- Research Article
- Open Access
Physical Layer Security Game: Interaction between Source, Eavesdropper, and Friendly Jammer
© Zhu Han et al. 2009
- Received: 31 December 2008
- Accepted: 9 November 2009
- Published: 6 January 2010
Physical layer security is an emerging security area that achieves perfect secrecy data transmission between intended network nodes, while malicious nodes that eavesdrop the communication obtain zero information. The so-called secrecy capacity can be improved using friendly jammers that introduce extra interference to the eavesdroppers. We investigate the interaction between the source that transmits the useful data and friendly jammers who assist the source by "masking" the eavesdropper. To obtain distributed solution, we introduce a game theoretic approach. The game is defined such that the source pays the jammers to interfere the eavesdropper, therefore, increasing the secrecy capacity. The friendly jammers charge the source with a certain price for the jamming, and there is a tradeoff for the price. If too low, the profit of the jammers is low; and if too high, the source would not buy the "service" (jamming power) or would buy it from other jammers. To analyze the game outcome, we investigate a Stackelburg type of game and construct a distributed algorithm. Our analysis and simulation results show the effectiveness of friendly jamming and the tradeoff for setting the price. The distributed game solution is shown to have similar performances to those of the centralized one.
- Channel Gain
- Malicious Node
- Centralize Scheme
- Stackelberg Equilibrium
- Secrecy Capacity
The future communication systems will be decentralized and adhoc, therefore allowing various types of network mobile terminals to join and leave. This aspect makes the whole system vulnerable and susceptible to attacks. Anyone within communication range can listen and possibly extract information. While these days we have numerous cryptographic methods with high level security, there is no system with perfect security on physical layer. Therefore, the physical layer security is regaining a new attention. The main goal of this paper is to design a decentralized system that will protect the broadcasted data and make it impossible for the eavesdropper to receive the packets even if it knows the encoding/decoding schemes used by the transmitter/receiver. In approaches where physical layer security is applied, the main objective is to maximize the rate of reliable information from the source to the intended destination, while all malicious nodes are kept as ignorant of that information as possible. This maximum reliable rate is known as secrecy capacity.
This line of work was pioneered by Wyner, who defined the wiretap channel and established the possibility to create almost perfect secure communication links without relying on private (secret) keys . 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 nonzero rate. The main idea proposed by him is to exploit the additive noise impairing the eavesdropper by using a stochastic encoder that maps each message to many codewords according to an appropriate probability distribution. With this scheme, a maximal equivocation (i.e., uncertainty) is induced at the eavesdropper. In other words, a maximal level of secrecy is obtained. By ensuring that the equivocation rate is arbitrarily close to the message rate, one can achieve perfect secrecy in the sense that the eavesdropper is now limited to learn almost nothing about the source-destination messages from its observations. Follow-up work by Leung-Yan-Cheong and Hellman characterized the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel . In their landmark paper, Csiszar and Korner generalized Wyner's approach by considering the transmission of confidential messages over broadcast channels . Recently, there have been considerable efforts on generalizing these studies to the wireless channel and multiuser scenarios (see [2, 4–11] and references therein). Jamming [12–14] has been studied for a long time to analyze the hostile behaviors of malicious nodes. Recently, jamming has been employed to physical layer security to reduce the eavesdropper's ability to decode the source's information . In other words, the jamming is friendly in this context. Moreover, the friendly helper can assist the secrecy by sending codewords, and bring further gains relative to unstructured Gaussian noise [15–17].
Game theory  is a formal framework with a set of mathematical tools to study some complex interactions among interdependent rational players. During the past decade, there has been a surge in research activities that employ game theory to model and analyze modern distributed communication systems. Most of these works [19–22] concentrate on the distributed resource allocation for wireless networks. As far as the authors' knowledge, the game theory has not yet been used in the physical layer security.
In this paper, we investigate the interaction between the source and its friendly jammers using game theory. Although the friendly jammers help the source by reducing the data rate that is "leaking" from the source to the malicious node, at the same time they also reduce the useful data rate from the source to the destination. Using well chosen amounts of power from the friendly jammers, the secrecy capacity can be maximized. In the game that we define here, the source pays the jammers to interfere the malicious eavesdropper, and therefore, to increase the secrecy capacity. The friendly jammers charge the source with a certain price for their service of jamming the eavesdropper. One could notice that there is a tradeoff for the proposed price. If the price of a certain jammer is too low, its profit is also low; if its price is too high, the source will buy from the other jammers. In modeling the outcome of the above games our analysis uses the Stackelberg type of game. Initially, the existence of equilibrium will be studied. Then, a distributed algorithm will be proposed and its convergence will be investigated. The outcome of the distributed algorithm will be compared to the centralized genie aided solution. Some implementation concerns are also discussed. From the simulation results, we can see the efficiency of friendly jamming and tradeoff for setting the price, the source prefers buying service from only one jammer, and the centralized scheme and the proposed game scheme have similar performance.
The rest of the paper is organized as follows. In Section 2, the system model of physical layer security with friendly jamming users is described. In Section 3, the game models are formulated, and the outcomes as well as properties of the game are analyzed. Simulation results are shown in Section 4, and conclusions are drawn in Section 5.
where . Both and are decreasing and convex functions of jamming power . However, might not be a monotonous and convex function.(Minus of two convex functions is not a convex function anymore.) This is because the jamming power might decrease faster than . As a result, might increase in some region of value . When further increases, both and approach zero.As a result, approaches zero. So, the questions are whether or not can be increased, and how to control the jamming power in a distributed manner so as to achieve the maximal . We will try to solve the problems in the following section using a game theoretical approach.
In this section, we study how to use game theory to analyze the physical layer security. First, we define the game between the source and friendly jammers. Next, we optimize the source and jammer sides, respectively. Then, we prove some properties of the proposed game. Furthermore, a comparison with the centralized scheme is constructed. Finally, we discuss some implementation concerns.
3.1. Game Definition
In the next two subsections, we analyze the optimal strategies for the source and friendly jammers to maximize their own utilities.
3.2. Source (Buyer) Side Analysis
Because of the complexity of the closed form solution of the quartic equation in (14), we also consider two special cases: low interference case and high interference case.
3.2.1. Interference at the Destination Is Much Smaller than the Noise
Finally, by comparing with the power under the boundary conditions ( , , and ), the optimal in the low SNR region can be obtained.
3.2.2. One Jammer with Interference That Is Much Higher than the Noise but Much Smaller than the Received Power at the Destination and the Malicious Node
From this equation we get the optimal closed-form solution , and similarly by comparing with the power under the boundary conditions ( , , and ), we can obtain the optimal solution for the this special case.
3.3. Friendly Jammer (Seller) Side Analysis
Notice that should be positive. Otherwise, the friendly jammer would not play.
In this subsection, we prove some properties of the proposed game. First, we prove that the power is monotonous function of the price under the two extreme cases. The properties can help for the proof of equilibrium existence in the later part of this subsection.
Under the two special cases, the optimal power consumption for friendly jammer is monotonous with its price , when the other friendly jammers prices are fixed. The proof is straightforward from (18) and (22).
We investigate the following analysis of the relation between the price and the power. We find out that the friendly jammer power bought from the source is convex in its own price under some conditions. To prove this we need to check whether the second derivative
The above equation is greater than zero when is small. This means when the interference is small and the price is small, the power is convex as a function of the price.
This means when the interference is severe, the power is a convex function of the price.
Next, we investigate the equilibrium of the proposed game. At the equilibrium, no user can improve its utility by changing its own strategy only. We first define the Stackelberg equilibrium as follows.
Finally, from the analysis in the previous two subsections, we can show the following property for the proposed game.
The pair of in (14) and in (26) is the Stackelberg equilibrium for the proposed game.
Notice that there might be multiple roots in (11), as a result, there might be multiple Stackelberg equilibria. In the simulation results shown in later section, we will show that the proposed scheme can still achieve the equilibria with better performances than those of the no-jammer case.
3.5. Distributed Algorithm and Convergence
where , and the iteration is from time to time . Next we show that the convergence of the proposed scheme by proving that the price update function in (32) is a standard function  defined as follows.
Monotonicity: if , then , or .
Scalability: for all ,?? .
In , it has been proved that the price will converge to the fixed point (i.e., the Stackelberg equilibrium in our case) from any feasible initial price vector. The positivity is very easy to prove. If the price goes up, the source would buy less from the friendly jammer. As a result, in (23) is negative, and we prove positivity .
which is monotonically increasing in and scalable.
For more general cases, the analysis is tractable. In the simulation section later, we employ the general simulation setups. The simulation results show that the proposed scheme can converge and outperform the no-jammer case.
3.6. Centralized Scheme
Using the KKT condition theorem , the final solution would be obtained by comparing the boundary conditions (i.e., , , and ).
Notice that our proposed algorithm is distributive, in the sense that only the pricing information needs to be exchanged. In the simulation results, we compare the proposed game theoretical approach with this centralized scheme.
Finally, from the simulation results in the next section, we see that the distributed solution and the centralized solution are asymptotically the same if is sufficiently large (the source cares more about the secrecy capacity than for the payment, i.e., the source is sufficiently rich).
3.7. Implementation Discussion
There are several implementation concerns for the proposed scheme. First, the channel information from the source to the malicious eavesdropper might not be known or accurately known. Under this condition, the secrecy capacity formula should be rewritten considering the uncertainty. If the direction of arrival is known, multiple antenna techniques can be employed such as in . Second, the proposed scheme needs to iteratively update the price and power information. A natural question arises if the distributed scheme has less signalling than the centralized scheme. The comparison is similar to distributed and centralized power control in the literature [23, 25]. Since the channel condition is continuously changing, the distributed solution only needs to update the difference of the parameters such as power and price to be adaptive, while the centralized scheme requires all channel information in each time period. As a result, the distributed solution has a clear advantage and dominates the current and future wireless network designs. For example, the power control for cellular networks, the open loop power control is done only once during the link initialization, while the close loop power control (distributed power allocation such as ) is performed 1500 times for UMTS and 800 times for CDMA2000. Finally, for the multisource multidestination case, there are two possible choices to solve the problem. First, we can use clustering method to divide the network into sub-networks, and then employ the single-source-destination pair and multiple-friendly-jammer solution proposed in this paper. If we believe that the jamming power can be useful for multiple eavesdroppers, some techniques such as double auction could be investigated. The detailed discussion is beyond the scope of this paper and would be considered in our future research.
The simulation is set up as follows. The source and friendly jammer have power of 0.02, the bandwidth is 1, the noise level is , the propagation loss factor is , and AWGN channel is assumed. The source, destination, and eavesdropper are located at the coordinates (0,0), (100,0), and (50,50), respectively. Here we select for the friendly jammer utility in (7).
Physical layer security is an emerging security technique that is an alternative for traditional cryptographic-based protocols to achieve perfect secrecy capacity as eavesdroppers obtain zero information. Jamming has been shown in the literature to effectively improve secrecy capacity. In this paper, we investigate the interaction between the source and friendly jammers using the game theory in order to have a distributed solution. The source pays the friendly jammers to interfere the malicious eavesdropper such that the secrecy capacity is increased, and therefore the security of the network. The friendly jammers charge the source with a price for the jamming. To analyze the game outcome, we investigate the Stackelburg game and construct a distributed algorithm. Some properties such as equilibrium and convergence are analyzed. From the simulation results, we conclude the following. First, there is a tradeoff for the price: If the price is too low, the profit is low; and if the price is too high, the source would not buy or buy from other jammers. Second, for the multiple jammer case, the source would buy service from only one jammer. Third, the centralized scheme and distributed scheme have similar performance, especially when is sufficiently large. Overall, the proposed game theoretical scheme can achieve a comparable performance with distributed implementation.
This work was supported by NSF CNS-0910461 and NSF CNS-0905556 and was supported by the Research Council of Norway through the project entitled "Mobile-to-Mobile Communication Systems (M2M)."
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