Joint subcarrier and power allocation for physical layer security in cooperative OFDMA networks
 An Wang^{1},
 Jin Chen^{1},
 Yueming Cai^{1}Email author,
 Chunxiao Cai^{1},
 Wendong Yang^{1} and
 Yunpeng Cheng^{1}
https://doi.org/10.1186/168714992013193
© Wang et al.; licensee Springer. 2013
Received: 4 December 2012
Accepted: 9 July 2013
Published: 18 July 2013
Abstract
In this paper, a joint subcarrier and power allocation algorithm is proposed to improve the physical layer security in cooperative orthogonal frequency division multiple access (OFDMA) networks, where several sourcedestination pairs and one untrusted relay are involved. The relay is friendly and intends to help these pairs to enhance their communications. However, it may be overheard by a malicious eavesdropper at the same time. To optimize the joint subcarrier and power allocation with low complexity, we divide the optimization problem into two simpler subproblems. Firstly, the subcarriers are assigned to the sourcedestination pairs by employing the dual approach under the assumption that the relay power is equally allocated to all the subcarriers. Then, the relay power is allocated to the subcarriers based on the alternative ascending clock auction mechanism. In addition, we prove that the two subproblems can converge after a finite number of iterations. We also find that the proposed auction is cheatproof and, thus, can avoid the cheating behaviors in the auction process. Numerical results demonstrate that our proposed algorithm can effectively improve the system sum secrecy rate, and the convergence performance is also desirable.
Keywords
1 Introduction
Cooperative relaying with orthogonal frequency division multiple access (OFDMA) has recently emerged as a promising technology to achieve the virtual spatial diversity in the wireless networks, which has been adopted in the fourthgeneration mobile communication standard. However, the broadcast nature of wireless communication makes it difficult to ensure reliable and secure message transmission in the presence of passive eavesdroppers. Consequently, physical layer security has aroused growing attention during the recent years. The basic idea of physical layer security is to exploit the physical characteristics of the wireless channels to guarantee secure communication. Physical layer security is quantified by the secrecy capacity, which was pioneered by Wyner in [1]. He also points out that the condition for secure communication is that the secrecy capacity is larger than zero.
Motivated by the fact that careful resource management can remarkably ameliorate the performance of cooperative OFDMA networks, resource allocation has been extensively employed to tackle the challenges of physical layer security [2–8]. In [2], the source and relay power allocation problem is considered in a twohop wireless relay network, where the secrecy rate is improved through choosing proper amount of power to transmit jamming signals for both the source and the relay. In [3], an outage probabilitybased power distribution algorithm between data and artificial noise is proposed to improve physical layer security in the multipleinputsingleoutput system.
Taking the multiuser communications scenario into consideration, the distributed resource management approaches are more desired. Game theory offers a novel perspective and an effective mathematical tool to investigate the interactions among rational players [9]. Recently, the distributed game approaches have been widely employed to develop distributed and flexible resource management mechanisms in order to avoid the high complexity and excessive energy consumption of centralized methods [4–8, 10, 11]. In [4], Han et al. investigated the interaction among the source and the friendly jammers to increase the secrecy capacity using the Stackelberg game. Physical layer security is also improved by utilizing jamming power allocation for the twoway untrusted relaying based on the Stackelberg game in [5]. In [6], the coalitional game is employed to enhance the physical layer security. Auction game [12] has been widely investigated as an efficient tool for resource allocation, such as in [7, 10], and [11]. In [7], the physical layer security is ameliorated using two auctions: the traditional ascending clock auction (ACAT) and the alternative ascending clock auction (ACAA). The literature mentioned above mostly focus on the power allocation (the relay’s or the jammer’s power). However, effective subcarrier assignment can also improve the performance of the OFDMA systems, which has not drawn sufficient attention [8]. In [8], the authors formulated an analytical framework for subcarrier and power allocation in a downlink OFDMAbased broadband network with coexistence of secure users and normal users. The average aggregate information rate of all the normal users was maximized via dual decomposition while maintaining an average secrecy rate for each secure user.
In this paper, a cooperative OFDMA network is considered, where there exist several sourcedestination pairs and one untrusted relay. The case of untrusted relaying in physical layer security has been investigated in the previous literature [13–15]. The untrusted relay in this paper is friendly and intends to help these user pairs to enhance their communications, which is different from the traditional ones in [13]. It is untrusted because it may be overheard by a malicious eavesdropper at the same time. Moreover, the eavesdropper can just passively listen to the relay, and it is not capable of disturbing the normal communication process. As a result, we can try to improve the physical layer security of the network by jointly optimizing the subcarrier and relay power allocation policies. To avoid the high complexity resulting from the optimal joint subcarrier and power allocation, we decompose it into two simpler subproblems instead. Firstly, the subcarriers are assigned to the sourcedestination pairs by employing the dual approach under the assumption that the relay power is equally allocated to all the subcarriers. Then, the relay power is allocated to the subcarriers according to the ACAA mechanism. In the proposed auction game, the relay is modeled as the auctioneer and the subcarriers are regarded as the bidders. However, due to the fact that the subcarriers are not entities, it is the corresponding sources that submit the bids instead of the subcarriers. The main contribution of this paper can be summarized as follows:

A novel joint subcarrier and power allocation algorithm is proposed to improve the physical layer security for the cooperative OFDMA networks. This algorithm reduces the complexity of optimal joint resource allocation and can ensure all the users’ secrecy rate effectively.

A nonconvergent infinite series is designed to ensure the convergence of the proposed dualbased subcarrier assignment algorithm. For the proposed ACAAbased power allocation algorithm, we prove the existence of the equilibrium and the cheatproof property.
The rest of the paper is organized as follows. System model and the assumptions are introduced in Section 2. Detailed description of the dualbased subcarrier assignment is presented in Section 3. Section 4 describes the ACAAbased power allocation and investigates some properties of the proposed auction, including the convergence performance and the cheatproof property. Section 5 shows and discusses the simulation results, and it is followed by the conclusion in Section 6.
2 System model
We assume that all the sourcedestination pairs share the total N available subcarriers, and each subcarrier n, $n\in \mathcal{N}=\{1,\dots ,N\}$, can only be exclusively allocated to one user pair. Assume that the instantaneous channel state information (CSI) of any user pair is perfectly known at the corresponding source node. For the broadband channel model, we consider the slow and flat Rayleigh fading and long path loss. For the n th subcarrier, the channel coefficients between the source S_{ m } and the destination D_{ m }, between the source S_{ m } and the relay R, and between the relay R and the destination D_{ m } are denoted by ${h}_{d,m}^{n}$, ${h}_{a,m}^{n}$, and ${h}_{b,m}^{n}$, respectively. For the long path loss, a path loss exponent β is assumed. Without loss of generality, we assume that the thermal noise at each node is independent and has the same variance σ^{2}.
We assume that all the nodes operate in the halfduplex mode, and the untrusted relay employs the amplifyandforward strategy. We take the m th sourcedestination pair, for example, to describe the communication process. For the m th user pair, the complete transmission process can be divided into two phases. In phase 1, the source S_{ m } broadcasts its data to the relay R and its intended destination D_{ m }. During phase 2, the relay R amplifies its received signal to the destination D_{ m }. We also assume that the destination can perfectly combine the signals received from the source and the relay.
where p_{ s } is the transmit power of the source, and ${p}_{m}^{n}$ is the power that the relay allocates to the m th user pair on the n th subcarrier. In this paper, we just focus on the relay power allocation and simply assume that all the source nodes transmit with the same power p_{ s }, 0 ≤ p_{ s } ≤ p_{max}, where p_{max} is the peak power of all the nodes in the network.
where (x)^{+} represents max{x, 0}.
In this paper, our primary goal is to maximize the available secrecy rate through careful resource allocation approaches. Joint subcarrier and power allocation is carefully considered to meet the secure requirements of the cooperative OFDMA network. As we know, the optimal subcarrier and power allocation is an NPhard problem and will become extremely complex as the number of subcarriers gets large. For simplicity, we divide the original optimization problem into two progressive subproblems, that is, the subcarrier assignment and power allocation are separately optimized. Firstly, we assign the subcarriers using the dual approach under the assumption that the relay power is equally allocated to all the subcarriers. After that, the relay power is allocated to the subcarriers according to the ACAA mechanism. The distributed auction can not only reduce the complexity of solving the power allocation problem but also can ensure all the users’ secrecy rate. Therefore, the auctionbased power allocation can effectively avoid the high complexity and the unfairness, which are both the drawbacks of the centralized methods. In the following two sections, the dualbased subcarrier assignment and the ACAAbased power allocation are respectively introduced.
3 Dualbased subcarrier assignment
As illustrated above, our first task is to assign the subcarriers to the sourcedestination pairs. Here, we can express the subcarrier assignment by the binary assignment variables ${c}_{m}^{n}$. If ${c}_{m}^{n}=1$, it implies that the n th subcarrier is assigned to the m th sourcedestination pair; and if ${c}_{m}^{n}=0$, otherwise. Also, the binary assignment variables form the subcarrier assignment matrix C_{N × M}. As a result, we can treat the subcarrier assignment problem as a 01 integer programming problem.
where RS is the system sum secrecy rate and is the optimization goal in this paper. The constraints 5a) and 5b) are used to guarantee that each subcarrier can only be exclusively assigned to one user pair, and the last constraint 5c) indicates that the secrecy rate of each user pair must be larger than a predefined threshold to ensure the secure communication.
It is not difficult to find that the optimization problem defined in (5) satisfies the timesharing condition which was introduced in [17], that is, the objective function is concave and the constraint 5c) is convex given that ${\text{RS}}_{m}^{n}$ is concave in ${p}_{m}^{n}$ and that the integral preserves concavity. As a result, we can employ the dual approach to solve the subcarrier assignment problem, and the duality gap becomes asymptotically zero for a large enough number of subcarriers.
where t is the iteration number, and α(t) represents the proper step sizes. The above update is guaranteed to converge to the optimal dual variables as long as the step sizes follow a diminishing step size rule.
Theorem 1.
[[19]] If $\sum _{t}\alpha \left(t\right)\to \infty $ and α(t) → 0 as t → ∞, then the optimizing goal can converge to the optimal value.
According to Theorem 1, the optimal solution can be obtained as long as we design a nonconvergent infinite series α(t), whose items decrease to zero as the iteration number goes to infinite. Therefore, in this paper, we let $\alpha \left(t\right)=\frac{1}{t}$ and the optimal solution can be achieved.
Finally, the dualbased subcarrier assignment algorithm can be summarized as follows:
 (S1)
Initialize λ(0). The user pairs feedback RS_{ m } and CSI to the relay.
 (S2)Given λ(t), for each subcarrier n, the relay
 (a)
calculates the secrecy rate ${\text{RS}}_{m,t}^{n}$,
 (b)
solves the assignment variables ${c}_{m,t}^{n}$ according to (eq10),
 (c)
broadcasts the subcarrier assignment matrix C _{ t } of this iteration to all the sources.
 (a)
 (S3)
The sources update the dual variables λ(t) according to (11) and then set t = t + 1.
 (S4)
Return to (S2) until convergence is reached.
4 ACAAbased power allocation
In Section 3, all the subcarriers have been carefully assigned to the user pairs based on the dual approach. The following key issue is how to allocate the available relay power efficiently to all the subcarriers in a distributed way. In this paper, we adopt the ACAA mechanism [12] to optimize the relay power allocation to improve the system sum secrecy rate. On one hand, the distributed auction can reduce the complexity of solving the power allocation problem. On the other hand, the auction can also ensure the competitive fairness among all the bidders, and thus, all the users’ secrecy rate can be guaranteed.
In the proposed ACAA model, we try to maximize the secrecy rate on each subcarrier, and finally the system sum secrecy rate is maximized. Taking the structure of the cooperative OFDMA network into consideration, we define the auction elements as follows: the good for sale is the total relay power, the relay is the auctioneer, and the subcarriers are the bidders. However, the subcarriers are not authentic entities and are not capable of reporting its optimal demands to the auctioneer in the auction process, but we should notice that the relay has broadcasted the final subcarrier assignment matrix to all the sources in the former step, which implies that each source knows which subcarriers belong to it. As a result, it is the corresponding source node that interacts with the relay instead of the subcarrier in the auction process. Therefore, although we model the subcarriers as the bidders, it is actually the sources that participate in the auction.
We can easily see that the relay’s utility function is monotonically increasing with the unit price μ and the total power consumption $\sum _{m=1}^{M}\sum _{n=1}^{N}{p}_{m}^{n}$. We should note that there should be a reserve price μ^{0} in the trade, which is set equal to the average cost of transmitting unit power, i.e., ${\mu}^{0}=\mathcal{C}\text{ost}/{p}_{\text{max}}$, where $\mathcal{C}\text{ost}$ denotes the basic cost of the relay. Then, we can always keep the relay in the trade if the asking price μ is larger than μ^{0}. In the next subsection, the ACAAbased power allocation algorithm is carried out and analyzed in detail.
4.1 ACAAbased power allocation algorithm
The detailed steps of the proposed ACAAbased power allocation algorithm are summarized as follows:
 (S1)
Given the available relay power p_{max}, the price step δ > 0 and the iteration index t = 0. The relay initializes the asking price with the reserve price μ^{0} and broadcasts it to all the sources.
 (S2)
For each subcarrier n, its corresponding source S_{ m } computes ${p}_{m,0}^{n}=arg\phantom{\rule{.5em}{0ex}}\underset{{p}_{m}^{n}}{\text{max}}\phantom{\rule{.5em}{0ex}}{U}_{m}^{n}\left({p}_{m}^{n},{\mu}^{0}\right)$ and submits its optimal bid ${p}_{m,0}^{n}$ to the relay.
 (S3)The relay sums up all the bids from the sources ${p}_{\text{total},0}=\sum _{n=1}^{N}\sum _{m=1}^{M}{p}_{m,0}^{n}$ and compares it with p_{max}:
 (1)
If p _{total,0} ≤ p _{max}, the relay concludes the auction and chooses to quit the trade.
 (2)Else, update μ ^{t+1} = μ ^{ t } + δ, t = t + 1, and repeat:
 (a)
The relay announces μ ^{ t } to all the sources.
 (b)
For each subcarrier n, its corresponding source S _{ m } computes ${p}_{m,t}^{n}=arg\phantom{\rule{.5em}{0ex}}\underset{{p}_{m}^{n}}{\text{max}}\phantom{\rule{.5em}{0ex}}{U}_{m}^{n}\left({p}_{m}^{n},{\mu}^{t}\right)$ and submits its optimal bid ${p}_{m,t}^{n}$ to the relay.
 (c)The relay sums up all the bids from the sources ${p}_{\text{total},t}=\sum _{n=1}^{N}\sum _{m=1}^{M}{p}_{m,t}^{n}$ and compares it with p _{max}:

If p_{total,t} > p_{max}, first compute ${\mathrm{F}}_{m,t}^{n}={\left({p}_{\text{max}}\sum _{i\ne m}\sum _{j\ne n}{p}_{i,t}^{j}\right)}^{+}$, then set μ^{t+1} = μ^{ t } + δ, t = t + 1 and continue the auction.

Else, set T = t and compute ${\mathrm{F}}_{m,T}^{n}={p}_{m,T}^{n}+\frac{{p}_{m,T1}^{n}{p}_{m,T}^{n}}{\sum _{m}\sum _{n}{p}_{m,T1}^{n}\sum _{m}\sum _{n}{p}_{m,T}^{n}}\left({p}_{\text{max}}\sum _{m}\sum _{n}{p}_{m,T}^{n}\right)$, conclude the auction and allocate ${p}_{m}^{{n}^{\ast}}={\mathrm{F}}_{m,T}^{n}$ to the n th subcarrier.

 (a)
 (1)
 (S4)
Finally, the utility of the n th subcarrier is ${U}_{m}^{{n}^{\ast}}\left({p}_{m}^{{n}^{\ast}},{\mu}^{T}\right)={\text{RS}}_{m}^{n}\left({p}_{m}^{{n}^{\ast}}\right)\mathcal{P}({p}_{m}^{{n}^{\ast}},{\mu}^{T})$, where ${\mathcal{P}}^{\ast}({p}_{m}^{{n}^{\ast}},{\mu}^{T})$ is the payment of the n th subcarrier and can be expressed as $\mathcal{P}({p}_{m}^{{n}^{\ast}},{\mu}^{T})={\mu}^{0}{\mathrm{F}}_{m,0}^{n}+\sum _{t=1}^{T}{\mu}^{t}\left({\mathrm{F}}_{m,t}^{n}{\mathrm{F}}_{m,t1}^{n}\right)$.
4.2 Properties of the ACAAbased power allocation algorithm
In this subsection, we analyze some properties of the proposed ACAAbased power allocation algorithm: the existence of the equilibrium and the cheatproof property. The cheatproof property implies that the cheating behaviors can be effectively avoided in the auction process. If an auction is cheatproof, it means that in every iteration, the mutually best response of each bidder is to submit its true optimal bid rather than any other bid value. Therefore, no bidder has the incentive to cheat in the auction procedure because any cheating will lead to the loss of its ultimate utility value.
Theorem 2.
The proposed ACAAbased power allocation algorithm converges after a finite number of iterations and exists at least one equilibrium point.
Proof.
Therefore, there exists a finite positive iteration index T, T < K p_{ s } / δ, satisfying the condition $\sum _{n=1}^{N}\sum _{m=1}^{M}{p}_{m,T}^{n}<{p}_{\text{max}}$. So, we can conclude that the proposed ACAA algorithm converges after a finite number of iterations and exists at least one equilibrium point. Therefore, Theorem 2 is proved. □
Theorem 3.
The proposed ACAAbased power allocation algorithm is cheatproof and no bidder cheats in the auction.
Proof.
 1.If T _{2} < T _{1}, then ${\mu}^{{T}_{2}}<{\mu}^{{T}_{1}}$, and we have$\begin{array}{ll}{U}_{m,{T}_{1}}^{n}{U}_{m,{T}_{2}}^{n}& ={\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{1}}^{n}\right){\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{2}}^{n}\right)\\ \phantom{\rule{1em}{0ex}}\sum _{t={T}_{2}+1}^{{T}_{1}}{\mu}^{t}\left({\mathrm{F}}_{m,t}^{n}{\mathrm{F}}_{m,t1}^{n}\right)\\ >{\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{1}}^{n}\right){\mu}^{{T}_{1}}{p}_{m,{T}_{1}}^{n}\\ \phantom{\rule{1em}{0ex}}{\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{2}}^{n}\right)+{\mu}^{{T}_{1}}{p}_{m,{T}_{2}}^{n}\\ ={U}_{m}^{n}\left({p}_{m,{T}_{1}}^{n},{\mu}^{{T}_{1}}\right){U}_{m}^{n}\left({p}_{m,{T}_{2}}^{n},{\mu}^{{T}_{1}}\right)\\ \ge 0\end{array}$(25)
 2.If T _{2} ≥ T _{1}, then ${\mu}^{{T}_{2}}\ge {\mu}^{{T}_{1}}$, and we have$\begin{array}{ll}{U}_{m,{T}_{1}}^{n}{U}_{m,{T}_{2}}^{n}& ={\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{1}}^{n}\right){\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{2}}^{n}\right)\\ \phantom{\rule{1em}{0ex}}+\sum _{t={T}_{1}+1}^{{T}_{2}}{\mu}^{t}\left({\mathrm{F}}_{m,t}^{n}{\mathrm{F}}_{m,t1}^{n}\right)\\ \ge {\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{1}}^{n}\right){\mu}^{{T}_{1}}{p}_{m,{T}_{1}}^{n}\\ \phantom{\rule{1em}{0ex}}{\text{RS}}_{m}^{n}\left({\mathrm{F}}_{m,{T}_{2}}^{n}\right)+{\mu}^{{T}_{1}}{p}_{m,{T}_{2}}^{n}\\ ={U}_{m}^{n}\left({p}_{m,{T}_{1}}^{n},{\mu}^{{T}_{1}}\right){U}_{m}^{n}\left({p}_{m,{T}_{2}}^{n},{\mu}^{{T}_{1}}\right)\\ \ge 0\end{array}$(26)
From (25) and (26), we can conclude that ${U}_{m,{T}_{1}}^{n}\ge {U}_{m,{T}_{2}}^{n}$ always holds in both cases. As a result, if all the other bidders submit their true optimal bid values, the best response of the bidder n is also to report its true optimal bid value at every iteration. In other words, to submit the optimal bid is the mutually best response for all the bidders. In such auction, no bidder intends to cheat in the auction procedure, and any cheating may lead to the loss of its ultimate utility value. Therefore, the proposed that ACAA algorithm is cheatproof, and Theorem 3 is proved. □
5 Simulation results and discussions
In this section, some simulation results done on the MATLAB platform are carried out to verify the performance of our proposed joint subcarrier and power allocation algorithm in this paper. The simulation assumptions and parameters are set up as follows [7]. We assume that there are totally M sourcedestination pairs randomly locating around the untrusted relay. These sourcedestination pairs share the total N available subcarriers. All the source nodes transmit with the power p_{ s } = 1 W. For all the channels, a slow and flat Rayleigh fading environment with unitary power is assumed, where the channel coefficients consist of the Rayleigh fading and the long path loss; the path loss factor is β = 2. The thermal noise variance at each node is σ^{2} = 10^{−12} W. We set the reserve price μ^{0} to 0.1 and set the price step δ to 0.01 in the simulation.
At the very beginning, we verify that our proposed that joint subcarrier and power allocation algorithm can actually ameliorate the physical layer security for the cooperative OFDMA network. In Figures 2 and 3, four different algorithms are simulated respectively:

DSA+ACAA. This represents our proposed joint subcarrier and power allocation algorithm in this paper.

DSA+EPA. The subcarriers are assigned to the user pair based on the dual approaches [22], and the relay power is equally allocated to all the subcarriers.

SSA + ACAA. In this algorithm, the subcarriers are randomly assigned to the pairs, and the relay power is allocated using the ACAA mechanism [7].

DSA + SG. This means the dualbased subcarrier assignment and the Stackelberg gamebased relay power allocation in [4].
Figure 2 depicts the relationship between the system sum secrecy rate and the number of subcarriers, and Figure 3 shows how the system sum secrecy rate changes with the total relay power. From Figures 2 and 3, we can easily draw the following conclusions:

Our proposed ‘DSA+ACAA’ algorithm can far outperform the other algorithms as the number of the subcarriers and the total relay power increase. This implies that physical layer security can be meliorated through felicitous subcarrier and power allocation in this paper.

In the Stackelberg gamebased power allocation algorithm, the information exchange indeed includes the optimal power and the optimal price. In each iteration of our proposed ACAAbased power allocation algorithm, each source needs to submit the optimal bid, and the relay only needs to broadcast the updated price. The two gamebased algorithms could both improve the physical layer security with less information exchange. This implies that our proposed power allocation algorithm can achieve better performance than the Stackelberg gamebased algorithm with almost the same information exchange.

We can also find that the security performance of the system will become much worse when the subcarriers are randomly allocated or the power is equally allocated. The simulation results demonstrate that the subcarrier assignment and power allocation are both very important to the improvement of the physical layer security. More importantly, we can conclude that joint resource management far outperforms the separate subcarrier assignment or power allocation.
Number of iterations of the dualbased subcarrier assignment algorithm
Number of subcarriers  Iteration times 

4  50 
8  29 
16  78 
24  71 
32  59 
40  79 
48  79 
56  76 
64  72 
6 Conclusion
In this paper, we develop a joint subcarrier and power allocation algorithm to improve physical layer security in cooperative OFDMA networks. Specifically, we assign the subcarriers to the sourcedestination pairs by utilizing the dual approach under the assumption that the power is equally allocated to all the subcarriers. Then, the relay power is allocated to the subcarriers according to the ACAA mechanism. We prove that both of the subproblems can converge in a finite number of iterations. We also found that the proposed auction is cheatproof and, thus, can avoid cheating behaviors in the auction process. Numerical results also demonstrate that our proposed algorithm can effectively increase the system sum secrecy rate.
Physical layer security is a potential supplement for the cryptographic methods and an effective technique to achieve perfect secrecy rate against eavesdropping. In the future, we will try to develop more effective and simpler resource allocation algorithms for the cooperative OFDMA networks to gain the capabilities against eavesdropping.
Declarations
Acknowledgements
This study is supported by the National Natural Science Foundation of China (no. 61001107), the China Postdoctoral Science Foundation under grant no. 2013T60912, and the Jiangsu Provincial Natural Science Foundation of China under grant no. BK2013105.
Authors’ Affiliations
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