Optimal Energy Scheduling Method for Active Eavesdroppers under Remote State Estimation Systems

Abstract

The present invention proposes an optimal energy scheduling method for an active eavesdropper under a remote state estimation system. For a class of active eavesdroppers that can implement attacks and eavesdropping with a finite number of energy options to achieve different eavesdropping performances and legitimate estimator receiving performances, by minimizing the eavesdropping cost and estimation error of the eavesdropper within a finite time, and at the same time maximizing the estimation error of the estimator, an eavesdropping model of the active eavesdropper under the remote state estimation system is established. It is theoretically proved that when the estimation error of the eavesdropper is given, there is a certain threshold characteristic between any two different energy options, so the optimal strategy between any two strategies can be obtained, and then the optimal energy strategy of the active eavesdropper under the remote state estimation system can be determined. The present invention conducts strategy analysis from the perspective of the active eavesdropper, provides a reference for the defense of cyber-physical systems (CPS), and is conducive to the upgrade of defense mechanisms and strategies.

Description

Technical Field

[0001] The present invention belongs to the technical field of wiretapping; in particular, it relates to an optimal energy scheduling method for active eavesdroppers under a remote state estimation system. Background Art

[0002] The system state information of a remote state estimation system needs to be measured by sensors and sent to a remote estimator through an untrusted channel, and the channel may be attacked or wiretapped by an adversary. The security issue of remote state estimation is an important security issue in a cyber-physical system (CPS). A remote state estimation system generally suffers from active attacks or passive wiretapping, where active attacks include denial-of-service attacks, jamming, data integrity attacks, etc.

[0003] From the perspective of legitimate sensors, for passive wiretapping, the sensors can achieve optimal transmission energy scheduling by selecting transmission options to cope with the wiretapping problem of eavesdroppers and energy-constrained scenarios. For example, when the sensors have two options of transmitting and not transmitting, the estimation error of the legitimate estimator can be bounded while the estimation error of the eavesdropper is unbounded; in the case of energy constraint, the sensors can choose not to transmit, transmit in plaintext, or transmit encrypted to achieve the optimal secrecy effect, etc.

[0004] However, eavesdroppers can also use various means to carry out attacks and improve their wiretapping performance. Moreover, the above optimal transmission energy scheduling is also obtained based on the assumption that the eavesdropper adopts a constant wiretapping strategy, which underestimates the ability of the eavesdropper. Therefore, it is necessary to perform scheduling from the perspective of the eavesdropper, considering how to schedule energy to carry out attacks and wiretapping throughout the working time, so as to achieve the best wiretapping effect or attack effect at the lowest cost, etc.

[0005] For an active eavesdropper, the energy used to carry out attacks and wiretapping is related to the attack cost of the eavesdropper. Therefore, it is necessary to comprehensively consider the specific situation of the system at a specific time and the benefits of its previous attacks to determine its attack and wiretapping energy, so as to minimize the attack cost and estimation error of the active eavesdropper within a limited time and maximize the estimation error of the estimator. Summary of the Invention

[0006] The present invention needs to determine which option an active eavesdropper should choose at each time step within a limited time to make the comprehensive index of the attack cost, estimation error of the active eavesdropper, and estimation error of the estimator meet the interests of the active attacker.

[0007] To solve the above problems, the present invention discloses an optimal energy scheduling method for an active eavesdropper in a remote state estimation system. By implementing pilot contamination attacks with different energies, the eavesdropping performance of the active eavesdropper and the receiving performance of the estimator can be changed. Specifically, the higher the energy used for the pilot contamination attack, the higher the packet reception success rate of the eavesdropper, while the packet reception success probability of the estimator decreases. Thus, different attack options are formed for different pilot contaminations and eavesdropping energies.

[0008] The optimal energy scheduling method for an active eavesdropper in a remote state estimation system specifically includes the following steps:

[0009] S1: Define basic elements;

[0010] S2: Establish an optimal eavesdropping energy scheduling model for a finite time;

[0011] S3: Given the eavesdropping estimation error at the previous time step, prove the threshold characteristics between any two options;

[0012] S4: Determine the optimal eavesdropping energy.

[0013] Further preferably, the specific definition of the basic elements in step S1 is as follows:

[0014] K is the total number of time steps, P k , are the estimation error of the estimator and the estimation error of the eavesdropper at time step k, respectively; the set is all possible values; for simplicity, we also use to represent the estimation error of the estimator and the estimation error of the eavesdropper at time step k, and are equivalent. If then n k = n, The relationship between is similar; θ k represents the selected strategy at time step k, where the available active eavesdropping energy options at each time step are respectively (θ 1 , θ 2 ,..., θ m ), and the corresponding increasing costs and the packet reception success rates of the eavesdropper are respectively (c 1 , c 2 ,..., c m ), (λ 1 , λ 2 ,..., λ m ), and the decreasing packet reception success rates of the estimator (μ 1 , μ 2 ,..., μm )。

[0015] Further preferably, in step S2, an optimal eavesdropping energy scheduling model with a limited time is established:

[0016]

[0017] where β ∈ [0, 1] is a regulation factor, which is used to control the roles played by the estimation errors of the eavesdropper and the estimator; P k , is the estimator error and the eavesdropper error at time step k; c(θ k ) is the cost of using a certain strategy at time step k; the establishment of this model considers minimizing the energy consumption of the active eavesdropper, minimizing the estimation error of the active eavesdropper, and maximizing the estimation error of the estimator.

[0018] Its equivalent form is given in an iterative form:

[0019]

[0020] where k = K, K - 1, …, 1. The first line of the above formula must be set for iterative progress. The second line is divided into three parts. The first part c(θ k ) represents the transmission cost of this time step. The second part minimizes the eavesdropper error at this time and maximizes the estimator estimation error. The third part represents the cost required for subsequent time steps.

[0021] Further preferably, in step S3, given the eavesdropping estimation error of the previous time step, the threshold characteristics between any two options are proved:

[0022]

[0023] This means that after the eavesdropping estimation error of the previous time step is given, between any two options, according to the comparison of the eavesdropping estimation error of the previous time step with the threshold, it can be judged which of the two options is better.

[0024] Or using the equivalent n k to represent the estimation error, there is

[0025]

[0026] where i < j.

[0027] Or is the threshold between the two options, and its calculation method is:

[0028]

[0029] or use the equivalent n k represent the estimation error to obtain the threshold

[0030]

[0031] This is given based on the threshold characteristics, and the three cases of calculation respectively correspond to the attached Figures 1-3 .

[0032] Further preferably, the determination of the optimal eavesdropping energy in step S3:

[0033]

[0034]

[0035] That is, according to the nature of the threshold characteristics in step S3, one inferior option is excluded each time. Since the number of energy options is limited, the optimal option can be finally selected.

[0036] Advantages of the present invention:

[0037] By considering minimizing the attacks and eavesdropping costs of active eavesdroppers, minimizing the estimation error of active eavesdroppers, and simultaneously maximizing the estimation error of the estimator, when the estimation error of the eavesdropper in the previous time step is given, there is a certain threshold characteristic between any two options, and thus the optimal option for each time step is determined. The present invention finally achieves as good eavesdropping and estimation effects as possible at the minimum energy cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figures 1-3 For for P k-1 Schematic diagrams of three cases.

[0039] Figure 4 Schematic diagram of comparison between the optimal scheduling strategy and the cost index under a single strategy in a certain case.

[0040] Figure 5 Schematic diagram of the optimal strategy at k = 3 time when the total number of time steps is 10 in a certain case.

[0041] Figure 6 Schematic diagram of the optimal strategy at k = 7 time when the total number of time steps is 10 in a certain case.

[0042] Figure 7 Schematic diagram of the optimal strategy at k = 10 time when the total number of time steps is 10 in a certain case;

[0043] Figure 8 Schematic diagram of the feasible optimal scheduling strategy. DETAILED DESCRIPTION OF THE INVENTION

[0044] The present invention will be further illustrated below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and not to limit the scope of the present invention.

[0045] For an active eavesdropper, taking the implementation of a pilot contamination attack as an example, the following parameters are defined in the present invention: K is the total number of time steps, P k , are the estimation error of the estimator and the estimation error of the eavesdropper at time step k, respectively; the set is the set of all possible values of P k , ; for simplicity, we also use n k , to represent the estimation error of the estimator and the estimation error of the eavesdropper at time step k. The relationship between n k , and P k , is similar. If , then n k =n, The relationship between is similar; θ k represents the selected strategy at time step k. The available active eavesdropping energy options at each time step are respectively (θ 1 , θ 2 ,..., θ m ). The corresponding increasing costs and the packet reception success rates of the eavesdropper are respectively (c 1 , c 2 ,..., c m ), (λ 1 , λ 2 ,..., λ m ), and the decreasing packet reception success rates of the estimator are (μ 1 , μ 2 ,..., μ m ).

[0046] First, based on minimizing the attack and eavesdropping costs of the active eavesdropper, minimizing the estimation error of the active eavesdropper, and maximizing the estimation error of the estimator, the optimal energy scheduling problem of the active eavesdropper in the remote state estimation system is established as follows:

[0047]

[0048] where β ∈ [0, 1] is a regulation factor that is used to control the roles played by the estimation errors of the eavesdropper and the estimator; P k , are the estimator error and the eavesdropper error at time step k; c(θ k) is the cost of using a certain strategy at time step k; the establishment of this model considers minimizing the energy consumption of the active eavesdropper, minimizing the estimation error of the active eavesdropper, and maximizing the estimation error of the estimator.

[0049] This problem can be transformed into its equivalent iterative form:

[0050]

[0051] where k = K, K - 1,..., 1. The first line of the above formula is what must be set for iterative progress. The second line is divided into three parts. The first part c(θ k ) represents the transmission cost of this time step. The second part minimizes the eavesdropper error at this time and maximizes the estimator estimation error. The third part represents the cost required for subsequent time steps.

[0052] We define So the error between using different options θ i , θ j is: Given the estimation error of the eavesdropper at the previous time step, it can be proved that is monotonically increasing with respect to the estimation error P k-1 of the estimator at the previous time step. Therefore, the difference between the two options θ i , θ j (i < j) is as shown in the case of Appendix Figures 1-3 . Therefore, we can determine the threshold or

[0053]

[0054] or

[0055]

[0056] Subsequently, determine the better one of the two options and exclude the inferior option:

[0057]

[0058] or

[0059]

[0060] For any two options θ i , θ jWhen (i < j), we can exclude the inferior options. Therefore, for m options, we only need to compare them pairwise and exclude one inferior option each time. Eventually, we will surely obtain the optimal option at that time.

[0061] According to the optimal option, we are based on Equation

[0062]

[0063]

[0064] Update Subsequently, we can solve the optimal option for each time step in turn until we obtain the energy option that the active eavesdropper should choose at each time step within a finite time, thus realizing the optimal energy scheduling.

[0065] We use the iterative formula as an index to measure the effect, where the parameters are all taken as the initial values. See specifically Figure 2 .

[0066] If the active eavesdropper adopts a single strategy, then it may incur a large attack cost but obtain a small benefit. As Figure 4 shown, for different trade-off parameters β, the cases where the active eavesdropper adopts a single strategy and the optimal strategy scheduling are respectively compared. From the evaluation index, the optimal eavesdropping strategy scheduling is superior to the other 5 strategy schedulings. Because the value of the optimal eavesdropping strategy scheduling is always the lowest.

[0067] The total time window given in the present invention is K = 10, and the eavesdropper has 5 eavesdropping strategies. The physical process of the remote state estimation system is linear time-invariant, and the process is as follows:

[0068] x k+1 = Ax k + ω k

[0069] y k = Cx k + ν k

[0070] where the parameters are given as:

[0071]

[0072] After the sensor and the eavesdropper successfully receive the data packet, their estimation error is Otherwise, their estimation error is h(P), h(P e ), where h(X) = AXA T + Q, X is a positive definite matrix, and the estimation errors in this specification all fall within the set in

[0073] For five kinds of transmitted energy, given their consumption, the packet reception success rate of the eavesdropper, and the packet reception success rate of the estimator are respectively: (10, 30, 60, 100, 150), (0.5, 0.7, 0.8, 0.85, 0.9), (0.9, 0.6, 0.5, 0.4, 0.3). Given β = 0.5 and the policy scheduling is obtained. Figures 5-7 The policy scheduling situations at time steps 3, 7, and 10 are respectively given. For each time step, for different estimator errors and different eavesdropper errors, the energy options to be used in this situation can be given.

[0074] The technical means disclosed in the solution of the present invention are not limited to the technical means disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features.

Claims

1. Optimal Energy Scheduling Method for Active Eavesdroppers in a Remote State Estimation System Characterized in that, Specifically includes the following steps: S1: Define basic elements; S2: Establish an optimal eavesdropping energy scheduling model for a finite time; The step S2 establishes an optimal eavesdropping energy scheduling model for a finite time: where β ∈ [0, 1] is a regulation factor that controls the roles played by the estimation errors of the eavesdropper and the estimator; P k , is the estimator error and the eavesdropper error at time step k; c(θ k ) is the cost of using a certain strategy at time step k; the model is established considering minimizing the energy consumption of the active eavesdropper, minimizing the estimation error of the active eavesdropper, and maximizing the estimation error of the estimator; Give its equivalent form in an iterative form: where k = K, K-1, …, 1; the first line of the above formula is necessary for iterative progress, and the second line is divided into three parts. The first part c(θ k ) represents the transmission cost of this time step. The second part minimizes the eavesdropper error at this time and maximizes the estimator error. The third part represents the cost required for subsequent time steps; S3: Given the eavesdropping estimation error at the previous time step, prove the threshold characteristics between any two options; S4: Determination of the optimal eavesdropping energy.

2. The optimal energy scheduling method for active eavesdroppers in a remote state estimation system according to claim 1, Characterized in that, The specific definition of the basic elements in step S1 is as follows: K is the total number of time steps, and P k , are the estimation errors of the estimator and the eavesdropper at time step k, respectively; the set is P k , all possible values; for simplicity, we also use n k , to represent the estimation errors of the estimator and the eavesdropper at time step k. n k , and P k , are equivalent. If then n k = n, and the relationship between is similar; θ k represents the selected strategy at time step k. The available active eavesdropping energy options at each time step are respectively (θ 1 , θ 2 ,..., θ m ). The corresponding increasing costs and the packet reception success rates of the eavesdropper are respectively (c 1 , c 2 ,..., c m ), (λ 1 , λ 2 ,..., λ m ), and the decreasing packet reception success rates of the estimator are (μ 1 , μ 2 ,..., μ m ).

3. The optimal energy scheduling method for active eavesdroppers in a remote state estimation system according to claim 1, Characterized in that: The step S3 given the eavesdropping estimation error at the previous time step, prove the threshold characteristics between any two strategies: Where i < j; This means that after the eavesdropping estimation error at the previous time step is given, between any two options, according to the comparison of the eavesdropping estimation error at the previous time step with the threshold, it can be judged which of the two options is better; or use the equivalent n k denotes the estimation error and there is Where i < j; or is the threshold between two options and is calculated as follows: Or This is given based on the threshold characteristics.

4. The optimal energy scheduling method for active eavesdroppers in a remote state estimation system according to claim 3, Characterized in that, The determination of the optimal eavesdropping energy in the step S3: That is, according to the nature of the threshold characteristics of the step S3, each time a suboptimal option is excluded; Since the number of energy options is finite, the optimal option can be finally selected.

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