph.

[0094] Assumption 4. Assumption when or At that time, random noise and They are independent of each other.

[0095] in, Represents intelligent agents exist The random noise used at each time step.

[0096] Definition 1. Mean-square asymptotic convergence is defined as follows: if there exists a random variable... ,satisfy: , ,in If the expected value is expressed, then the multi-agent system is said to have achieved mean-square asymptotic consistency.

[0097] Theorem 1. Under the conditions that Assumptions 1 to 4 hold, multi-agent systems can achieve mean-square asymptotic convergence. That is, there exists a random variable... , so that:

[0098] .

[0099] The system dynamic equations are obtained as follows:

[0100] .

[0101] make:

[0102] .

[0103] .

[0104] .

[0105] in, This indicates that all intelligent agents are in The overall system state vector, composed of the actual states at each moment; This indicates that all intelligent agents are in The overall system state vector composed of the perturbation states at any given time; This indicates that all intelligent agents are in The overall system state vector is composed of the random noise added at each time step.

[0106] Therefore, the system equation is:

[0107] ;

[0108] in, This indicates that all intelligent agents are in The overall system state vector, composed of the actual states at each moment. Represents the identity matrix.

[0109] Define Lyapunov functions for: .

[0110] ,in This represents a column vector consisting entirely of 1s. This represents a row vector consisting entirely of 1s.

[0111] get:

[0112] .

[0113] in, express The Lyapunov function value at time t.

[0114] And thus obtain Multi-agent systems controlled by distributed controllers can achieve mean-square asymptotic convergence.

[0115] In step 4 of this embodiment, after completing the design of the encryption pairing algorithm, a convergence accuracy analysis is also performed on the multi-agent system controlled by the distributed controller:

[0116] Definition 2. Convergence accuracy is defined as: for a multi-agent system controlled by a distributed controller, the convergence accuracy is: and If mean-square asymptotic consistency holds, and satisfies Then the multi-agent system can achieve Accuracy, among which Represents probability. and Represents a constant.

[0117] Theorem 2. Under the condition that Assumptions 1 to 4 hold, the control method of the present invention is implemented. Accuracy, if the parameters satisfy:

[0118] .

[0119] According to Chebyshev's inequality , It is a constant, and , Represents random variables The variance; substituting ,get:

[0120] .

[0121] in, .

[0122] make ,but ,get:

[0123] .

[0124] .

[0125] because ,so .

[0126] when At that time, a multi-agent system controlled by a distributed controller is implemented. Precision.

[0127] The method of this invention adjusts the time-varying step size and noise parameters Different convergence accuracies can be achieved. As can be seen from the above formula, when the step size varies... and noise parameters The smaller the value, the higher the convergence accuracy.

[0128] In step 4 of this embodiment, after completing the design of the encryption pairing algorithm, a privacy protection analysis is also performed on the multi-agent system controlled by the distributed controller:

[0129] Definition 3. The adjacency dataset is defined as: preset Initial state set and They are adjacent and exist. This makes the following equation true:

[0130] .

[0131] That is, it exists , making hour , hour Established. Among them, Representing adjacent datasets Middle intelligence agent The state value at the initial moment.

[0132] Definition 4. Differential privacy is defined as: preset If for any two Adjacency Initial State Set and and observation set ,in express 3D real vector space, Represents all numbers from natural numbers Mapped to The set of is:

[0133] .

[0134] in, Representation mechanism Effect on dataset superior, Representation mechanism Effect on dataset Above; then mechanism yes Differential privacy mechanism. Specifically, this refers to a mechanism designed to protect privacy. In this embodiment, noise is added to achieve privacy protection; therefore, the mechanism... This refers to adding noise to a real-world environment.

[0135] Definition 5. Sensitivity is defined as: defining the dataset and and observation set Noisy sequences exist. and trajectory ,in Indicates in the dataset and observation set The first generated under the condition An intelligent agent in The state value at any given time.

[0136] mechanism In time Sensitivity to time for:

[0137] .

[0138] in, This represents the set of all adjacent dataset pairs that differ only in the initial information of a single agent. Indicates in the dataset and observation set The system trajectory generated under the conditions is The value at time, Indicates in the dataset and observation set The system trajectory generated under the conditions is The value at time.

[0139] Theorem 3. Under the condition that Assumption 3 holds, the sensitivity of the control method proposed in this invention is... satisfy:

[0140] ;

[0141] in, express The step size of time.

[0142] definition and As a set of trajectories.

[0143] The probability density function is and .

[0144] For dataset have:

[0145] .

[0146] in, for Step size of time, Indicates in the dataset and observation set The first generated under the condition An intelligent agent in The state value at time t, Represents intelligent agents exist Perturbation data with added noise transmitted in real time.

[0147] For dataset have:

[0148] .

[0149] in, In the dataset and observation sequence Under the conditions, the generated first An intelligent agent in The state value at time 10:00 In the dataset and observation sequence Under the conditions, the generated first An intelligent agent in The state value at any given time.

[0150] Due to the observation set For dataset and They are the same, among which Represents intelligent agents exist Perturbation data with added noise transmitted in real time.

[0151] therefore .

[0152] in, Indicates in the dataset and observation set The first generated under the condition The state values ​​of each agent at the initial moment; Indicates in the dataset and observation set The first generated under the condition The state value of each agent at the initial moment.

[0153] Through iteration, we obtained:

[0154] hour, ; hour, .

[0155] Theorem 3 shows that in the method of the present invention, with time... As the initial difference increases, its impact is gradually diminished, thus ensuring that the strength of differential privacy is controllable over time.

[0156] In differential privacy, the goal of privacy protection is to ensure that attackers cannot accurately infer sensitive information in the input dataset by observing the output. Sensitivity It directly reflects how sensitive the output is to changes in the input, and is therefore a key parameter for designing privacy protection mechanisms. It measures time At that time, two adjacent datasets and In the same observation set The maximum value of the trajectory difference generated below. Sensitivity The larger the value, the greater the impact of small changes in the dataset on the trajectory, thus requiring stricter privacy constraints to achieve privacy protection.

[0157] Theorem 4. Under the condition that Assumption 3 holds, the control method proposed in this invention is effective over any time range. Above is Differential privacy, among which .

[0158] Random noise follows a Laplace distribution, with probability density function for:

[0159] .

[0160] in, For a set, , Total time; For index pairs, index pairs middle Represents intelligent agents , , express time, ; Indicates in the dataset and observation set Intelligent agents generated under the conditions The trajectory in The value at time; yielding:

[0161] .

[0162] Theorem 4 shows the level of differential privacy With sensitivity and noise parameters Furthermore, the time-varying step size can be derived from the expression for sensitivity. The larger the size, the higher the level of privacy. The smaller the value, the better the privacy protection; similarly, the noise parameter... The larger the size, the higher the level of privacy. The smaller the step size, the better the privacy protection. Therefore, the method of this invention can adjust the time-varying step size. and noise parameters To achieve the expected level of privacy protection.

[0163] Criterion 1. Select a time-varying step size that satisfies the assumptions. and noise parameters They are respectively:

[0164] , .

[0165] in, , , Indicates a parameter, and , , .

[0166] When satisfied At that time, a multi-agent system controlled by a distributed controller can achieve infinite time. Differential privacy, in which Indicates the level of privacy protection The upper boundary.

[0167] When satisfied A multi-agent system controlled by a distributed controller can achieve convergence with a preset accuracy while maintaining differential privacy over an infinite time range.

[0168] Furthermore, this embodiment also balances convergence accuracy with privacy protection level.

[0169] According to Theorem 2 and Theorem 4, when the step size... and noise parameters The smaller the step size, the higher the convergence accuracy; and noise parameters The larger the area, the higher the level of privacy protection. The smaller the value, the better the privacy protection. This is because the noise parameter... The larger the value, the greater the noise injected into data transmission, and therefore the better the privacy protection effect; however, when the noise parameter... As the value increases, the introduced randomness increases, and the convergence accuracy decreases.

[0170] Specifically, select a step size of , , . for One of the parameters, when the noise parameter The larger the area, the higher the level of privacy protection. A smaller value enhances privacy protection but reduces convergence accuracy. Therefore, this can be addressed by adjusting the parameter. This allows us to select appropriate convergence accuracy and privacy protection levels, achieving a balance between the two.

[0171] Figure 3 and Figure 4 The parameters were studied. The relationship between convergence accuracy and privacy protection level.

[0172] Step 5. Using the distributed controller designed in Step 3 and the encrypted pairing algorithm designed in Step 4, implement consensus control of the multi-agent system against replay attacks.

[0173] The principle of the method of the present invention will be explained in detail below.

[0174] Differential privacy protection: The method of this invention protects privacy through intelligent agents. Transmit its status Add Laplace noise before giving it to your neighbors. This ensures that even if an attacker intercepts the perturbation data... It is also difficult to make an accurate inference. The original value is obtained, thus achieving differential privacy. This is achieved through noise parameters. The scale of noise control affects the strength of privacy protection.

[0175] Time-varying noise variance: Unlike existing technologies, the method of this invention allows for noise parameters Corresponding noise variance Over time, this design avoids the problem of the state trajectory being exposed when the noise decays to zero, thus providing stronger privacy protection.

[0176] Replay attack detection: The method of this invention also couples the time-varying key with the transmitted perturbation data. After coupling, communication is achieved through network transmission, and the existence of a replay attack can be detected by comparing the keys. When a replay attack is detected, the control input is set to 0 to ensure that the system is not disturbed by replay data; when no replay attack is detected, normal communication and state iteration between systems are performed.

[0177] Random approximation and convergence: The method of this invention uses a time-varying step size. Furthermore, by incorporating stochastic approximation theory, mean square asymptotic uniform convergence can be guaranteed even with increasing noise variance.

[0178] Parameter Design and Trade-offs: This invention provides a method for rationally selecting time-varying step sizes. and noise parameters This invention achieves predefined consistent convergence accuracy and differential privacy levels. Furthermore, theoretical analysis reveals the trade-off between convergence accuracy and privacy protection: increasing noise variance typically enhances privacy but may slow down the convergence rate.

[0179] The method of the present invention will be illustrated below with examples.

[0180] In a numerical example, consider a multi-agent system with six agents, whose communication topology is as follows: Figure 5 As shown, Figure 5 The numbers 1 to 6 in the diagram represent the first to sixth agents in the multi-agent system, respectively. The method of this invention achieves high accuracy. Convergence and privacy protection level Privacy protection, including distinguishing the precision parameters used in the theory. and In the simulation results, use and The parameter represents the convergence accuracy. The results obtained using the method of this invention are as follows: Figures 6 to 9 As shown. Figure 6 middle to These represent the actual states of the first to sixth agents in a multi-agent system. Figure 7 middle to These represent the perturbation states used for transmission by agents 1 through 6 in a multi-agent system. For clearer observation, Figure 8 The real state of agent 1 is given separately. and disturbance state The comparison Figure 8 Agent 1 in the multi-agent system is the first agent. Figure 9 middle to These represent the control inputs of the first to sixth agents in a multi-agent system.

[0181] The results obtained by comparing the noise mechanism of variance attenuation are as follows Figure 10 and Figure 12 As shown, it is evident that although both methods achieve uniform convergence, the decaying noise mechanism only provides protection for the initial state. As time increases, the added noise gradually decreases, causing the perturbed state to approximate the true state. Therefore, the comparison demonstrates that the variance-increasing noise employed in this invention provides stronger privacy protection.

[0182] Furthermore, the method of the present invention has a certain degree of flexibility, and the following alternative or extended solutions can be considered:

[0183] Imbalanced network structure: The method of the present invention is not only applicable to structurally balanced networks, but can also be extended to structurally unbalanced signed networks. With appropriate modifications, the method of the present invention can still achieve mean square asymptotic uniform convergence.

[0184] Binary Consistency: By dividing multiple agents into cooperative and competitive relationships, this invention can achieve binary consistency convergence.

[0185] Local Differential Privacy: The differential privacy concept of this invention can be extended to local differential privacy scenarios. In this case, each agent can independently select and set different noise parameters according to its own privacy needs, thereby achieving a personalized level of privacy protection.

[0186] Other noise distributions: Besides the Laplace distribution, other distributions that satisfy differential privacy requirements, such as the Gaussian distribution, can also be considered for privacy noise. As long as their variance characteristics and co-design with the step size can guarantee convergence and privacy levels, they can be used as alternatives.

[0187] Alternative forms for step size and noise parameters: Step size and noise parameters can take other functional forms that satisfy the convergence conditions, such as exponential decay forms or other polynomial forms, as long as they meet the convergence requirements in the assumptions.

[0188] Other cyberattacks: Other forms of attacks can be considered, such as spoofing attacks and DoS attacks.

[0189] This invention addresses the limitations of existing methods, such as noise form constraints, insufficient convergence analysis, and unclear parameter design. It provides a more general, robust, and adjustable mean-square consistency control method for multi-agent systems under differential privacy protection. Specifically, it relates to a method for achieving differential privacy mean-square asymptotic consistency in undirected networks against replay attacks. This technical field can be applied to scenarios requiring the protection of sensitive agent information such as location and private data, as well as defense against replay attacks, including distributed decision-making, cooperative control, and social network analysis.

[0190] This invention proposes a novel differential privacy mean-square consistency control method. This method allows the addition of privacy noise with time-varying increasing variance, thereby avoiding direct exposure of agent state information and achieving stronger privacy protection. Under this variance-increasing privacy noise, a time-varying step size satisfying a stochastic approximation method is introduced to ensure mean-square consistency and provides its convergence rate. Furthermore, to resist replay attacks, this invention employs an encrypted pairing algorithm to ensure data security. Before transmitting the perturbation state, a dedicated pairing key is generated, and the perturbation state to be transmitted is coupled with this key. After coupling, communication is achieved through network transmission. In addition, this invention provides a systematic method for designing the step size and noise parameters, revealing the inherent trade-off between accuracy and privacy to achieve the required consistency accuracy and a predefined level of differential privacy protection.

[0191] The method of this invention has the following innovative features:

[0192] First, this invention introduces time-varying privacy noise. It proposes a mechanism to add time-varying privacy noise during information transmission. This differs significantly from existing technologies where noise variance must decay exponentially to zero or remain constant. This effectively avoids the risk of directly inferring the state trajectory over time and provides more robust and durable global privacy protection.

[0193] Secondly, regarding the convergence guarantee under the privacy noise mechanism with increasing variance, this invention adopts a time-varying step size design based on a stochastic approximation method, theoretically proving that even if the variance of the privacy noise increases with time, mean square consistency can still be achieved, and the corresponding convergence rate is given.

[0194] Third, the invention employs an encrypted pairing method to detect replay attacks. During the transmission of perturbation data, a time-varying key is coupled with the transmitted perturbation data. After coupling, communication is achieved via network transmission. The presence of a replay attack can be detected by comparing the keys. When a replay attack is detected, the control input is set to 0 to ensure the system is not interfered with by replay data; when no replay attack is detected, normal inter-system communication and state iteration proceed.

[0195] Fourth, this invention addresses the trade-off between convergence accuracy and privacy protection. By designing the step size and noise parameters, it achieves the desired convergence accuracy and a preset level of differential privacy protection. Furthermore, the method explicitly reveals the inherent trade-off between accuracy and privacy, providing quantitative guidance for parameter selection in practical applications. In addition, this invention proposes a criterion, Criterion 1, which can maintain differential privacy over an infinite time range while achieving convergence with a preset accuracy.

[0196] Example 2

[0197] This embodiment 2 describes a computer device that includes a memory and one or more processors.

[0198] The memory stores executable code, which, when executed by the processor, implements the steps of the differential privacy consistency control method for multi-agent systems against replay attacks in Embodiment 1 above.

[0199] In this embodiment, the computer device can be any device or apparatus with data processing capabilities, and will not be described in detail here.

[0200] Example 3

[0201] This embodiment 3 describes a computer-readable storage medium storing a program that, when executed by a processor, implements the steps of a differential privacy consistency control method for a multi-agent system resistant to replay attacks.

[0202] The computer-readable storage medium can be an internal storage unit of any device or apparatus with data processing capabilities, such as a hard disk or memory, or an external storage device of any device with data processing capabilities, such as a plug-in hard disk, smart media card (SMC), SD card, flash card, etc.

[0203] 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. A differential privacy consistency control method for multi-agent systems facing replay attacks, characterized in that, Includes the following steps: Step 1. Establish a mathematical model of the multi-agent system; Step 2. Add randomly growing noise to the real state information that needs to be transmitted, so that the multi-agent system transmits the disturbed data with added noise during the communication process; Step 3. Based on the mathematical model of the multi-agent system established in Step 1, design a distributed controller; By employing a time-varying step size and combining it with stochastic approximation theory, mean-square asymptotic uniform convergence is guaranteed even with increasing noise variance. Step 4. Design an encrypted pairing algorithm to defend against replay attacks; Step 5. Using the distributed controller designed in Step 3 and the encrypted pairing algorithm designed in Step 4, implement consensus control of the multi-agent system against replay attacks.

2. The differential privacy consistency control method for multi-agent systems resistant to replay attacks as described in claim 1, characterized in that, Step 1 specifically involves: A collection of intelligent agents Represented as: ; edge set Represented as: ; in, Represents intelligent agents Received from intelligent agent Information; intelligent agent Neighbor set Represented as: ; Adjacency Matrix Represented as: ; Adjacency matrix The element, when hour ,otherwise ; Degree matrix Represented in diagonal matrix form: ,in For nodes The sum of the weights of all edges in the connected component; the degree matrix The maximum and minimum eigenvalues ​​are denoted as follows: and ; Laplace matrix Defined as: ; and These are the Laplace matrices. The maximum and minimum non-zero eigenvalues; Given an undirected connected graph, consider using... In a multi-agent system network consisting of [number] agents, each agent is considered a node, and the dynamic equation for each agent is: ； in, Represents intelligent agents exist The state at any given moment, Represents intelligent agents exist The state at any given moment; Represents intelligent agents exist Time-based control input; intelligent agent The initial state is denoted as , , , All are intelligent agents The true state information.

3. The differential privacy consistency control method for multi-agent systems resistant to replay attacks according to claim 2, characterized in that, Step 2 specifically involves: Information on the actual state Add random noise intelligent agent Transmitting disturbed data with added noise when communicating with neighbors : ； Among them, intelligent agents exist Random noise added at any time Follows a Laplace distribution: Added random noise They are mutually independent; This indicates that the mean is 0 and the variance is 0. Laplace distribution, This represents a time-varying, increasing noise parameter.

4. The differential privacy consistency control method for multi-agent systems resistant to replay attacks according to claim 3, characterized in that, In step 3, the distributed controller is designed as follows: ； in, Indicates time-varying step size, Represents intelligent agents exist Perturbation data with added noise transmitted in real time.

5. The differential privacy consistency control method for multi-agent systems resistant to replay attacks according to claim 4, characterized in that, Step 4 specifically involves: A replay attack model is constructed in which the attacker, after intercepting legitimate network communication data of a multi-agent system, repeatedly sends the captured data to the target node. The attack time period is defined as: ; in, to These represent the start times of the first to the sth attacks, respectively. to These represent the time lengths of the first to the sth attacks, respectively. Total duration of replaying the attack for: And the total duration of the replay attack. It is finite; intelligent agent Send to neighboring smart agents The data encryption pairing algorithm is designed as follows: ； in, Represents intelligent agents Send to neighboring intelligent agents Data, It is a non-singular matrix. Represents intelligent agents The key used to encrypt data; intelligent agent Received from neighboring intelligent agents The data decryption algorithm is designed as follows: ； in, Represents intelligent agents From neighboring intelligent agents The data received at the location Represents intelligent agents The key used to decrypt the data; From neighboring intelligent agents Data obtained from Afterwards, the intelligent agent Data based on decryption algorithm Decryption is performed by comparing the keys. and To detect replay attacks; when When a replay attack is detected, it is determined that a replay attack has been detected; when At that time, it was determined that no replay attack was detected; If a replay attack is detected, then ;otherwise, ; The control input is rewritten as: ； Among them, when a replay attack is detected When no replay attack was detected .

6. The differential privacy consistency control method for multi-agent systems facing replay attacks according to claim 5, characterized in that, In step 4, after completing the design of the encryption pairing algorithm, a convergence analysis is performed on the multi-agent system controlled by the distributed controller: Assuming time-varying step size Satisfy the random approximation conditions: , , ; in, Indicates time-varying step size The supremum; Assuming time-varying step size and noise parameters satisfy: Approaching hour, Approaching , , Approaching 0; Hypothesis diagram It is an undirected connected graph; Assuming when or At that time, random noise and They are independent of each other, among which Represents intelligent agents exist Random noise used at any given time; Mean-square asymptotic convergence is defined as follows: if there exists a random variable ,satisfy: , ,in If the expected value is expressed, then the multi-agent system is said to have achieved mean-square asymptotic consistency. Under the assumption that this holds, multi-agent systems can achieve mean-square asymptotic convergence. The system dynamic equations are obtained as follows: ； make: ； ； ； in, It means that all intelligent agents are in The overall system state vector, composed of the actual states at each moment; This indicates that all intelligent agents are in The overall system state vector composed of the perturbation states at any given time; It means that all intelligent agents are in The overall system state vector composed of the random noise added at each time step; The system equations are: ； in, It means that all intelligent agents are in The overall system state vector, composed of the actual states at each moment. Represents the identity matrix; Define Lyapunov functions for: ; ,in This represents a column vector consisting entirely of 1s. This represents a row vector consisting entirely of 1s; get: ； in, express The Lyapunov function value at time t; And thus obtain Multi-agent systems controlled by distributed controllers can achieve mean-square asymptotic convergence.

7. The differential privacy consistency control method for multi-agent systems facing replay attacks according to claim 6, characterized in that, In step 4, after completing the design of the encryption pairing algorithm, a convergence accuracy analysis is performed on the multi-agent system controlled by the distributed controller: Convergence accuracy is defined as: for a multi-agent system controlled by a distributed controller, the convergence accuracy is: and If mean-square asymptotic consistency holds, and satisfies Then the multi-agent system can achieve Accuracy, among which Represents probability. and Represents a constant; The parameters satisfy: ； According to Chebyshev's inequality , It is a constant, and , Represents random variables The variance; Substitution ,get: ； in, ; make ,but ,get: ； ； because ,so ; when At that time, a multi-agent system controlled by a distributed controller is implemented. Precision.

8. The differential privacy consistency control method for multi-agent systems resistant to replay attacks according to claim 7, characterized in that, In step 4, after completing the design of the encryption pairing algorithm, a privacy protection analysis is performed on the multi-agent system controlled by the distributed controller: The adjacency dataset is defined as: preset Initial state set and They are adjacent and exist. , making hour , hour Established, among which Representing adjacent datasets Middle intelligence agent The state value at the initial moment; Differential privacy is defined as: preset If for any two Adjacency Initial State Set and and observation set ,in express 3D real vector space, Represents all numbers from natural numbers Mapped to The set of is: ,in Representation mechanism Effect on dataset superior, Representation mechanism Effect on dataset Above, then the mechanism yes Differential privacy; Sensitivity is defined as: defining the dataset and and observation set Noisy sequences exist. and trajectory ,in Indicates in the dataset and observation set The first generated under the condition An intelligent agent in The state value at any given moment, the mechanism In time Sensitivity to time for: ； in, This represents the set of all adjacent dataset pairs that differ only in the initial information of a single agent. Indicates in the dataset and observation set The system trajectory generated under the conditions is The value at time, Indicates in the dataset and observation set The system trajectory generated under the conditions is The value at time; Under the assumption that this holds, sensitivity satisfy: ； in, express The step size of time; definition and As a set of trajectories; The probability density function is and ; For dataset have: ； in, for Step size of time, Indicates in the dataset and observation set The first generated under the condition An intelligent agent in The state value at time 10:00 Represents intelligent agents exist Perturbation data with added noise transmitted in real time; For dataset have: ； in, In the dataset and observation sequence Under the conditions, the generated first An intelligent agent in The state value at time 10:00 In the dataset and observation sequence Under the conditions, the generated first An intelligent agent in The state value at any given time; Due to the observation set For dataset and They are the same, among which Represents intelligent agents exist Perturbation data with added noise transmitted in real time; therefore ; in, Indicates in the dataset and observation set The first generated under the condition The state values ​​of each agent at the initial moment; Indicates in the dataset and observation set The first generated under the condition The state values ​​of each agent at the initial moment; Through iteration, we obtained: hour, ; hour, ; Under the assumption that holds, within any time range Above is Differential privacy, among which ; Random noise follows a Laplace distribution, with probability density function for: ； in, For a set, , Total time; For index pairs, index pairs middle Represents intelligent agents , , express time, ; Indicates in the dataset and observation set Intelligent agents generated under certain conditions The trajectory in The value at time; yielding: ； Choose a time-varying step size that satisfies the assumptions. and noise parameters They are respectively: , ; in, , , Indicates a parameter, and , , ; When satisfied At that time, a multi-agent system controlled by a distributed controller can achieve infinite time. Differential privacy, in which Indicates the level of privacy protection The upper bound; When satisfied A multi-agent system controlled by a distributed controller can achieve convergence with a preset accuracy while maintaining differential privacy over an infinite time range.

9. A computer device comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that, When the processor executes the executable code, it implements the steps of the differential privacy consistency control method for multi-agent systems against replay attacks as described in any one of claims 1 to 8.

10. A computer-readable storage medium having a program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the differential privacy consistency control method for multi-agent systems against replay attacks as described in any one of claims 1 to 8.

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