Bargainingbased jammer power allocation for dynamic eavesdropping scenario
 Duan Bowen^{1},
 Cai Yueming^{1},
 Zheng Jianchao^{1}Email author,
 Yang Weiwei^{1} and
 Yang Wendong^{1}
https://doi.org/10.1186/168714992014186
© Bowen et al.; licensee Springer. 2014
Received: 30 May 2014
Accepted: 27 October 2014
Published: 12 November 2014
Abstract
This paper proposes a bargainingbased jammer power allocation scheme for multisource multidestination wireless network in the presence of a friendly jammer and a malicious node which eavesdrops erratically. We formulate the erratic behavior of the eavesdropper as a novel model where the eavesdropper wiretaps the message of the legitimate sources with a certain probability in a time slot. Moreover, in order to obtain a fair and efficient solution, the jammer power allocation problem is modeled as a Nash bargaining game under the constraint of maximum transmit power of a friendly jammer, which is a convex optimization problem. Then, the closed form of the Nash bargaining solution (NBS) is derived, and a simple but effective centralized algorithm is proposed. Besides, we find that the even power allocation solution and the sumsecrecyrate optimal solution are the special cases of the NBS, when the bargaining power is properly selected. Simulation results demonstrate that the NBS achieves a good performance in terms of both effectiveness and fairness.
Keywords
1 Introduction
The shared nature of the wireless communication channel poses numerous security challenges, one of which is making wireless communications susceptible to eavesdropping [1–4]. Privacy and security are the key aspects in wireless communication systems while the traditional encryption cannot ensure that the message of legitimate source is absolutely infeasible to be deciphered especially when the mismanagement of key occurs. Fortunately, many researches find that the physical layer of wireless communications also enables novel ways to defend against the eavesdroppers [1–5]. Therefore, physical layer security approaches are gaining extensive attention.
Friendly jamming is one of the techniques in physical layer security, whose idea is to produce the artificial noise to deteriorate the eavesdropping channels to such an extent that successful decoding of the legitimate messages becomes infeasible [3–13]. The works [3–8] investigate the jammerassisted communication system in terms of the design of transmission plan and the analysis of performance. [3] investigate the impact of the jammer’s antenna number on system secrecy rate, and they find that only the eavesdropper’s channel will be degraded when the jammer uses multiple antennas to generate artificial noise. Moreover, Tippenhauer et al. in [4] consider the reliability of the jammerassisted communication system and illustrate the limitation of the jammer in terms of maintaining confidentiality from a perspective of attacker. Besides, Tang et al. in [6] analyze the secrecy performance of the jammerassisted communication system in the case of the discrete memoryless channels and the Guassian channels. The secrecy capacity and the achievable secrecy rate are given for both channels.
Specifically, it is pioneered by [9–13] in terms of the jammer power allocation [9] and [10] use the auction theory to design the jammer power allocation schemes. The friendly jammer and the sources act as the auctioneer and the bidders, respectively. A distributed solution is obtained, and the properties such as convergence and equilibrium are analyzed. Han et al. in [11] introduce the Stackelberg game to investigate the interaction between the source and the friendly jammers. Through exchanging the ‘price’ between the sources and the friendly jammer repeatedly, a distributed solution with desirable performance is obtained. In [12, 13], the authors design a Stackelberg game framework to encourage terminals to act as the jamming node by compensating them with an opportunity to access the channel of legitimate parties. This cooperative mechanism cannot only increase the spectral efficiency but also improve secrecy rate meaningfully.
For the works above, it is worth mentioning that the scenario they consider is the case that the eavesdropper works all the time. However, in some practical systems, the eavesdropper is likely to wiretap erratically or has different preferences to different legitimate source’s message, especially when the antenna number of the eavesdropper is limited and its eavesdropping ability tends to be reserved for the more valuable legitimate users. In this paper, this erratic nature of eavesdropper is referred to as the dynamic nature of the eavesdropper and this kind of scenario is called dynamic eavesdropping scenario. Undoubtedly, one of the results that the dynamic nature of eavesdropper brings about is the change on the strategy of jammer power allocation. If the network operator implements the jammer power allocation regardless of the dynamic nature of eavesdropper, it is likely that some sources experiencing securer communication link are allocated redundant jammer power while the packets of other sources endangered may be received by the eavesdropper. To the best of our knowledge, there is no any research on the scenario where the eavesdroppers work intermittently. Therefore, a study on the model which can reflect the dynamic nature of the eavesdroppers and the corresponding optimization schemes is urgently required.
In addition, the objective of the aforementioned works is to improve the secrecy rate or maximize the secrecy capacity, while seldom works consider the fairness among the sources. The works [9–13] have made progress in terms of the interaction between the sources and the jammer, but the fairness among the sources has not been taken into consideration, and thus, that kind of solution will generally suffer from severe fairness problem. Although [14] has investigated the similar physical layer security scenario with consideration of fairness, the author only compared the proposed scheme with the even power allocation scheme. Thus, the effectiveness of that scheme is questionable. Fortunately, cooperative game theory often acts as a powerful tool to investigate how the sources negotiate to achieve their conflicting objectives. Specifically, the Nash bargaining solution (NBS), a core concept in the cooperative game theory, possesses the NBS fairness and Pareto optimality, and thus, it is introduced to increase efficiency while maintaining fairness in the jammer power allocation.
Briefly, the contributions and novelty of our work are summarized below:

To the best of our knowledge, this is the first paper that studies the jammer power allocation in the dynamic eavesdropping scenario. Furthermore, this novel model is a generalized physical layer security model. When the malicious node is ensured to eavesdrop all the time, the generalized model will degenerate to the classic physical layer security model [9–13].

We take both the sumsecrecyrate and the fairness among the sources into account by proposing a NBSbased optimization scheme. Moreover, the closedform NBS is derived so that the iteration progress in the Stackelberg game and auction game can be replaced by a simple but effective centralized implementation.

The impact of bargaining power on system performance is investigated. Theoretical analysis and numerical results indicate that the even power allocation solution and the sumsecrecyrate optimal solution are the special cases of the NBS, when the bargaining power is properly selected.
The remainder of the paper is organized as follows. The system model and utility function are presented in Section 2. In Section 3, the optimization problem in dynamic eavesdropping scenario is modeled as the NBSbased power allocation problem. The NBS is derived, and the effect of bargaining power on network performance is investigated. Section 4 presents our simulation results, and Section 5 concludes the paper.
2 System model and utility function
2.1 System model
Obviously, the Γ_{ i }, Γ i′, and Γ_{i,0} represent how secure the received signal is and is closely related to the security performance. It can be seen from Equation 4 that the more jammer power to be allocated to a source, the more seriously the eavesdropper’s ability will be impaired and the larger security factor can be obtained. However, since the jammer power is limited, a competition among sources for the jammer power will exist inevitably. Moreover, to maximize the security performance of the network, considering the requirement of sources in the static scenario is not enough, the jammer has to take the dynamic nature of the eavesdropper into account. To resolve the conflict among sources and describe the scenario more precisely, the interaction among the sources will be modeled as a Nash bargaining game and the utility function will be designed according to the whole probability formula.
2.2 Utility function
Formally, a bargaining game is defined by a set of feasible utilities U and a disagreement point u_{0}∈U. The set U contains utility vectors u=(u_{1}u_{2}⋯u_{ N }), denoting the payoff to each source, for all possible strategies the players may implement. The disagreement point u_{0} represents the ‘status quo’ prior to bargaining or the worst possible outcomes for the sources. In our model, although the strategies are the power contributed by jammer to help each source instead of the sources action, it can be interpreted as an outcome of the bargaining among the sources. In other words, sources cooperatively choose a compromise point. That is, rather than individually focusing on payoff maximization, sources jointly choose a mutually agreeable utility vector and agree to implement the strategies at the jammer.
3 NBSbased jammer power allocation
3.1 Nash bargaining game
The NBS is a core solution concept in cooperative game theory, and we choose it as the bargaining game solution among many bargaining games for the following reasons: First, since the NBS is obtained based on a certain set of axioms representing the fairness of the solution, it can improve the sources’ utility and maintain the fairness among sources at the same time. Second, the NBS has flexibility in bargaining power selection. This property makes it available to balance between the optimal sumsecrecyrate and the optimal fairness among sources, which is preferred on system design perspective. Third, due to the property of the Nash bargaining game, an efficient and fair solution can be obtained readily, with no need for the iteration used for converging to the equilibrium in the noncooperative game. Thus, the pricing mechanism which will result in heavy overheads to the network can be avoided.
In the Nash bargaining game, given the disagreement point u_{0} and the feasible utility set U, the Nash bargaining solution provides a fair and efficient method to distribute jammer power among all the sources. Suris et al. [16] have made it clear that the convexity of the utility space (U) is a sufficient condition to guarantee that the optimization problem of the Nash product (NP) is a Nash bargaining game. In order to obtain the NBS, the conclusion in [16] will be applied to prove the following theorem first.
Theorem 1
The game $\left\{\mathcal{N},U\right\}$ is a Nash bargaining game.
Proof
The game $\left\{\mathcal{N},U\right\}$ is a bargaining game if and only if U is a closed and convex subset of ${U}^{\mathcal{N}}$[17]. It is obvious that the utility set U is closed, we only need to check whether the convexity of set U is met, which means for any 0≤θ≤1, if ${U}^{a}=\left({U}_{1}^{a},\cdots \phantom{\rule{0.3em}{0ex}},{U}_{2}^{a}\right)\in \mathbb{U}$ and ${U}^{b}=\left({U}_{1}^{b},\cdots \phantom{\rule{0.3em}{0ex}},{U}_{2}^{b}\right)\in \mathbb{U}$, then $\theta {U}^{a}+\left(1\theta \right){U}^{b}\in \mathbb{U}$.
where the last constraint means that 0≤P_{ i }<∞,∀i. To make the problem solvable, we define ${U}_{1}\triangleq \left\{\text{u}\left{u}_{i}\ge {u}_{i,0},i=1,\dots ,N\right.\right\}$ and ${U}_{2}\phantom{\rule{2.77626pt}{0ex}}\triangleq \phantom{\rule{2.77626pt}{0ex}}\left\{\text{u}\left\phi \left(\text{u}\right)\le {P}_{J},{u}_{i}<{P}_{s}{\gamma}_{\mathit{\text{sd}}}^{i},i=1,\dots ,N\right.\right\}$. Hence, we have U=U_{1}∩U_{2}. Since U_{1} is a convex set obviously, to prove the utility set U is convex, we only need to prove that U_{2} is a convex set.
Since ${u}_{i}<{P}_{s}{\gamma}_{\mathit{\text{sd}}}^{i}$ for any finite P_{ i }, we have $\frac{{\partial}^{2}\phi \left(\text{u}\right)}{\partial {u}_{i}^{2}}>0$ for all i=1,…,N, which means the Hessian matrix of φ(u) is positive definite. Therefore, φ(u) is a convex function and for set $\theta {U}_{2}^{a}+(1\theta ){U}_{2}^{b}$, we have $\phi \text{(}\theta {U}_{2}^{a}+(1\theta ){U}_{2}^{b}\text{)}\le \mathrm{\theta \phi}\left({U}_{2}^{a}\right)+(1\theta )\phi \text{(}{U}_{2}^{b})\le \theta {P}_{J}+(1\theta ){P}_{J}={P}_{J}$, which means $\theta {U}_{2}^{a}+(1\theta ){U}_{2}^{b}\in \mathbb{U}$. As a result, U_{2} is a convex set [18], and thus, the game $\left\{\mathcal{N},U\right\}$ is a Nash bargaining game. This completes the proof.
We can see that if α changes from 0 to ∞, the left side of Equation 20 will monotonically decrease from ∞ to 0. Hence, it is ensured that a unique positive solution α exists for Equation 20, and the optimal α can be found using bisection method. Then, the Nash bargaining solution can be obtained using Equation 19 and the optimal α.
3.2 Analysis of bargaining power
The bargaining power of the source S_{ i } represents the priority of the source, which can be tuned by the jammer to achieve a balance between the optimal global security performance and the optimal fairness among sources. In the respect of the fairness among sources, since the optimal secrecyrate fairness, which depends on the topology of the network, is not always achievable, here, we only derive the relationship between the NBS and the optimal power fairness solution. The following theorem shows the effect of the bargaining power analytically.
Theorem 2.
where ${q}_{i}^{E}=\frac{({K}_{i}{P}_{J}+{L}_{i}N)\left({I}_{i}{P}_{J}+{J}_{i}N\right){H}_{i}{\left({I}_{i}{P}_{J}+{J}_{i}N\right)}^{2}}{\left({K}_{i}{J}_{i}{I}_{i}{L}_{i}\right)}$ and ${q}_{i}^{\text{O}}=\frac{\frac{\left({K}_{i}{P}_{i}^{\text{O}}+{L}_{i}\right)}{\left({I}_{i}{P}_{i}^{\text{O}}+{J}_{i}\right)}{H}_{i}}{ln2\left(1+\frac{\left({K}_{i}{P}_{i}^{\text{O}}+{L}_{i}\right)}{\left({I}_{i}{P}_{i}^{\text{O}}+{J}_{i}\right)}\right)}$.
Proof.
Thus, when ${\beta}_{i}=\frac{{q}_{i}^{E}}{{\sum}_{i=1}^{N}{q}_{i}^{E}}$, the NBS coincides with the even power allocation solution.
the NBS coincides with the sumsecrecyrate optimal solution. This completes the proof.
3.3 Centralized implementation
Since the Stackelberg game and auction game require many iterations before converging to the Nash equilibrium, the exchange of information in the iteration progress will give rise to heavy overheads to the network. In this paper, we propose a centralized algorithm, which is simple but efficient thanks to the closed form of the NBS. Nevertheless, it requires enough computational ability and global channel state information (CSI) at the jammer.
In the centralized algorithm, we assume that the jammer have global and perfect CSI to compute the NBS proposed before. To get the NBSbased power allocation, each source informs the jammer of the eavesdropping probability θ_{ i } through backhaul communication, according to the evaluation of the security of their message by themselves. Then, the jammer uses the bisection method to find the α that satisfies Equation 20 and computes the NBSbased power allocation solution by using Equation 19. In this way, without any iteration and resulting heavy overheads to the network, the NBS scheme can obtain its solutions precisely.
4 Simulation results
In this section, we evaluate the performance of the NBSbased jammer power allocation by MATLAB simulations. The global security performance and the fairness among sources will be demonstrated and analyzed as the eavesdropping probability increases. We compare the three optimization schemes (sumsecrecyrate optimal (SSRO), even power allocation (EPA), and NBS) and evaluate the four parameters: network sumsecrecyrate, individual achievable secrecy rate, Jain’s fairness index (JFI) of the secrecy rate and the JFI of the allocated power [20]. Note that a larger JFI indicates a fairer solution. For all the channels, the path loss coefficient is α=4 and the static channels only with path loss is assumed due to the consideration of the average performance during a given time slot.
In Figure 7, we compare the fairness among four schemes. In Figure 7a, we find that the NBS2 is the best and NBS1 is the worst among all four schemes in terms of secrecyrate fairness. Although the EPA is a powerfairest scheme, which, however, cannot achieve more desirable fairness than NBS2 in the simulated eavesdropping probability range. In Figure 7b, with the eavesdropping probability increasing, the power fairness of NBS2 is approaching that of EPA (e.g., J F I=1). Since the NBS1 more focuses on improving sumsecrecyrate, its power fairness is the worst in high eavesdropping probability region.
5 Conclusion
In this paper, we investigated the security performance and designed the corresponding optimization scheme in the dynamic eavesdropping scenario. The eavesdropping probability was defined to characterize the erratic behavior of the eavesdropper, and the bargaining theory was introduced to analyze the negotiation among the sources on jammer power allocation. In order to address both the fairness among sources and global security performance, the optimization problem was formulated as a Nash bargaining game, which is a convex optimization problem. The closedform NBS was derived, and the effect of the bargaining power was investigated analytically. Simulation results showed that the effectiveness and fairness of the proposed NBSbased resource allocation are desirable.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 61371122 and No. 61301162) and the Jiangsu Natural Science Foundation (BK20130067).
Authors’ Affiliations
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