A Location Prediction-Based Helper Selection Scheme for Suspicious Eavesdroppers

This paper aims to improve security performance of data transmission with a mobile eavesdropper in a wireless network. The instantaneous channel state information (CSI) of the mobile eavesdropper is unknown to legitimate users during the communication process. Different from existing work, we intend to reduce power consumption of friendly jamming signals. Motivated by the goal, this work presents a location-based prediction scheme to predict where the eavesdropper will be later and to decide whether a friendly jamming measure should be selected against the eavesdropper. The legitimate users only take the measure when the prediction result shows that there will be a risk during data transmission. According to the proposed method, system power can be saved to a large degree. Particularly, we first derive the expression of the secrecy outage probability and set a secrecy performance target. After providing a Markov mobile model of an eavesdropper, we design a prediction scheme to predict its location, so as to decide whether to employ cooperative jamming or not, and then design a power allocation scheme and a fast suboptimal helper selection method to achieve targeted and efficient cooperative jamming. Finally, numerical simulation results demonstrate the effectiveness of the proposed schemes.


Introduction
As a promising technology, an Internet of Things (IoT) network offers opportunities to directly transform physical things into information world without human interventions [1][2][3].It may be composed of billions of low-end devices that connect everyday objects and surrounding environments.These devices equipped with various sensors and actuators can be connected to the Internet via heterogeneous wireless networks.We can exploit the devices to collect meaningful and suitable data conveniently to achieve information sharing, computing, and controlling remotely.Obviously, these data contain sensitive and private information such as social relationships and financial transactions [4].As a result, the security of IoT networks is of critical importance for the wide deployment and acceptance of big data services in the future.
Due to properties of broadcast communication and signal superposition in wireless networking scenarios, it is difficult to shield transmitted signals from unauthorized receivers as well as protect legitimate receivers from unintended overlapping of multiple signals.These facts make security become a vital issue, especially in the openness of the wireless medium.As a result, many works have been done to meet security requirements.These works mainly exploit cryptographic techniques at the upper layers of wireless networks [5][6][7].As a complement to the measures at the upper layers, the idea of physical layer security (PLS) is proposed and has been widely discussed in recent years.To be specific, PLS is to exploit channel characteristics to enhance secure performance of data transmission, which means the inherent randomness of the noise and communication channels are used to limit the amount of information to be extracted by unauthorized receivers [8].A number of studies in this field propose to address the problem of either active or passive attacks in wireless networks [9].As for passive attacks, eavesdropping is a well-known security risk in the whole communication process.The strategies of antieavesdropping, as Wyner described in his classic wiretap channel model [10], have recently regained substantial research attention [11][12][13][14][15].
To the best of our knowledge, many studies aiming at eavesdropping usually design different schemes to ensure secure transmission according to whether the channel state information (CSI) of the eavesdropper is known to legitimate users.For one thing, it is possible to design a targeted and efficient security scheme when the CSI of the eavesdropper is known to legitimate users, which means the exact location of the eavesdropper is known.Because of knowing the eavesdropper's CSI, the equation of the secrecy performance metric can be deduced into a convex optimization problem that can be solved.For another, however, the design of a secure scheme becomes challenging if the CSI of the eavesdropper is partially or even totally unknown to legitimate users.The general solution is to guarantee the robust secrecy performance in the worst case and to design a suboptimal algorithm.Also, there exist other tactical methods, such as the artificial noise alignment schemes [16] that do not use the knowledge of the eavesdroppers channel gains.Among these schemes, few of them consider a network model with a mobile eavesdropper, whose CSI is unknown to legitimate users because of the unexpected movement.
A mobile eavesdropper is like an unexpected risk to the communication process of legitimate users.Due to the continuous random movement in the network all the time, it is difficult to decide when the eavesdropper will move close to the legitimate transmitter and start to wiretap information.This kind of wiretapping exists widely in our real life, especially mobile social networks.Data transmission among legitimate users should be kept from information stealing by an unauthorized passerby.Regarding this issue, many PLS-based security schemes are designed by transmitting jamming signals during the whole communication process.With enough and accurate CSI of an eavesdropper, these schemes are achievable and effective at the expense of a large amount of power consumption.Different from these schemes, we want to deal with this secure issue in a more practical situation.The legitimate users do not have to act in a defensive way during the whole communication period.They only have to take a security measure against the eavesdropper when they find out the risk of privacy disclosure.Such behavior can save system power to a great degree.
To address these challenges, we have proposed a novel risk prediction scheme in our conference paper [17].This scheme is aimed at the network model with a mobile eavesdropper, whose CSI is unknown to the legitimate users.After studying the mobile eavesdropping model, we analyze the network model and derive an expression of the secrecy outage probability as the security metric and set a target secrecy outage probability for the risk decision in this paper.Considering the mobile path prediction of the eavesdropper, next, a Markov chain is exploited to set up a Markov mobile model of the eavesdropper.According to the mobile model, we perform the prediction by exploiting the history movement information of the eavesdropper.After predicting the location where the eavesdropper will be, we exploit the prediction results to decide whether the eavesdropper might be harmful to data transmission later or not.If it will be, the corresponding security measures will be taken against it, and the power allocation scheme is designed to achieve maximum secrecy capacity.At last, this paper also demonstrates the effectiveness of our scheme via a series of simulations.The main contributions of this paper can be summarized as follows: (i) We propose a Markov chain-based location prediction scheme for a mobile eavesdropper by its history movement information.(ii) Based on prediction results, we formulate an optimization problem to allocate power for transmitter and helper so as to obtain the maximum achievable secrecy rate after selecting a suitable helper.(iii) In order to enhance the friendly jamming efficiency of a helper, we design a fast suboptimal helper selection algorithm that takes into account both algorithm complexity and secrecy performance.
The rest of the paper is organized as follows.The related works are described in Section 2. We present the system model and derive the expression of the secrecy outage probability in Section 3, followed by the detailed illustration of our location prediction-based helper selection scheme in Section 4.Moreover, the numerical simulation is shown and analyzed in Section 5. Finally, we conclude this article in Section 6.

Related Work
Jamming is generally treated as an unfavorable factor in wireless communications [18].It may overlap with information signals, which finally impacts decoding performance.In spite of the negative side, some studies suggested that friendly jamming can be used as an effective tool to protect information signals from malicious adversaries.From the perspective of the wiretap channel model, perfect secrecy can be achieved when channel condition of legitimate users is better than that of eavesdroppers.Accordingly, the basic idea of friendly jamming strategies is to degrade the wiretap channel quality of eavesdroppers.
This idea was first introduced in [19].Negi and Goel attempted to exploit artificially generated noise to degrade the eavesdroppers channel but not to affect the information signals.They discussed the secrecy capacity over the multiple transmit antennas scenario and the multiple helpers scenario.Following this work, a number of studies have been performed.Wang et al. proposed a targeted jamming scheme against the eavesdropper in [20].In the work, they designed an asymptotic power allocation method to solve an achievable secrecy rate maximization problem.Besides, they provided a jammer selection method to make a decision for reducing the abuse rate of jammers.Similarly, Zhang et al. also investigated secure communications for cooperative cognitive radio networks in [21].They studied a joint time and power allocation scheme to achieve the maximum secrecy rate for relay-jammer scenario and then presented a weight and time allocation strategy for cluster-beamforming scenario.All these proposed schemes are designed under the assumption that the eavesdropper channel condition is known.In that case, the jammer among legitimate users can make target jamming and optimizing power allocation is possible to achieve.
Yet, a more practical assumption is the unknown CSI of eavesdroppers.Obviously, it is more difficult to achieve target jamming and performance.Some researchers exploited the secrecy outage probability and -outage secrecy capacity to describe the system secrecy performance.According to the above two indicators, a series of suboptimal algorithms were studied.In [22], Li and Ma considered a worst-case robust secrecy rate maximization problem with incomplete Eve's CSI.They presented a suboptimal but safe solution to an outage-constrained robust secrecy rate maximization problem.In [23], Jiang et al. derived closed-form expressions of secrecy indicators for unknown CSI of eavesdroppers.Also, they developed a joint zero-forcing and successive interference cancellation method to analyze the individual secrecy performance for a multiple access wiretap channel.A friendly cooperative jamming strategy for IoT networks with imperfect eavesdropper's CSI was also investigated in [24].The authors in [24] transformed this challenge into the worstcase eavesdropper's CSI and formulated a two-stage robust optimization problem to find the optimal solution.
Although there still exist numerous studies on the unknown or imperfect CSI of eavesdroppers [25][26][27], all of these only focused on the wiretap by static eavesdroppers.The more general case is that eavesdroppers may have other behaviors.For instance, they can still wiretap information signals when they are in mobile state.In that case, the CSI of mobile eavesdroppers may be changeable with the changing location.In this paper, we intend to analyze and predict the mobility of eavesdroppers.With the mobility prediction, we can decide whether it may steal the information via the legitimate channel.In terms of the prediction of the mobile path, the Markov chain is a good way to realize it.In [28], Fazio and Marano employed a distributed set of hidden Markov chains to predict the probable cells that a mobile node may visit in the future.Besides, [29] also formulated a Markov-history model for realistic mobility of nodes in a network.All these studies inspire us to predict the location (i.e., CSI) of a mobile eavesdropper.According to this, a power allocation and jammer selection scheme is put forward.To the best of our knowledge, this work is the first one to investigate the PHY layer security issue in the mobile eavesdroppers scenario.

System Model
Considering a typical wireless wiretapping network, there exists a legitimate transceiver pair (called Alice and Bob), an eavesdropper (Eve), and several helpers, shown in Figure 1.Each of them in the network is equipped with a single omnidirectional antenna.Alice sends its message to Bob via a legitimate channel ℎ  .At that time, Eve, a random mobile passive adversary, is likely to come within the area around Alice to wiretap Alice's message through a wiretapping channel ℎ  .To prevent this wiretapping attack, a helper may be selected to broadcast jamming signals (e.g., artificial noises) to degrade the reception quality of Eve, which is also called a friendly jamming strategy.
In this scenario, we define   as the distance between Alice and Bob and   as the distance between Alice and Eve.The legitimate channel (from Alice to Bob) is denoted by ℎ  , and the wiretap channel (between Alice and Eve) is denoted by ℎ  .We assume both channels are modeled as Rayleigh fading channels.Signal to interference plus noise ratio (SINR) of both legitimate and unauthorized users is decided by pathloss and fading effects [30].Then, we can first characterize channel vectors as follows: where  is the path-loss coefficient and () is the channel power fading coefficient.Here,  can be characterized by the path-loss exponent  and the distance  between two communicating parties; that is, where  is the path-loss constant.In addition, () follows exponential distribution; that is, Without loss of generality, the coefficient  is modeled as a random variable with unit mean, and hence  = 1.
When Alice selects a helper to be the friendly helper to preserve its privacy transmission to Bob, the received signals of Bob and Eve are respectively, where   and   are information signal power and jamming signal power, respectively.   is the beamforming vector at Alice to transmit the jamming signal, which is used to eliminate the interference of the jamming signal at Bob.   and   are the additive white Gaussian noise (AWGN) with zero mean and variance  2   at Bob and Eve, Wireless Communications and Mobile Computing respectively.The SINR at Bob and at Eve can be, respectively, presented as Accordingly, the secrecy capacity at Bob can be deduced as follows, which is defined as the difference between the mutual information of the legitimate channel and that of the wiretap channel.
where {} + = max[, 0].(;   ) and (;   ) denote the mutual information of the channels between Alice and Bob and between Alice and Eve, respectively.Secrecy capacity is the maximum achievable rate between the legitimate transmitter and receiver that can guarantee perfect secrecy.It gives the upper bound of the transmission rate subject to constraints of unauthorized users.Obviously, the communication link is secure when   > 0. On the contrary, it has the risk of information leakage because Eve experiences a better channel condition than Bob.However, it is hard for Alice to be aware of the channel information of Eve, ℎ  , in a passive eavesdropping mode.As a result, we can characterize the secrecy outage probability.It is the probability that the instantaneous secrecy capacity is less than a target secrecy rate   : According to [31], we get the following expression: Here,  out is a function of SINR  and SINR  .Since SINR is a function of  as shown in (1) to (3), we can deduce that  out (  ) is a function of   and   .Taking the locations of Alice, Bob, and Eve into account, the secrecy outage regions corresponding to different secrecy outage probabilities should be calculated.
Since Alice and Bob are communicating with their locations settled, secrecy outage probabilities of other locations in the network can be calculated.If we can predict the location where the randomly moving eavesdropper will be later, we can calculate the secrecy outage probability of the predicted location and decide whether it may steal the information from Alice or not.With the decision, we can take measures to guarantee security beforehand.The criterion of the decision is based on specific secure requirements.Here, we set a target secrecy outage probability  th to define whether Eve will be harmful to the communication later.The value of  th is set based on actual requirements of users or administrators in the network.For various network scenarios, this value can be set variously.When  out (  ) <  th , we consider that the communication is secure.Conversely, when  out (  ) >  th , we consider that the communication is suffering from the risk of being eavesdropped and we need to take some security measures.

A Location Prediction-Based Helper Selection Scheme
In this section, we propose a prediction scheme to predict where Eve will be in its later movement, so as to decide whether security measures should be taken against Eve to guarantee security.As we discussed in Section 2, the Markov chain shows effectiveness for mobility prediction in such network scenarios.Hence, we decide to exploit it to achieve our prediction.We first present a detailed illustration of our scheme.Then, we introduce the metrics of our scheme.

The Prediction Scheme.
To give a clear illustration of our scheme, we first introduce some definitions.As we assume that Eve keeps moving in an area all the time, its mobile path is continuous.We predict Eve's location at intervals of one moment, where one moment is set to be one length of time.That is to say, the mobile path that we predict is a discrete one.We consider that the area is composed of an infinite number of points (locations), and Eve moves from one location to another from the current moment to the next moment.This is called one-step movement.For example, we can define  − 1 as the current moment,  as the next moment, and so on.Because Eve may keep moving randomly in the area all the time, the history information of its movement can be of great usefulness to the prediction.We assume that the history movement information of Eve is known.Note that when there is a newly coming eavesdropper that provides no pattern for mobile path, the prediction scheme is unsuitable.We just treat it as a threat and make the jammer selection as the method shown in Sections 4.2 and 4.3.
In the prediction scheme, we first use the Markov chain to set up a mobile model of Eve, and then try to extract some characteristics from the statistics of history movement information.This is called the transition matrix in the Markov model.Finally, we calculate and compare the probabilities of locations at the next moment to decide which location Eve will move to.
As for setting up a Markov mobile model, it is obvious that every location in the area can be seen as a state.However, it is invalid to carry out the prediction scheme in the case of infinite number of locations as the state space.It inevitably results in high computation complexity, which may cost huge resources.Thus, to improve the efficiency of our prediction scheme, we divide the whole area into an  ×  gridding, as shown in Figure 2. We define each grid as a state.When Eve is moving in the same grid, we consider that it stays at the same state.The length of every grid is set to be 1, and the distance between each pair of adjacent grids is set to be 1.And we assume that the grid is the minimum unit of the area.
To better elaborate the process of modeling, we use a 3 × 3 gridding as an example.The example gridding is shown  in Figure 3. Here, we number the grids as  1 , where  V (1 ≤  ≤ 9, 1 ≤ V ≤ 9) denotes the transition probability in the th row and the th column of the transition matrix  e.g. , which is the probability that Eve is in grid   the former moment and chooses to move to grid   the latter moment.For the one-step movement, Eve can only move from its current grid to the adjacent grids or stay still (which means it is moving in the same grid).Thus, some of the probabilities in the transition matrix are meant to be 0.For example, if Eve stays in  1 at the current moment, it can only be in  1 ,  2 ,  4 , or  5 at the next moment and cannot be in  3 ,  6 ,  7 ,  8 , or  9 .So, the probabilities  13 ,  16 ,  17 ,  18 , and  19 in the transition matrix  e.g. are 0. Now back to the  ×  gridding Markov mobile model, we are going to discuss how to get the transition matrix of it.We use  total to denote the total number of grids, where  total =  × .Like the example above, the state space of this  ×  gridding model is SP = { 1 ,  2 , . . .,   total }.And the corresponding transition matrix is an  total ×  total matrix.We use   (1 ≤  ≤  total , 1 ≤  ≤  total ) to denote the transition probability in the th row and the th column of the transition matrix .Every   can be calculated based on the statistics of history movement information.We use   (1 ≤  ≤  total , 1 ≤  ≤  total ) to denote the total number of times that Eve moves from grid   to grid   in its movement history; thus,   can be expressed as After we get the transition matrix , which is also called the one-step transition matrix, we can further derive the step transition matrix ().By exploiting  −  equation, we can see that Thus, We consider that the location of Eve at the next moment is related to the history movement information, that is, the transition matrix and its states of former  steps of movements.It is obvious that if the moment of the state is nearer to the next moment, this state may have more influence on Eve's next moment movement.And the states of the far past moments can be negligible.There exists an optimal value of  which can lead to the best prediction performance.We can obtain the value by simulation experiences.Based on the foregoing analysis, we decide to use a weighted way to calculate the probabilities of every location at the next moment; that is, where () is a 1 ×  total matrix, containing the probabilities of all states.() ( −  ≤  ≤  − 1) is a set containing states' information.It represents the state Eve was at the former th moment before the next moment.It is also a 1 ×  total matrix.Its value at the first row with the th column is 1, while other values are 0.  1 ,  2 , . . .,   are weighted coefficients, representing different influence degrees that movements at the former 1st, 2nd, . .., th moment before the next moment have on the next moment's movement, respectively.Note that we consider that the influence degree is a relative value.Thus, the summation of  1 ,  2 , . . .,   is not 1.

Initialization:
= { When obtaining (), we can compare these probabilities to decide which state Eve will be at the next moment.As the previous assumption of Eve's movement state (move to the adjacent grids or stay still), we can set the probabilities of those unreachable states to be 0 and only have to compare the probabilities of potential states.The state which the maximum probability is corresponding to is where Eve will be at the next moment.Noting that the result is the location we predict, it may not be the location which Eve actually moves to at the next moment.Every time Eve performs a movement, we need to count   again to get the new statistics of the history movement information and to modify the transition matrix, so as to get a more accurate prediction result next time.
Since the location of Eve at the next moment is predicted, we can use this result to decide whether we should take measures against Eve.The way is basically mentioned in Section 3. In particular, we first substitute the predicted location of Eve to the calculation process of the secrecy outage probability  out (  ) and then check whether  out (  ) is smaller than the threshold probability  th .If the answer is yes, we consider that the communication is secure; else, we take security measures.The whole decision process is summarized as Algorithm 1.

Power Allocation for Cooperative Friendly Jamming.
According to the above discussion, Alice may be aware of the predicted CSI of Eve.Assuming that the helper has been selected, we would like to design a prediction-based power allocation algorithm to optimize the proportion of   and   in the total power   .The goal of the allocation algorithm is to achieve maximum secrecy rate at Bob; that is, max In general, jamming signals are deliberately designed in nullspace of the legitimate channel.In this manner, the selected helper may adjust its transmit covariance matrix to jam Eve and simultaneously null out interference at Bob.As a result, the reception SINR at Bob can be described as follows: Accordingly, if we assume that   and   are much greater than the noise power Obviously, ( 16) is a convex analysis optimization.Here, we employ the method of Lagrange multipliers and the Karush-Kuhn-Tucker (KKT) conditions to provide a closedform solution of this mathematical optimization.We first introduce an auxiliary function of the optimization objective in (16), where  is a Lagrange multiplier, and then solve the corresponding gradient expressions Initialization: The predicted Eve's CSI;  helper = { 1 ,  2 , . . .,   };    (1 ≤  ≤ ).(1) Determine whether there exist a risk of secure communication between Alice and Bob based on Algorithm 1; (2) Calculate secrecy rates {   } for all helpers based on the predicted Eve's CSI; (3) Find a helper  * as a jammer by comparing every    ; (4) Calculate information signal power   and jamming signal power   by ( 19) and (20) for the  * helper, (5) The  * helper broadcasts artificial noises to jam Eve.
Algorithm 2: The fast helper selection scheme.
Note that (18) amounts to solving three equations in three unknowns.As a result, we can find the following optimal solutions: Finally, we can calculate the maximum achievable rate,   , by the following equation:

A Fast Suboptimal Helper Selection Scheme.
In order to ensure secure communication from Alice to Bob, a suitable helper may be selected to broadcast jamming signals.We assume there are several helpers in the network, and one of them can satisfy the security requirements if it broadcasts jamming signals.Obviously, this helper should not be selected randomly due to the requirement of secrecy rate.On the contrary, the selected helper should have enough jamming power to prevent Eve from getting information illegally.Here, we would like to design a fast suboptimal helper selection scheme for the mobile Eve.
In the above subsection, the maximum achievable secrecy rate has been calculated via (22).For every helper, we are able to obtain different   according to different Eve locations.Intuitively, Alice may select the helper as a jammer that can help Alice obtain the maximum achievable secrecy rate when Eve is in a certain location.Thus, the helper selection scheme can be described as a mathematical expression; that is, where  * represents the selected helper and  helper is the set of all candidate helpers in the network.Accordingly, the secrecy rate at Bob can be computed as follows: where ℎ  *  and   * denote the channel state vector and the beamforming vector between the selected helper  * and Eve, respectively. *  and  *  are the information signal power and the jamming signal power that are calculated by the optimal power allocation algorithm.
Obviously, we can exploit the exhaustive search method to find the optimal helper  * .This method is suitable for the scenario of static eavesdropping.Nevertheless, Eve is a mobile passive eavesdropper in our system model.The process of searching for and selecting the optimal jammer needs to be repeated, which may undoubtedly result in high computational complexity.To deal with the challenge, we intend to design a suboptimal helper selection scheme.
According to (24), we are aware that   is mainly affected by ℎ  .Also, the channel state vector ℎ  is inversely proportional to the distance from the jammer to Eve.As a result, we can select the nearest helper from Eve as the jammer without hesitation; that is, The whole process of helper selection can be summarized as Algorithm 2.
Remark 1.There is still an extreme case where there are a huge number of helpers in the network.To reduce the computational complexity, Alice may stop searching once it finds a helper that satisfies the security requirement.In other words,  out (  ) that a helper provides is lower than  th .The feasible solution is as follows:

Production Evaluation Metrics.
As for a prediction scheme, obviously, an important issue is the accuracy.When it comes to our system model, what we care about most is the accuracy of the risk prediction, not simply the accuracy of the position prediction.We want to know how possible it is when the prediction tells us there are or there are not any risks in the communication process.Actually, we can employ a sum of two different errors to describe the accuracy of prediction.The first error is caused by misdetection.In that case, Eve is actually in the secrecy outage region although the prediction result says the communication is secure, which will lead to information leakage.Second, a false alarm also introduces prediction errors.This may let the system take unnecessary security measures and waste system power.According to the above description, we define an index   to represent the error level of our prediction scheme; that is, where 1 means true (there exist risks in the communication process) and 0 means false. 1 denotes that the decision result is true, and  0 denotes that the decision result is false. 1 denotes that the actual result is true, and  0 denotes that the actual result is false.( | ) denotes the amount of the states with the actual state  and the predicted state . count denotes the total time of the prediction.And the simulation results in Section 5 show the error level of our scheme.

Prediction Performance Analysis.
In this subsection, simulation results are shown to verify the effectiveness of our prediction scheme (here, we first employ the transition probability of Eve's location to simulate its movement in a predefined secrecy outage region; this probability can be computed by the historical data collected by Alice, which will be presented in our future work; also, in the future work, we will provide real-life datasets based on typical social applications, e.g., WeChat or Facebook, to conduct further experiments and analyses).We observe the error probability   in scenarios with different settings of parameter values, which are the number of grids of the area  total , the number of the considered former steps of movements before the next moment , and the number of history movements .Those parameters may cause varying degrees of influences on the error probability of the prediction scheme.Therefore, we intend to divide the area into a 10 × 10, 20 × 20, and 30 × 30 gridding, respectively.In other words, the area, respectively, consists of 100, 400, 900 grids.Besides, we set  to be 3, 5, 10 and  to be 2000, 4000, 8000 and set the target secrecy outage probability  th to be 0.8, whose value can be adjusted based on actual requirements.Figure 4 provides a description of   with different values of  total and different values of .As for the relationship between   and  total , it is shown that   is monotonously decreasing with  total , indicating that as the division unit of the area goes smaller, the accuracy degree of the prediction scheme goes higher.There exist two reasons to illustrate the result.One is that when the division unit of the area is large, this grid division will lead to a take-security-measure decision if Eve moves around the edge of the target secrecy outage region.Both the neighboring area in the target secrecy outage region and the neighboring area out of the target secrecy outage region can be in the same grid, which impacts the decision.Another reason is that when the division unit of the area is smaller, the number of paths in the same grid goes smaller.As a result, the history movements can provide more information for setting up the corresponding Markov mobile model, so as to perform a more accurate prediction.Besides, as for the relationship between   and , the result in Figure 4 also demonstrates that as the value of  increases, the value of   goes down at first and then goes up.Obviously, the value of   is at the minimum if  = 5.The result indicates that in the prediction process it may give rise to a better accuracy performance without employing more previous steps of movements.When too much former steps of movements are considered, which are not that related to the movement of Eve at the next moment, this consideration will impact the prediction result reversely.
In Figure 5, we present a description of   with different values of  total and different values of .It is shown that   is monotonously decreasing with , indicating that the error probability of the prediction scheme decreases along with the increase in the number of history movements.This is reasonable because the history movements are crucial to the generation of the transition matrix of the Markov chain.More history movements can provide more information about the mobile characteristics of Eve.Thus, more historical movement information data should make the transition matrix more specific to represent those mobile characteristics.In one word, the prediction result with more history movement information can be more accurate.From these simulation results, we can see that the value of the error probability in our proposed risk prediction scheme is in a relatively acceptable range; that is to say, the secrecy performance of this risk prediction scheme can be guaranteed.If we want to achieve a better secrecy performance, we need to divide the area into more grids and to gather and utilize more history movement information, thus to lower the value of the error probability.

Secrecy Performance Analyses.
In this section, we conduct several simulations to verify the secrecy performance of the proposed power allocation scheme as well as the corresponding helper selection method.Without loss of generality, we assume that channels among all nodes in the network are modeled as Rayleigh fading channels.Also, there exists the additive white Gaussian noise (AWGN) with mean 0 and variance  2  .The CSI of legitimate nodes is known to each other, while that of Eve has been estimated by the location prediction scheme.
We first study the secrecy performance of the optimal helper selection scheme and the suboptimal scheme.Supposing Eve is at a position in the secrecy outage region, we conduct 30 simulations to compare the difference between two schemes in the case of various distributions of helpers.
It can be seen from Figure 6 that the difference of secrecy rate between two selection schemes is not significant.And they even have the same secrecy rate sometimes (i.e., the two schemes choose the same helper as a jammer).Yet, compared with the optimal helper selection scheme, the suboptimal one can drastically reduce the computational complexity of the selection procedure in the case of similar security rate.
Next, we investigate the effect of total system power on the secrecy rate when the number of helpers is 10.Here, we   The optimal power allocation The average power allocation assume the total power   = 10, 20, 30, 40, 50 dB.In Figure 7, we are aware that the secrecy rate increases as the total power grows for two schemes.This is because the stronger jamming power may further deteriorate the receiving signal quality of Eve.Besides, it is obvious that the proposed optimal power allocation scheme has better secrecy performance than the average method.
As a benchmark, we also derive the effect of the number of helpers on the secrecy rate by running the experiment with  helper = 5, 10, 20, 30, 40, 50 to compare the performance of our power allocation scheme and the average scheme.In Figure 8, along with the increase in the number of nodes in The optimal power allocation The average power allocation the network, we can see that the secrecy rates of both schemes are growing fast at first and then flatten.The reason is that there may be some adjacent helpers as the number of helpers in the network increases.These adjacent helpers may have a similar impact on Eve.
According to these simulation results, we can find that the proposed schemes are effective in achieving the requirement of secrecy rate at the legitimate receiver.Also, the proposed suboptimal helper selection scheme can save the system resources while ensuring the secure communication of legitimate channel.

Conclusion
This paper proposes a location prediction-based helper selection scheme to address physical layer security in the communication scene with a suspicious mobile eavesdropper, the case where the eavesdropper's CSI is unknown to the legitimate users.In this scheme, we exploit the secrecy outage probability as the security metric, set a target secrecy outage probability for the risk decision, and perform the prediction by using the Markov chain.With a Markov mobile model of the eavesdropper set up, the history movement information of the eavesdropper is employed to form the transition matrix.Besides, the position of the eavesdropper at the next moment we want to predict is related to both the history movement information and its own former states.Based on this, a weighted method is used to do the prediction.Next, a power allocation scheme and a fast suboptimal helper selection method are developed to interfere with a mobile eavesdropper.In order to demonstrate the effectiveness and the secrecy performance of our scheme, a set of simulations are conducted.These simulation results illustrate that the prediction scheme is with low error probability.Also, secrecy performance analyses demonstrate that the optimal power allocation with suboptimal helper selection scheme can achieve the requirement of secrecy rate.
Note that, in this paper, we only discuss the mobile eavesdropper in a basic single-antenna network model.As for future work, we are going to reinvestigate such problem in a MIMO (multiple-input-multiple-output) system and come up with a specific physical layer security strategy.

Figure 1 :
Figure 1: Description of the network layout.

Figure 2 :
Figure 2: The grid division of the network.
simulationsSecrecy rate (bps/Hz)The optimal helper selection scheme The suboptimal helper selection scheme

Figure 6 :
Figure 6: The instantaneous secrecy rate between two selection schemes, with  helper = 10 and   = 30 dB.