CUSUM detection-oriented covert communication performance evaluation and optimization method

Abstract

The application discloses a kind of hidden communication performance evaluation and optimization method for CUSUM detection, comprising the following steps: modeling the process of CUSUM detection method in sequence change point detection;Based on the principle that CUSUM test is repeated SPRT application in the same process, analyze each parameter in SPRT;The upper limit of excess expectation is obtained by using Lorden inequality, and the limit of each parameter is obtained by scaling different parameters, and then the closed-form approximate solution of average false alarm running time and average detection delay is obtained;The limit of each parameter obtained is substituted into the closed-form solution of average false alarm running time and average detection delay to obtain the characteristic function and approximate solution of average false alarm running time and average detection delay;Optimize the detection threshold, minimize the detection delay under the condition of meeting the false alarm constraint;The application gives a new method for performance evaluation of hidden communication system under CUSUM test.

Description

Technical Field

[0001] This invention relates to the field of communication technology, and more specifically to a method for evaluating and optimizing the performance of covert communication for CUSUM detection. Background Technology

[0002] Covert wireless communication is a special communication method that conceals communication activities, and it has significant application prospects in military and other fields with high requirements for communication security. Unlike traditional encrypted communication, covert communication can hide communication activities, making it impossible for the enemy to determine whether communication has occurred. Traditional covert communication is essentially a physical problem; any adversary with sufficient computing power can decrypt encrypted communication. Covert communication, however, is a more advanced physical layer security implementation. In the context of covert communication, the transmission power is typically required to be very low, making the communication undetectable to the adversary. Currently, the mainstream method for evaluating covert performance is offline detection, using KL divergence to characterize the error probability. Given a constant, if the adversary's missed detection probability and false alarm probability are greater than a certain constant, covert communication can be considered to have been achieved.

[0003] In offline detection methods, Willie's detection performance is affected by the detection window and Alice's transmission time. If Alice transmits information at some point within Willie's detection window, the sequence observed by Willie will be a mixture of noise and transmitted symbols, thus degrading detection performance. Only when the timing of Willie's detection window and Alice's transmission is perfectly matched can the observed sequence be either entirely noise or entirely transmitted symbols.

[0004] Furthermore, traditional offline detection suffers from significant latency; Willie can only perform detection after receiving all the symbols. Alice has already transmitted all the information, so even if Willie successfully detects the communication at the end, it cannot prevent the communication from occurring. Existing detection methods cannot perform real-time detection when communication occurs, making it impossible for the detector to take action against Alice's communication behavior to prevent her from continuing to communicate. Summary of the Invention

[0005] Purpose of the invention: The purpose of this invention is to provide a method for evaluating and optimizing the performance of covert communication for CUSUM detection, and to provide an online detection method in addition to existing offline detection methods, which makes decisions in real time based on observations.

[0006] Technical solution: The present invention provides a method for evaluating and optimizing the performance of covert communication for CUSUM detection, comprising the following steps:

[0007] (1) The CUSUM detection method process in sequence change point detection is modeled, and the covert transmission time and covert performance of covert communication are described by average false alarm runtime, average detection delay, average false alarm runtime and average detection delay ratio.

[0008] (2) Based on the principle that CUSUM test is a repeated SPRT application in the same process, analyze the parameters in SPRT; use the Lorden inequality to obtain the upper bound of the excess expectation, and scale different parameters to obtain the limits of each parameter, and then obtain the closed approximate solution of the average false alarm runtime and the average detection delay.

[0009] (3) Substitute the obtained parameter limits into the closed-form solution of the average false alarm duration and the average detection delay to obtain the characteristic function and approximate solution of the average false alarm duration and the average detection delay.

[0010] (4) Based on the approximate expressions of the average false alarm runtime and average detection delay obtained, the detection threshold is optimized to minimize the detection delay while satisfying the false alarm constraint.

[0011] Furthermore, step (1) includes the following steps:

[0012] (11) Model the codebook;

[0013] (12) Model the CUSUM detector;

[0014] (13) Model the stopping time;

[0015] (14) Model the average runtime;

[0016] (15) Model the average detection delay;

[0017] (16) Convert the modeling problem in steps (11) to (15) into a mathematical problem.

[0018] Furthermore, in step (11), input symbols are randomly selected from the discrete codebook; the encoded message is an independent and identically distributed random codeword, and the selected codebook includes Gaussian codebook or BPSK codebook.

[0019] Furthermore, in step (12), the CUSUM detector codebook is modeled: the detector samples a symbol at each time point and uses the log-likelihood ratio to accumulate statistics; based on the received symbols and the codebook composition, the detector sets up a binary hypothesis test; and uses the average detection delay and average false alarm runtime to characterize the concealment.

[0020] Furthermore, step (2) includes the following steps:

[0021] (21) Based on the Lorden inequality, the upper bound of the excess expectation is obtained; the KL divergence is approximated by a Taylor series and expanded for transmission power to obtain the first-order expansion of the KL divergence.

[0022] (22) Using the upper bound of the obtained excess expectation and the inequality, the bounds of the other parameters are obtained;

[0023] (23) Substitute the obtained parameter limits into the closed-form solution of the average false alarm running time and the average detection delay to obtain the approximate solution and the characteristic function.

[0024] Furthermore, step (4) includes the following steps:

[0025] (41) Based on the approximate expressions for the average false alarm runtime and average detection delay obtained, the threshold optimization problem is transformed into a joint optimization problem of the power coefficient and the threshold.

[0026] (42) Based on the derivative of the characteristic function, the unique solution of the power coefficient and threshold that satisfy the conditions is obtained, thus obtaining the concealment performance expression when the detection performance is optimal.

[0027] The present invention discloses a covert communication performance evaluation and optimization system for CUSUM detection, comprising:

[0028] Description module: Used to model the CUSUM detection method process in sequence change point detection, and uses average false alarm runtime, average detection delay, average false alarm runtime and average detection delay ratio to describe the covert transmission duration and covert performance of covert communication;

[0029] Approximate Solution Module: Based on the principle that CUSUM test is a repeated SPRT application in the same process, it analyzes the parameters in SPRT; it uses the Lorden inequality to obtain the upper bound of the excess expectation, and scales different parameters to obtain the limits of each parameter, thereby obtaining closed-form approximate solutions for the average false alarm runtime and average detection delay.

[0030] Closed-form solution module: Used to substitute the obtained parameter limits into the closed-form solution of average false alarm duration and average detection delay to obtain the characteristic function and approximate solution of average false alarm duration and average detection delay;

[0031] The optimization module is used to optimize the detection threshold based on the approximate expressions of the obtained average false alarm runtime and average detection delay, and to minimize the detection delay while satisfying the false alarm constraint.

[0032] An electronic device according to the present invention includes a memory and a processor, wherein the memory stores a computer program, and the processor executes the program to implement the method described herein.

[0033] The present invention discloses a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the method described herein.

[0034] Beneficial Effects: Compared with existing technologies, this invention has the following significant advantages: This invention provides approximate solutions for the average false alarm run length and average detection delay of infinite covert communication under discrete codebooks in the CUSUM test. It presents a characterization of covert communication performance in the CUSUM test, using simple statistics to accurately characterize the average false alarm run length and average detection delay, with errors controlled within a very small range. A novel method is presented for performance evaluation of covert communication systems under the CUSUM test. Attached Figure Description

[0035] Figure 1 This is a flowchart of the present invention. Detailed Implementation

[0036] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0037] like Figure 1 As shown, this embodiment of the invention provides a method for evaluating and optimizing the performance of covert communication for CUSUM detection, including the following steps:

[0038] (1) Model the process of the CUSUM detection method in sequence change point detection; characterize the concealment performance by using the average false alarm runtime and average detection delay and their ratio; and use Gaussian codebook to encode the covert communication sender messages into a codebook. ,in These are codewords generated independently and randomly with identical distributions (iID); each symbol detected by the receiver may be noise or a superposition of noise and transmitted symbols, as shown in the following formula:

[0039]

[0040] Each observation symbol may follow one of two distributions. or , or It follows a Gaussian distribution, as shown in the following formula:

[0041] ;

[0042] There is also the following definition:

[0043] ;

[0044] ;

[0045] ;

[0046] ;

[0047] ;

[0048] ;

[0049] Inspired by Lorden's minimax detection method and the optimality of the CUSUM test, the same problem setting is used, and the CUSUM detector is employed. The sign of each sample is the log-likelihood ratio before and after the change. The statistic for the CUSUM test is the sum of the log-likelihood ratios. Unlike the simple summation of log-likelihood ratios, the CUSUM test uses a zero-based approach, as shown in the following formula:

[0050] ;

[0051] ;

[0052] The stopping time of the CUSUM test is defined as:

[0053] ;

[0054] The expected stopping time under both hypothesis tests is defined as follows: and The average false alarm runtime and average detection delay are defined as follows:

[0055] ;

[0056] Because the detector selects the optimal threshold while satisfying the false alarm constraint. To minimize detection latency, the wireless covert communication problem under the CUSUM test can be described as follows:

[0057] ;

[0058] Concealment performance can be characterized as:

[0059]

[0060] (2) Since the CUSUM test is a repeated SPRT application in the same process, the parameters in the SPRT are analyzed first; the upper bound of the excess expectation is obtained based on the Lorden inequality, and the bounds of each parameter are obtained by scaling for different parameters and then substituted into the equation. and In the closed-form expression, an approximate solution is obtained;

[0061] The stop time of SPRT is defined as:

[0062] ;

[0063] SPRT has two types of error probabilities, namely the Type I error probability. Type II error probability The expected values ​​and parameters of the remaining excesses are defined as follows:

[0064] ;

[0065] ;

[0066] ;

[0067] ;

[0068] ;

[0069] ;

[0070] ;

[0071] ;

[0072] ;

[0073] ;

[0074] Excess and The expectation can be obtained from the Lorden inequality, which can be expressed as:

[0075] ;

[0076] in It is an excess, and I is the KL divergence. The moments of the excess can be obtained from the Lorden inequality, especially when... Sometimes, the expected value of the excess can be directly obtained using simple statistics. When, we can obtain from Jensen's inequality:

[0077] ;

[0078] ;

[0079] because ,and There are no simple expressions for solving the moments of each order, so we can... Scaling is performed on the moments of each order.

[0080] when When it is an odd number, We can use integral inequalities to expand and contract the integral inequalities. The solution for even-order moments is transformed into a simpler method. Solving for the moments:

[0081] ;

[0082] In the covert communication problem, the excess expectation of the classic SPRT can be derived from... The solution is found, but the excess expectation obtained in the context of covert communication is too loose. Improvement is needed. The value, due to When it is odd, it can be complex The solution of the moments is transformed into a simpler The solution for the moments, therefore in Solving at that time, we can obtain:

[0083] ;

[0084] ;

[0085] From Wald's log-likelihood ratio equation, we can further derive two additional upper bounds on the excess expectation:

[0086] ;

[0087] ;

[0088] Furthermore, the expected values ​​of the remaining parameters can be obtained:

[0089] ;

[0090] Similarly, we can obtain

[0091] ;

[0092] Type I error probability Type II error probability The following relationship must be satisfied:

[0093] ;

[0094] ;

[0095] ;

[0096] ;

[0097] Depend on , and , We can obtain the following respectively:

[0098] ;

[0099] ;

[0100] Furthermore, we can obtain:

[0101] ;

[0102] ;

[0103] There are several forms of probability for two types of errors; here we use... , and , Solve for the closed-form solutions of the probabilities of the two types of errors:

[0104] ;

[0105] In the classic CUSUM setup, the optimal reset threshold is 0. However, setting the corresponding threshold to 0 in SPRT will cause detection to fail. Degeneracy, i.e., assumption The fact that detection is always accepted with a probability of 1 renders it meaningless. To solve this problem, a new lower threshold is set. It can always be adjusted. The velocity approaches 0, thus making the limit hold in the asymptotic case, as shown in the formula:

[0106] ;

[0107] By combining the expected value of excess with the formula for error probability and the definition of parameters, we can obtain the range of each parameter:

[0108] ;

[0109] ;

[0110] ;

[0111] ;

[0112] ;

[0113] ;

[0114] ;

[0115] ;

[0116] The upper and lower bounds for the two types of errors are:

[0117] ;

[0118]

[0119] Among them, for two important ratios and The specific range can be obtained through Monte Carlo simulation:

[0120] ;

[0121] (3) Based on the obtained parameter range, combine and The closed-form solution can be obtained regarding Approximate expression:

[0122] and The closed-form solution expression is:

[0123] ;

[0124] ;

[0125] Substituting the range of all parameters yields:

[0126] ;

[0127] ;

[0128] From this, we can obtain and The characteristic function is:

[0129] ;

[0130] By differentiating the characteristic equation, we can obtain the equation that satisfies the conditions. There is only one solution. Define a new function as:

[0131] ;

[0132] when At that time, we can obtain:

[0133] ;

[0134] therefore It is monotonically decreasing. Because exist From time to time You can get when hour and hour Furthermore, due to and There exists a unique Make Therefore when hour and hour .because and Therefore, there exists a unique solution. Make .because It is a monotonically increasing function, therefore There is also a unique solution.

[0135] (4) Based on the obtained and An approximate expression is used to optimize the threshold, minimizing the detection delay while satisfying the false alarm constraint.

[0136] The detection team will choose a threshold that minimizes the detection latency while satisfying the false alarm constraint; therefore, the optimization problem is transformed into: ;

[0137] Because the given false alarm constraint is Combining monotonicity, the set of thresholds that satisfy the conditions can be obtained as follows: ,in .because about It is also a monotonically increasing function, so let The minimum detection threshold is .

[0138] Given concealment performance parameters When, by characteristic function and The optimal power coefficient can be obtained by combining the results, satisfying the following:

[0139] ; This refers to the maximum transmit power at the sender's disposal that enables covert communication. The optimal threshold and covert performance can be expressed as:

[0140] .

[0141] Table 1 Parameter Explanation 1

[0142] ;

[0143] Table 2 Parameter Explanation 2

[0144] 。

Claims

1. A method for covert communication performance evaluation and optimization oriented to CUSUM detection, characterized in that, The method comprises the following steps: (1) modeling the CUSUM detection method process in sequence change point detection, using the average false alarm running time, the average detection delay, the average false alarm running time and the average detection delay ratio to describe the covert transmission time and the covert performance of the covert communication; (2) based on the principle that the CUSUM test is repeated SPRT applied in the same process, analyzing the parameters in SPRT; using the Lorden inequality to obtain the upper bound of the excess expectation, and scaling different parameters to obtain the limits of the parameters, and then obtaining the closed-form approximate solution of the average false alarm running time and the average detection delay; (3) substituting the obtained parameter limits into the closed-form solution of the average false alarm running time and the average detection delay to obtain the characteristic function and the approximate solution of the average false alarm running time and the average detection delay; (4) based on the approximate expression of the average false alarm running time and the average detection delay, optimizing the detection threshold to minimize the detection delay under the condition of meeting the false alarm constraint.

2. The method of claim 1, wherein, Step (1) comprises the following steps: (11) modeling the codebook; (12) modeling the CUSUM detector; (13) modeling the stopping time; (14) modeling the average running time; (15) modeling the average detection delay; (16) converting the modeling problems of steps (11) to (15) into mathematical problems.

3. The method of claim 2, wherein, In step (11), an input symbol is randomly selected from a discrete codebook; the coded message is an independent and identically distributed random code word, and the selected codebook includes a Gaussian codebook or a BPSK codebook.

4. The method of claim 2, wherein, In step (12), the CUSUM detector codebook is modeled: the detection party samples a symbol at each time point, and uses the log-likelihood ratio for statistical quantity accumulation; Based on the received symbol and the codebook structure, the detection party sets up a binary hypothesis test; The average detection delay and the average false alarm running time are used to characterize the concealment.

5. The method of claim 1, wherein, Step (2) comprises the following steps: (21) obtaining the upper bound of the excess expectation based on the Lorden inequality; performing Taylor series approximation on the KL divergence, expanding the transmission power to obtain the first-order expansion of the KL divergence; (22) using the obtained upper bound of the excess expectation to obtain the limits of the remaining parameters combined with the inequality; (23) substituting the obtained parameter limits into the closed-form solution of the average false alarm running time and the average detection delay to obtain the approximate solution and the characteristic function.

6. The method of claim 1, wherein, Step (4) comprises the following steps: (41) based on the approximate expression of the average false alarm running time and the average detection delay, converting the threshold optimization problem into a joint optimization problem of the power coefficient and the threshold; (42) based on the characteristic function, the unique solution of the power coefficient and the threshold that meet the condition is obtained, so that the covert performance expression at the best detection performance is obtained.

7. A system for covert communication performance evaluation and optimization oriented to CUSUM detection, characterized in that, It comprises: A description module is used to model the CUSUM detection method process in sequence change point detection, and the average false alarm running time, the average detection delay, the average false alarm running time and the average detection delay ratio are used to describe the covert transmission time and the covert performance of the covert communication; The approximate solution module is configured to analyze parameters in the SPRT based on a principle that the CUSUM test is repeated SPRT application in the same process, obtain an upper limit of an excess expectation by using a Lorden inequality, scale different parameters, obtain a limit of each parameter, and further obtain a closed-form approximate solution of the average false alarm running time and the average detection delay; The closed-form solution module is configured to substitute the obtained limit of each parameter into the closed-form solution of the average false alarm running time and the average detection delay to obtain a characteristic function and an approximate solution of the average false alarm running time and the average detection delay. The optimization module is configured to optimize the detection threshold based on the obtained approximate expression of the average false alarm running time and the average detection delay, and minimize the detection delay under the condition of meeting a false alarm constraint. 8.An electronic device comprising a memory and a processor, the memory storing a computer program, wherein, The processor implements the method in any one of claims 1-6 when executing the program.

9. A computer readable storage medium storing a computer program, characterized in that, The program is executed by the processor to implement the method in any one of claims 1-6.

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