Physical Layer Security for Two-Way Untrusted Relaying with Friendly Jammers

Physical Layer Security for Two-Way Untrusted Relaying with Friendly Jammers

Rongqing Zhang1, Lingyang Song1, Zhu Han2, and Bingli Jiao1
This work is partially supported by US NSF CNS-0910401, CNS-0905556, and CNS-0953377. 1School of Electrical Engineering and Computer Science, Peking University, Beijing, China.
2Electrical and Computer Engineering Department, University of Houston, Houston, USA.
Abstract

In this paper, we consider a two-way relay network where two sources can communicate only through an untrusted intermediate relay, and investigate the physical layer security issue of this two-way relay scenario. Specifically, we treat the intermediate relay as an eavesdropper from which the information transmitted by the sources needs to be kept secret, despite the fact that its cooperation in relaying this information is essential. We indicate that a non-zero secrecy rate is indeed achievable in this two-way relay network even without external friendly jammers. As for the system with friendly jammers, after further analysis, we can obtain that the secrecy rate of the sources can be effectively improved by utilizing proper jamming power from the friendly jammers. Then, we formulate a Stackelberg game model between the sources and the friendly jammers as a power control scheme to achieve the optimized secrecy rate of the sources, in which the sources are treated as the sole buyer and the friendly jammers are the sellers. In addition, the optimal solutions of the jamming power and the asking prices are given and a distributed updating algorithm to obtain the Stakelberg equilibrium is provided for the proposed game. Finally, the simulations results verify the properties and the efficiency of the proposed Stackelberg game based scheme.

I Introduction

Traditionally security in wireless networks has been mainly considered at higher layers using cryptographic methods. However, recent advances in wireless decentralized and ad-hoc networking have led to an increasing attention on studying physical layer based security. The basic idea of physical layer security is to exploit the physical characteristics of the wireless channel to provide secure communication. The security is quantified by the secrecy capacity, which is defined as the maximum rate of reliable information sent from the source to the intended destination in the presence of eavesdroppers. This line of work was pioneered by Aaron Wyner, who introduced the wiretap channel and established fundamental results of creating perfectly secure communications without relying on private keys [1]. Wyner showed that when the eavesdropper channel is a degraded version of the main channel, the source and the destination can exchange perfectly secure messages at a non-zero rate. In follow-up work [2], the secrecy capacity of Gaussian wiretap channel was studied, and in [3] Wyner’s approach was extended to the transmission of confidential messages over broadcast channels. Recently, researches on physical layer security have generalized these studies to wireless fading channels [4, 5, 6, 7], MIMO channels [8, 9, 10, 11, 12], and various multiple access scenarios [13, 14, 15, 16, 17, 18].

Motivated the fact that if the source-wiretapper channel is less noisy than the source-destination channel, the perfect secrecy capacity will be zero [3], some recent work has been proposed to overcome this limitation using relay cooperation, which mainly consists of cooperative relaying [19, 20], and cooperative jamming [21, 22]. For instance, in [19] and [20], the authors proposed effective decode-and-forward (DF) and amplify-and-forward (AF) based cooperative relaying protocols for physical layer security, respectively. Cooperative jamming is another approach to improve the secrecy capacity by distracting the eavesdropper with codewords independent of the source messages. In [21] and [22], several cooperative jamming schemes were investigated for different scenarios, where classical relay strategies fail to offer positive performance gains. Relay channel with confidential messages was studied in [23] and [24], where the relay node acts both as an eavesdropper and a helper. In [25], it was established that cooperation even with an untrusted relay node could be beneficial in relay channels with orthogonal components. Then in [26], the authors considered a two-hop communication system using an untrusted relay and showed that a cooperative jammer enables a positive secrecy capacity which would be otherwise impossible.

Two-way communication is a common scenario where two terminals transmit information to each other simultaneously. Recently, the two-way relay channel [27, 28, 29, 30, 31] has attracted lots of interest from both academic and industrial communities due to its advantage in saving bandwidth efficiently. In [27] and [28], both AF and DF protocols for one-way relay channels were extended to general full-duplex discrete two-way relay channel and half-duplex Gaussian two-way relay channel, respectively. In [29], different relay strategies consisting of AF, DF, and EF (estimate-and-forword) for uncoded two-way relay channels were investigated. In [30], analogue network coding based two-way relay channel with linear processing was analyzed and an optimal relay beamforming structure was presented. In [31], a joint network-channel coding was proposed for the two-way relay channel, where channel codes are used at both the sources and a network code is used at the intermediate relay. Although two-way relay networks have received so much attention so far, the security issue about the relay, especially from the physical layer security point of view, has not been well investigated.

To improve the security in two-way relay channel, distributed protocols are desired. Game theory [32] offers a formal analytical framework with a set of mathematical tools to study the complex interactions among interdependent rational players. Recently, there has been significant growth in research activities that use game theory for analyzing communication networks, mainly due to the need for developing autonomous, distributed, and flexible mobile networks where the network devices can make independent and rational strategic decisions, as well as the need for low complexity distributed algorithms for competitive or collaborative scenarios [33]. In [34] and [35], the authors introduced some recent studies on signal processing and communication networks using game theory. In [36] and [37], the authors employed game theory to physical layer security to study the interaction between the source and the jammers who assist the source by distracting the eavesdropper, and got some distributed game solutions. For two-way relay networks, it is desirable to study the physical layer security problems with the aid of game theory similar to those for one-way relay cases.

In this paper, we investigate physical layer security issues in a two-way relay network with friendly jammers. The two sources can exchange information only through an untrusted relay, as there is no direct communication link between them. The untrusted AF relay acts as both an essential relay and a malicious eavesdropper that has the incentive to eavesdrop on the information transmission. For convenience and ease of comparison, we first study the system without friendly jammers as a special case. We find that a non-zero secrecy rate here is indeed available even without the help of friendly jammers. We also derive an optimal power vector of the relay and the sources by maximizing the secrecy rate. We further investigate the two-way relay secure communication with friendly jammers, and find that a positive gain can be obtained in the secrecy rate by utilizing proper jamming power from the friendly jammers. Then the problem comes to how to effectively utilize the jamming power from different friendly jammers to maximize the secrecy rate. Thus, we propose a Stackelberg game model between the sources and the friendly jammers as a power control scheme. In the defined game, the sources pay the friendly jammers for interfering the untrusted relay in order to increase the secrecy rate, while the friendly jammers charge the sources with a certain price for their service of jamming. In addition, the optimal solutions of the jamming power and the asking prices are given and a distributed updating algorithm to obtain the Stakelberg equilibrium is provided for the proposed game. Furthermore, a centralized scheme is also proposed for comparison with the distributed Stackelberg game based scheme. Finally, the proposed approaches and solutions are verified by simulations.

The rest of this paper is organized as follows. In Section II, the system model of two-way relay communication with friendly jammers is described and the corresponding secrecy rate is formulated. In Section III, a two-way relay system without jammers as a special case is investigated. In Section IV, we define a Stackelberg type of buyer/seller game to investigate the interaction between the sources and the friendly jammers, and analyze the optimization problem of physical layer security in the presence of friendly jammers. Simulation results are provided in Section V, and the conclusions are drawn in Section VI.

Ii System Model

As shown in Fig. 1, we consider a basic two-way relay network consisting of two source nodes, one untrusted relay node, and friendly jammer nodes, which are denoted by , , , and , , respectively. We denote by the set of indices . All the nodes here are equipped with only a single omni-directional antenna and operate in a half-duplex way, i.e., each node cannot receive and transmit simultaneously. Then the complete transmission can be divided into two phases. During the first phase, shown with solid lines, both source nodes transmit their information to the relay node. Simultaneously, the friendly jammers also transmit the jamming signals in order to distract the malicious relay. In the second phase, shown with dashed lines, the relay node broadcasts a combined version of the received signals to both source nodes. Note that in the system we investigate there is only one intermediate relay and we assume that no direct link exists between the two sources. Thus, the sole untrusted relay is necessary for the two-way relaying data transmission. A key assumption 111We can guarantee this by using some pseudo-random codes which are known to both the friendly jammers and the sources but not open to the malicious relay. Beyond this, we can also use some cryptographic signals at the friendly jammers for jamming, where the decryption book is a secret key only open to the sources. Then the sources can have a perfect knowledge of the jamming signals if each jammer sends some additional bits consisting of the information of the jamming signal transmitted (e.g., which code or which encryption method to be used). The information that needs to be sent is for one time, which will lead to trivial bandwidth cost. we make here is that the sources have perfect knowledge of the jamming signals transmitted by the friendly jammers, for they have paid for the service.

Let , , and , , denote the signal to be transmitted by the source , and the jammers , , respectively. Suppose source nodes and transmit with power and , and the channel gains from the source nodes to the relay node are denoted by , . Each friendly jammer node transmits with power , and the channel gain from it to the relay node is denoted by , . The channel gain contains the path loss and the Rayleigh fading coefficient with zero mean and unit variance. For simplicity, we assume that the fading coefficients are constant over one frame, and vary independently from one frame to another.

In phase , the received signal at the malicious relay can be expressed as

(1)

where denotes the thermal noise at the relay node , which is a zero mean Gaussian random variable with two sided power spectral density of , i.e., . Furthermore, we assume that the noises at , , and are independent and identically distributed (i.i.d.).

In phase , the malicious relay, which works in AF mode, amplifies the received signal by a factor and then broadcasts the signal to both and with power . The power normalization factor at the relay node can be written as

(2)

Then the corresponding signal received by the source , denoted by , can be written as

(3)

where , , , and . Similarly, the signal received by the source , denoted by , can be written as

(4)

where , , , and .

Assuming that both the source nodes and the jammer nodes are independent, from (1), in phase , using the matched filter (MF) 222For simplicity, we use the matched filter for signal detection [38] while many other advanced detectors can be applied and the analysis can be done in a similar way., the untrusted relay node has the capacity with respect to and as

(5)

and

(6)

where represents the channel bandwidth, , , and , .

In phase , at , as as well as is known to the source node, and thus we have

(7)

Then, the corresponding SNR for the transmission from to , denoted by , can be expressed as

(8)

where . Similarly, at , the received signal with and removed can be written as

(9)

The corresponding SNR for the transmission from to , denoted by , can be expressed as

(10)

where .

Capacities of two-way relay channel between the two sources are denoted by and , and we have

(11)

and

(12)

Then, the secrecy rate for and [5] can be defined as

(13)

and

(14)

where represents . According to [25] and [26], we have that the defined secrecy rate is achievable in a two-hop secure communication with an untrusted relay.

As a special case, if the jammers are not used, the jammers’ transmit power should be set to zero, . Then from the derivation above, we can get the corresponding secrecy rate in this case as

(15)

and

(16)

In the system we investigate, there is only one intermediate relay, thus this sole relay is necessary in our assumption for two-way relaying data transmission. Actually, the untrusted relay has the incentive to forward the signals from both the sources since it can eavesdrop on the information transmission through this kind of cooperation. If the relay is non-cooperative that it only receives but not relays the information, then the problem comes to deny-of-service attack. However, this can be easily detected by the sources, then the non-cooperative relay will be treated as a thorough eavesdropper and lose the good opportunity to eavesdrop on the information transmission. The sources will then turn to another intermediate relay for help to relay their information in a practical scenario where there exist multiple intermediate relays. In this paper, we focus on the studies how to prevent the untrusted but necessary intermediate relay from eavesdropping the information, and thus, for simplicity and without loss of generality, we assume that there is only one necessary intermediate relay in the system and the relay is cooperative.

Iii Secrecy Rate of Two-Way Relay Channel Without Jammers

For comparison and consistence, we first investigate the special case without the presence of jammers in this section. We prove that there indeed exists a positive secrecy rate for the two-way relay channel even without the help of friendly jammers distracting the malicious relay. Furthermore, we also obtain an optimal power allocation of the sources and the relay to maximize the secrecy rate. In the next section, we will compare the case with friendly jammers with this case to expect a positive performance gain in the secrecy rate.

Iii-a Existence of Non-zero Secrecy Rate

When the eavesdropper channels from the two sources to the malicious relay are degraded versions of the equivalent main two-way relay channel between and , the two sources can exchange perfectly secure messages at a non-zero rate. Firstly, we consider the transmission from to . In phase , the malicious relay receives the signal from , which consists of the information for . Meanwhile, also transmits the signal at the relay, which acts as both the information carrier for and a jamming signal that makes the eavesdropper channel from to the malicious relay getting worse. In phase , the combined signal consisting of and arrives at . As has a perfect knowledge of its own signal , the signal that jammed the malicious relay in phase has no such an effect on . Therefore, it makes possible that the eavesdropper channel is worse than the data transmission channel from to , which means that a non-zero rate for secure communication from to is available. It is the same situation in the transmission from to . From (15), (16) and the expressions of in (10) and in (8), we can write the probability of the existence of a non-zero secrecy rate as

(17)

where .

Considering the power constraints , , and , we can get that there exists at least one pair of that satisfies , under the channel condition of . Therefore, we have at some power vectors of , which actually indicates that a non-zero secrecy rate in the two-way relay channel is indeed available.

Iii-B Maximizing the Secrecy Rate

In this subsection, we try to get an optimal power vector of which maximizes the secrecy rate of the two-way relay channel. We can formulate the problem subject to the individual secrecy rate constraints and power constraints as

(18)
s.t.

From (III-A), we have that

(19)

From (15), (16), and (18), we can get that

(20)

where

(21)

As has the same monotonic property as under the conditions of (18), we can transform the optimization problem as

(22)
s.t.

It can be calculated that is always established under the conditions of (22), which implies that is a monotonically increasing function of . Therefore, when maximizing the secrecy rate , , where denotes the optimal relay power 333Note that here we calculate the optimal power solution of only from a mathematical perspective to maximize the secrecy rate. In fact, the intermediate relay has no incentive to transmit with the maximum power.. As a result, the problem can be further transformed into .

The optimal solutions of and when maximizing the secrecy rate can be easily obtained under different conditions (i.e., , , and ) through the Lagrangian method by solving the Karush-Kuhn-Tucker (KKT) conditions [43]. In this paper, subject to the space limit, we omit the detailed computing process and only give the results of the optimal solutions of and as:

  1. For the case that , it yields that . Meanwhile, if there exists a solution that satisfies the equation , then we have . Otherwise, we have . Here and denote the optimal power transmitted by and , respectively.

  2. For the case that , it yields that . Meanwhile, if there exists a solution that satisfies the equation , then we have . Otherwise, we have .

  3. For the case that , we have that , and .

In addition, we verify these optimal solutions by simulations in different cases which is shown in Fig. 2 and Fig. 3 in Section-V, and after further calculation based on the simulation results, we can get that the results agree with the optimal power solutions well.

Iv Physical Layer Security with Friendly Jammers

In this section, through further analysis, we first find that the secrecy rate of the sources can be effectively improved by utilizing proper jamming power from the friendly jammers. Then, the problem comes to how to control the jamming power from different friendly jammers when optimizing the secrecy rate of the sources. In general, in a cooperative wireless network with selfish nodes, nodes may not serve a common goal or belong to a single authority. Thus, a mechanism of reimbursement to the friendly jammers should be employed such that the friendly jammers can earn benefits from spending their own transmitting power in helping the sources for secure data transmission. For the source side, the sources aim to achieve the best performance of secrecy rate with the friendly jammers’ help with the least reimbursements to them. For the friendly jammer side, each friendly jammer aims to earn the payment not only covers its transmitting cost but also gains as many extra profits as possible. Therefore, we employ a Stackelberg game model [32] as a power control scheme jointly considering both the benefits of the sources and the friendly jammers. In the Stackelberg game model we proposed, the two sources as a unity is the sole buyer that starts the process of the proposed Stackelberg game, and the friendly jammers are the sellers, therefore, the sources are treated as leader while the friendly jammers are the followers. Furthermore, the optimal solutions of the jamming power and asking price are investigated and a corresponding distributed updating algorithm is provided. Finally, a centralized scheme is proposed for performance comparison.

Iv-a Secrecy Rate Improvement using Friendly Jammers

From (13) and (14), we have

(23)

and

(24)

From (IV-A) and (IV-A), we can see that both and , , are decreasing and convex functions of jamming power , . However, if decreases faster than as the jamming power increases, might increase in some region of value . But when further increases, both and will approach zero. As a result, approaches zero. Compared to (15) and (16), we can get that if and , , i.e., , the gain of the secrecy rate will be above zero in some region of the jamming power . Then the problem comes to how to utilize the jamming power from different friendly jammers effectively to maximize the secrecy rate. Thus, we propose a Stackelberg game model to achieve effective jamming power control in the following subsections.

Note that synchronization among the sources and the friendly jammers is important in the investigated system with friendly jammers. Many works have been devoted to the synchronization issues among distributed nodes in cooperative networks, for example in [44, 45], effective synchronization schemes among distributed sensors and cooperative relays with low complexity and good performance were proposed. Thus, the synchronization issue among the sources and the friendly jammers can be addressed effectively using methods similar to those proposed in [44, 45]. However, this is not the key investigated issue in this paper, therefore, we assume that perfect synchronization among the sources and the friendly jammers is implemented in the system.

Iv-B Source Side Game

We consider the two sources as two buyers who want to optimize their secrecy rates, while the cost paid for the “service”, i.e., jamming power , , should also be taken into consideration. For the source side we can define the utility function as

(25)

where is a positive constant representing the economic gain per unit rate of confidential data transmission between the two sources, and is the cost to pay for the friendly jammers. Here we have

(26)

where is the price per unit power paid for the friendly jammer by the sources, .

When considering the optimal transmitting power vector of source and , i.e., to achieve the maximum utility value in (25), we can treat the jamming power , , as constants since all the nodes transmit with independent power. Thus, we can obtain similar results of optimal power solutions as given in Subsection-III-B. But to obtain the optimal solutions of and is not our main purpose here. In this subsection, we formulate the source side game to study how to effectively utilize the jamming power from different friendly jammers in order to achieve the maximum utility value.

Then the source side game can be expressed as

(27)
s.t.

The goal of the sources as buyers is to buy the optimal amount of power from the friendly jammers in order to maximize the secrecy rate. From (IV-A), (IV-A), and (27), we have

(28)

where , , , , , , , and , .

By differentiating (28) with respect to , we have

(29)

Rearranging the above equation, when , we can get an eighth order polynomial equation as

(30)

where , , are formulae of constants , , , , and variables , , , but .

The solutions to the high order equation (IV-B) can be expressed in closed form, but the expressions of the solutions are extremely complex and have little necessity for our following work. Actually, what to our particular interest are not the closed-form expressions of the optimal jamming power, but the parameters that affect these optimal solutions. Thus, the optimal jamming power solution can be expressed as

(31)

which is a function of the friendly jammer’s price , the other jammers’ jamming power , and other system parameters. Noting that there may be up to eight roots of the polynomial equation (IV-B), the selected solution should be a real root and can lead to a higher value of in (28) than the other real ones. Subject to the power constraints in the game, we can get the optimal strategy as

(32)

If there are no real roots of the equation (IV-B), then the optimal strategy will be either or according to which one can achieve a larger when other parameters are settled.

Because of the high complexity of the solutions to the high order equation in (31), we further consider a special high interference case to obtain a simple expression of the optimal solution. In this special case, we assume that there is one jammer staying very close to the malicious relay, so that the interference from the jammer is much stronger than the power of the received signals from the sources at the relay. Meanwhile, we also assume that the received signal power is much higher than the additive noise, i.e., high signal-to-noise ratio, which means and . Then, we have , , , , and . We assume all the left sides of these inequalities which are much smaller than approach zero. Therefore, the utility function of the source side in (25) can be approximately calculated as

(33)

where and the second approximation comes from the Taylor series expansion when is small enough 444Here we say is small enough means the high order of approaches zero.. It can be easily observed that if , is a decreasing function of . As a result, can be optimized when , i.e., the jammer would not play in the game. If , in order to find the optimal power for the sources to buy, we can calculate

(34)

Hence, the optimal closed-form solution can be expressed as