# Physical Layer Security Game: How to Date a Girl with Her - UniK

код для вставки1 Physical Layer Security Game: How to Date a Girl with Her Boyfriend on the Same Table Zhu Han1 , Ninoslav Marina2 , MВґerouane Debbah3 , and Are HjГёrungnes2 1 Electrical and Computer Engineering Department, University of Houston, USA. 2 UNIK - University Graduate Center, University of Oslo, Norway. 3 Вґ Alcatel-Lucent Chair on Flexible Radio, SUPELEC, Gif-sur-Yvette, France. AbstractвЂ” Physical layer security is an emerging security concept that achieves perfect secrecy data transmission between the intended network nodes, while the eavesdropping malicious nodes obtain zero information. The so-called secrecy capacity can be improved using friendly jammers that introduce extra interference to the eavesdropping malicious nodes while the interference to the intended destination is limited. In this paper, we investigate the interaction between the source that transmits the desired data and friendly jammers who assist the source by вЂњdisguisingвЂќ the eavesdropper. In order to obtain a distributed solution, we introduce a game theoretic approach. The game is deп¬Ѓned in such a way that the source pays the friendly jammers to interfere the eavesdropper, and, therefore, increasing its secrecy capacity. Friendly jammers charge the source with a certain price for this вЂњjamming serviseвЂќ. There is a tradeoп¬Ђ for the price: If the price is too low, the proп¬Ѓt of the jammers is low; and if the price is too high, the source would not buy the вЂњserviceвЂќ (jamming power) or would buy it from other jammers. To analyze the game outcome, we deп¬Ѓne and investigate a Stackelberg game and construct a distributed algorithm. Our analysis and simulation results show the effectiveness of friendly jamming and the tradeoп¬Ђ for setting the price. The fancy title comes from the fact that it is similar to a scenario where the main fellow character (the source) tries to send a dating message to a lady (the intended destination), whose вЂњpoorвЂќ boyfriend plays the role of the eavesdropper that may hear the message. Friends of the source, the so-called вЂњfriendly jammers,вЂќ try to distract the boyfriend, so that the dating message can be secretly transmitted. The game is deп¬Ѓned in order to derive what is the optimal price that the friends can charge for this вЂњfriendlyвЂќ action. KeywordsвЂ”Physical Layer Security, Secrecy Capacity, Jamming, Game Theory, and Stackelberg game. I. Introduction The wireless networks of the future will be ad-hoc and decentralized, allowing various types of mobile terminals dynamically to join and leave. This aspect makes the network vulnerable and susceptible to attacks. Any node within a communication range of the tranmitting node will be able to listen and possibly extract information. While current wireless networks proп¬Ѓt from the use of numerous cryptographic methods with high level of security, there is no system with perfect security on physical layer. Hence, the physical layer security is regaining a new attention. The main goal of this article is to design a decentralized sysThis work was supported by NSF CNS-0910461, by the Research Council of Norway through the project 176773/S10 entitled вЂќOptimized Heterogeneous Multiuser MIMO Networks OptiMOвЂќ and the project 183311/S10 entitled вЂќMobile-to-Mobile Communication Systems (M2M)вЂќ, as well as the AURORA project entitled вЂќCommunications under uncertain topologiesвЂќ. tem that will protect the broadcasted data and make it impossible for the eavesdropper to receive the packets even if it is aware of the encoding/decoding schemes used by the transmitters and the intended receivers in the network. 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 Aaron Wyner, who deп¬Ѓned the wiretap channel and established the possibility to create almost perfect secure communication links without relying on private (secret) 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. The main idea proposed by Wyner was 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-YanCheong and Hellman [2] characterized the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. In their landmark paper [3], CsiszВґ ar and KВЁ orner generalized WynerвЂ™s approach by considering the transmission of conп¬Ѓdential messages over broadcast channels. Recently, there have been considerable eп¬Ђorts on generalizing these studies to the wireless channel and multi-user scenarios (see [2, 4, 6вЂ“14] and the references therein). Jamming [15вЂ“17] 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 eavesdropperвЂ™s ability to decode the sourceвЂ™s information [18]. In other words, the jamming is friendly in this context. Game theory [19] oп¬Ђers a set of mathematical tools to study the complex interactions among interdependent rational players. For more than half a century, game theory has led to revolutionary changes in economics, and 2 has found important applications in politics, sociology, psychology, transportation etc. During the past decade, there has been a surge in research activities based on game theory to model and analyze modern communication systems. This is mainly due to: (1) The emergence of the Internet as a global platform for communication and computation, which has sparked the development of large-scale, distributed and heterogeneous communication systems; (2) The deregulation of the telecommunication industry and the dramatic improvement in computation power, which makes it possible for various network entities to make independent and self-operational decisions; (3) The need for robust designs against uncertainties of malicious nature, that can be modeled as games between regular and malicious network nodes (players). Most of these works [21вЂ“24] concentrate on the distributed resource allocation for wireless networks. To the best of the authorsвЂ™ 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 вЂњleakingвЂќ 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 deп¬Ѓne 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 jamming the eavesdropper. One could notice that there is a tradeoп¬Ђ for the proposed price: If the price of a certain jammer is too low, its proп¬Ѓt 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. As for the fancy title, the source is the main character, the destination is a lovely girl, and her poor boyfriend resembles the eavesdropper. A friend (jammer) or several friends distract the boyfriend, so that the dating message can be secretly transmitted from the main character (the source) to the girl (the destination). The game is deп¬Ѓned as follows: How much these friends should charge the main character for this вЂњfriendlyвЂќ action. The authors hope the fancy title can attract more attentions on this research track for distributed solutions of physical layer security problems. The rest of the paper is organized as follows: In Section II, the system model of physical layer security with friendly jammers is described. Section III, formulates the game models and analyzes the outcomes as well as properties of the game. Simulation results are shown in Section IV and conclusions are drawn in Section V. Fig. 1. System model for physical layer security game. II. System Model We consider a network with a source S, a destination D, a malicious eavesdropper node M, and N friendly jammer nodes denoted J1 , J2 , . . . ,JN as shown in Figure 1. The malicious node tries to eavesdrop the transmitted data coming from the source node. When the eavesdropperвЂ™s channel from the source S to the malicious node M is worse than the main source-destination channel, the source and destination can exchange perfectly secure messages at a non-zero rate. By transmitting a message at a rate higher than the receiving rate of the malicious node, the malicious node can learn almost nothing about the messages from its observations. The maximum rate of secrecy information from the source to its intended destination is called secrecy capacity. Suppose the source transmits with power P0 . The channel gains from the source to the destination and from the source to the malicious node are Gsd and Gsm , respectively. Each friendly jammer Ji , i = 1, . . . , N transmits with power Pi and the channel gains from Ji to the destination and the malicious node, are denoted by Gid and Gim , respectively. For convenience, we denote by J the set of indices {1, 2, . . . , N }. If the path loss model is used, the channel gain is п¬Ѓxed and it is proportional to the distance to the negative power of the path loss coeп¬ѓcient ОІ. The variance of the thermal noise for each channel is Пѓ 2 = N0 W and W is the bandwidth. The channel capacity for the source to the destination is C1 = W log2 1 + Пѓ2 P0 Gsd + iв€€J Pi Gid . (1) The channel capacity from the source to the malicious node is P0 Gsm . (2) C2 = W log2 1 + 2 Пѓ + iв€€J Pi Gim In order to ensure that the eavesdropping malicious node can obtain zero mutual information from the source, the source should send its data with the secrecy capacity as Cs = (C1 в€’ C2 )+ (3) where (В·)+ max(В·, 0). We can see that with the increase of the jamming power Pi , both C1 and C2 are reduced. The 3 questions are whether or not Cs can be increased, and how to control the jamming power in a distributed manner. We will try to solve the problems in the following section using a game theoretical approach. It is worth mentioning that the system model used in this paper is additive white Gaussian noise (AWGN) channel, which can provide some insights on the game and interactions between the source and the friendly jammers. For more sophisticated scenarios such as Rayleigh fading, it is usually assumed that the source-destination channels are known but only the channel statistics of source-jamming channel are known. The problem is how to write (3), while the rest of derivations of this paper can be performed in a similar way. The games for the source and friendly jammer are similar to the games between buyers and sellers. In the next two subsections, we analyze the optimal strategies for the source and friendly jammers to maximize their own utilities. The analysis is similar to a Stackelberg game [19]. B. Source (Buyer) Side Analysis The goal of the source as a buyer is to buy the optimal amount of power from the friendly jammers so as to improve its secrecy capacity. Let A P0 Gsd , Пѓ2 (8) B P0 Gsm , Пѓ2 (9) Gid , Пѓ2 (10) III. Game for Physical Layer Security In this section, we study how to use game theory to analyze the physical layer security. First, we deп¬Ѓne 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. Finally, we discuss some implementation concerns. A. Game Definition ui and Gim ,i в€€ J, Пѓ2 where from (4), we have вЋ› вЋ› vi The source can be modeled as a buyer who wants to optimize its secrecy capacity minus cost by modifying the вЂњserviceвЂќ (jamming power Pi ) from the friendly jammers, i.e., max Us = max(aCs в€’ M ), (4) Us M= pi P i , (5) iв€€J where pi is the price per unit power for the friendly jammer, Pi is the friendly jammerвЂ™s power, and J is the set of friendly jammers. From (3) and (4) we note that the source will not participate in the game if C1 < C2 , or in other words, the secrecy capacity is zero. For each jammer, Ui (pi , Pi (pi )) is the utility function of the price and power bought by the source. For the jammerвЂ™s (sellerвЂ™s) utility, in this paper we assume the following utility Ui = pi Pici , (6) where ci в‰Ґ 1 is a constant to balance from the payment pi Pi from the source and the transmission cost Pi 1 . Notice that Pi is also a function of the vector of prices (p1 , . . . pN ) since the power that the source will buy also depends on the price that the friendly jammers ask. Hence, for each friendly jammer, the optimization problem is Friendly JammerвЂ™s Game: max Ui . pi 1 From (7) the simulation results, we realize that ci could not less than 1. Otherwise, the jammer would set arbitrary price, the jamming power is almost zero, and the system is balanced in trivial solution. в€’ вЋњ log2 вЋњ вЋќ1 + вЋћ A u j Pj 1+ вЋџ вЋџ вЋ jв€€J вЋ› s.t. 0 в‰¤ Pi в‰¤ Pmax , where a is the gain per unit capacity, Pmax is the maximal power that a jammer can provide, and M is the cost to pay for the other friendly jamming nodes. Here вЋњ вЋњ вЋњ = aW вЋњ вЋќlog2 вЋќ1 + (11) вЋћвЋћ+ B vj P j 1+ вЋџвЋџ вЋџвЋџ в€’ вЋ вЋ pj Pj .(12) jв€€J jв€€J For the source (buyer) size, we п¬Ѓrst analyze the case where C1 > C2 , i.e., the secrecy capacity is not zero before the friendly jammersвЂ™ participation. By diп¬Ђerentiating (12) with respect to Pi , we get в€‚Us =в€’ в€‚Pi (1 + A + + (1 + B + aW Aui / ln 2 jв€€J uj Pj )(1 + aW Bvi / ln 2 jв€€J vj Pj )(1 + jв€€J jв€€J vj P j ) u j Pj ) в€’ pi = 0. (13) Rearranging the above equation, we have a fourth order polynomial equation: Pi4 + Fi,3 Pi3 + Fi,2 (pi )Pi2 + Fi,1 (pi )Pi + Fi,0 (pi ) = 0, (14) where (2 + 2О±i + A)2 + (2 + 2ОІi + B)2 , (2 + 2О±i + A)(2 + 2ОІi + B) Fi,2 (pi ) = u i vi B Li Ki aW A + + 2 в€’ в€’ , 2 vi ui p i u i vi vi ui Li Ci + Ki Di aW (ADi в€’ BCi ) Fi,1 (pi ) = + , 2 2 u i vi pi u2i vi2 Ki Li aW (Aui Li в€’ Bvi Ki ) Fi,0 (pi ) = + , u2i vi2 pi u2i vi2 Fi,3 = (15) (16) (17) (18) 4 and О±i = Gjd Pj , Solving the above equation we obtain a closed-form solution (19) j=i ОІi = Gjm Pj , Piв€— = в€’ (20) (2 + 2ОІi + B)2 (1 + ОІi )(1 + B + ОІi ) aW B в€’ + 4vi2 vi2 pi vi ln 2 j=i + Ki Li Ci Di = (1 + О±i )(1 + О±i + A), = (1 + ОІi )(1 + ОІi + B), = ui (2 + 2О±i + A), = vi (2 + 2ОІi + B). (21) (22) (23) (24) = qi + wi + zi , pi (30) where The solutions of the quartic equation (14) can be expressed in closed form [20], but this is not the primary goal here. It is important that the solution we are interested in is given by Piв€— = Piв€— (pi , A, B, {uj }, {vj }, {Pj }j=i ) , 2 + 2ОІi + B 2vi 2 + 2ОІi + B (31) 2vi (2 + 2ОІi + B)2 (1 + ОІi )(1 + B + ОІi ) в€’ (32) 2 4vi vi2 aW B . (33) vi ln 2 qi = в€’ wi = zi = (25) which is a function of the friendly jammerвЂ™s price pi and the other system parameters. Note that 0 в‰¤ Pi в‰¤ Pmax . Since Pi satisп¬Ѓes the polynomial function, we can have the optimal strategy as Finally, by comparing Piв€— with the power under the boundary conditions (Pi = 0, Pi = Pmax and Cs = 0), the optimal Piв€— in the low SNR region can be obtained. Piв€— = min [max(Pi , 0), Pmax ] . B.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 (26) Because of the complexity of the closed form solution of (26), we also consider two special cases: Lower interference case and high interference case in the following two subsubsections. In this special case, the interference from one jammer is much higher than the additive noise but much smaller than the power of the received signal at the destination and the malicious node. In other words, that means 1 << u1 P1 << A and 1 << v1 P1 << B. Therefore, the utility function of the source can be approximated as B.1 Interference at the Destination is much Smaller than the Additive Noise Remember the deп¬Ѓnitions (8), (9), (10), and (11). Imagine a situation in which all jammers are close to the malicious node and far from the destination node. In that case the interference from the jammers to the destination is very small in comparison to the additive noise. Therefore, by omitting interfering terms, we get the following approximation Us в‰€ aW в€’ log2 (1 + A) в€’ log2 1+ jв€€J pj P j . vj P j Then, by diп¬Ђerentiating with respect to the power Pi that the source is willing to buy and setting the result to zero, we have в€‚Us aW Bvi / ln 2 = в€‚Pi (1 + B + vj Pj )(1 + jв€€J vj P j ) в€’ pi = 0. (28) в€‚Us aW B aW A + в€’ p1 = 0. =в€’ в€‚Pi u1 P12 ln 2 v1 P12 ln 2 2 + 2ОІi + B (1 + ОІi )(1 + B + ОІi ) Pi + vi vi2 aW B = 0. в€’ pi vi ln 2 (29) в€’ p1 P1 (34) (35) Hence, P1в€— = aW p1 ln 2 B A в€’ v1 u1 = D1 , p1 (36) where D1 = jв€€J Rearranging we get Pi2 + log2 1 + where the second approximation comes from the Taylor series expansion log2 (1+x) в‰€ x/ ln 2 when x is small enough. In order to п¬Ѓnd the optimal power to buy, similarly we calculate (27) jв€€J A B в€’ log 1 + u 1 P1 v1 P 1 aW B aW A в€’ в€’ p1 P 1 , u1 P1 ln 2 v1 P1 ln 2 в‰€ aW в‰€ + B 1+ Us aW ln 2 B A в€’ v1 u1 . (37) From (36), we get the optimal closed-form solution P1в€— , and, similarly, by comparing P1в€— with the power under the boundary conditions (P1 = 0, P1 = Pmax , and Cs = 0), we can obtain the optimal solution of the source for this special case. 5 C. Friendly Jammer (Seller) Side Analysis From (25), we can see that the power that a source would buy is related to the prices that the friendly jammers select. In this subsection, we study how the friendly jammers can set the optimal price to maximize their utility. By diп¬Ђerentiating the friendly jammerвЂ™s utility in (6) with respect to Pi and setting it to zero, we have в€‚Ui в€‚P в€— = (Piв€— )ci + pi ci (Piв€— )ci в€’1 i = 0. в€‚pi в€‚pi (38) This is equivalent to (Piв€— )ci в€’1 Piв€— + pi ci В· в€‚Piв€— в€‚pi = 0. (39) This equation is satisп¬Ѓed either if Piв€— = 0 (the source does not buy anything from the friendly jammer) or if Piв€— + pi ci В· в€‚Piв€— = 0. в€‚pi From the closed form solution of be a function given as pв€—i = Piв€— , the solution of (40) pв€—i will (41) pв€—i should be positive. Otherwise, the friendly Notice that jammer i would not play. D. Properties In this subsection, we prove some properties of the proposed game. First, we prove that the power is a monotonous function of the price under the two extreme cases (in Section III-B.1 and III-B.2). The properties can help for the proof of equilibrium existence in the later part of this subsection. Property 1: Under the two special cases, the optimal power consumption Piв€— of friendly jammer Ji is monotonous in its price pi , when the other friendly jammers prices are п¬Ѓxed. The proof is straightforward from (30) and (36). We investigate the following analysis of the relation between the price and power. We п¬Ѓnd out that the friendly jammer power Pi bought from the source is convex in its own price pi under some conditions. To prove this we need to check whether the second derivative satisп¬Ѓes в€‚ 2 Pi /в€‚p2i < 0. In the п¬Ѓrst special case, in which the interference is small, from (30) we have the п¬Ѓrst order derivative as i zi wi + 1 в€‚P1в€— =в€’ в€‚p1 2 в€’3/2 D1 p1 , (44) > 0. (45) and the second order derivative as в€‚ 2 P1в€— 3 = в€‚p21 4 в€’5/2 D1 p1 This means when the interference is severe, the power P1в€— is a convex function of the price p1 . Next, we investigate the equilibrium of the proposed game. In other words, no user can improve its utility by changing its own strategy only. We п¬Ѓrst deп¬Ѓne the Stackelberg equilibrium as follows: Definition 1: PiSE and pSE are the Stackelberg equilibi rium of the proposed game, if when pi is п¬Ѓxed, Us ({PiSE }) = sup Pmax в‰ҐPiSE в‰Ґ0,в€Ђiв€€J Us ({Pi }), в€Ђi в€€ J (46) and when Pi is п¬Ѓxed, pв€—i (Пѓ 2 , Gsd , Gsm , {Gid }, {Gim }). в€‚Piв€— =в€’ в€‚pi 2p2 In the second special case, in which the interference is severe, we have the п¬Ѓrst order derivative zi pi and the second order derivative as вЋ› вЋћ 2 в€— в€‚ Pi 1 zi вЋќ1 в€’ вЋ . = 1/2 pi wi в€‚p2i zi 3 4 1 + pi wi + pi zi (42) (43) The above equation is greater than zero. This means when the interference is small and the price is small, the power is convex as a function of the price. Ui (pSE i ) = sup Ui (pi ), в€Ђi в€€ J . (47) pi Finally, from the analysis in the previous two subsections, we conclude the following property for the proposed game. в€— N Property 2: The pair of {Piв€— }N i=1 in (26) and {pi }i=1 in (41) is the Stackelberg equilibrium for the proposed game. E. Distributed Algorithm and Convergence In this subsection, we study how the distributed game can converge to the Stackelberg equilibrium deп¬Ѓned in the above subsection. After rearranging (38), we have pi = Ii (p) в€’ (Piв€— ) ci в€‚Piв€— в€‚pi , (48) where p is the price vector p [p1 , . . . , pN ]T and Ii (p) is the price update function. Notice that the optimal power Piв€— is a function of the price vector p. The information for the update can be obtained from the source node. This is similar to the distributed power control [27]. The update of the friendly jammersвЂ™ prices can be written in a vector form as Distributed Algorithm: p(t + 1) = I(p(t)), (49) where I = [I1 , . . . , IN ]T , and the iteration is from time t to time t + 1. Next, we show that the convergence of the proposed scheme using the update in (49) by proving that the price update function in (49) is a standard function [25] deп¬Ѓned as Definition 2: A function I(p) is standard, if for all p в‰Ґ 0, the following properties are satisп¬Ѓed 1. Positivity: I(p) > 0, 2. Monotonicity: if p в‰Ґ p , then I(p) в‰Ґ I(p ), or I(p) в‰¤ I(p ), 3. Scalability: For all О· > 1, О·I(p) в‰Ґ I(О·p). 6 Secrecy Capacity as a Function of Jammer Power 1 Jammer Location (50,75) Jammer Location (10,75) 0.9 Secrecy Capacity C s 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 Jamming Power 0.015 0.02 Fig. 2. Secrecy capacity versus the jamming power. In [25], it has been proved that the price will converge to the п¬Ѓxed 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 pi increases, the source would buy less from the ith friendly в€‚P в€— jammer. As a result, в€‚pii in (38) is negative, and we prove positivity pi = Ii (p) > 0. For monotonicity and scalability, we can only show these properties for the two special cases. For the low interference case, from (30) it follows Ii (p) = в€’ = 2 (Piв€— ) ci (50) в€‚Piв€— в€‚pi wi p2i + zi pi (qi pi + ci zi wi p2i + zi pi ) , which is monotonically increasing in pi . For scalability, we have Ii (О·p) = О·Ii (p) wi p2i + zi pi /О·(qi pi + wi p2i + zi pi (qi pi + wi p2i + zi pi /О·) wi p2i + zi pi ) IV. Simulation Results < 1, (51) since О· > 1. For the large interference case, from (36), we have Ii (p) = в€’ (Piв€— ) в€‚P в€— ci в€‚pii = 2pi , ci 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. While the closed-form solution might be hard to п¬Ѓnd, it is possible to get some insight from the numerical results we obtained. Moreover, some side information can also be helpful. For example, if the direction of arrival is known, multiple antenna techniques can be employed such as in [12]. Secondly, the proposed scheme need iteratively updating the price and power information. A natural question arises that if the distributed scheme has less signalling than the centralized scheme. The comparison is similar to distributed and centralized power control in the literature [25,27]. Since the channel conditions are continuously changing, the distributed solution only needs to update the diп¬Ђerence 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 design. 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 [25]) is performed 1500 times for UMTS and 800 times for CDMA2000. An interesting scenario would be the multi-sourcedestination-pair multi-eavesdropper case. We see two possible choices to solve this problem. One choice is to use a 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. Another choice is to consider that the jamming power generates interference for multiple eavesdroppers. There, some techniques such as double auction can be investigated. The detailed discussion is beyond the scope of this paper and could be considered in our future research. (52) which is monotonically increasing in pi and scalable. Based on the above analysis, we can conclude that under the two special cases, the game will converge to the Stackelberg equilibrium from an arbitrary initial value. From the observation in the simulations, the price and power indeed converge. F. Implementation Discussion There are several implementation concerns for the proposed scheme. Firstly, the channel information from the The simulation parameters are set up as follows: The bandwidth is W = 1, the noise variance is Пѓ 2 = 10в€’8 , the path loss coeп¬ѓcient is ОІ = 3, AWGN channel is assumed. The source, destination, and eavesdropper are located at the coordinate (0,0), (100,0), and (50,50), respectively. Here we select a = 2 for the friendly jammer utility in (6). For the single friendly jammer case N = 1, we show the simulation with the friendly jammer located at two diп¬Ђerent locations: (50,75) and (10,75). In Figure 2, we show the secrecy capacity as a function of jamming power. We can see that with the increase of the jamming power, the secrecy capacity п¬Ѓrst increases and then decreases. This is because the jamming power has diп¬Ђerent eп¬Ђects on C1 and C2 . So there is an optimal point for the jamming power. Also the optimal point depends on the location of the friendly jammer, and the friendly jammer close to the eavesdropper is more eп¬Ђective to improve the secrecy capacity. Moreover, notice that the curve is not convex and 7 Source(0,0), Dest.(100,0), Malic.Node (50,90), User 1(50,50), User 2(50,75) How much Power Bought vs. Jammer Price 0.016 Jammer Location (50,75) Jammer Location (10,75) 0.014 в€’7 x 10 0.01 3 1 0.008 Source U Amount of Power Bought 0.012 0.006 2 150 1 0.004 100 0300 0.002 0 0 50 100 Jammer Price 150 250 200 User 1 Price p 200 150 50 100 50 2 User 2 Price p1 0 Fig. 5. U1 versus prices of the two users. Fig. 3. How much power the source would buy versus the price. Source(0,0), Dest.(100,0), Malic.Node (50,90), User 1(50,50), User 2(50,75) Source(0,0), Dest.(100,0), Malic.Node (50,90), User 1(50,50), User 2(50,75) в€’6 4 0.8 3 2 1 Source U Source Us x 10 0.6 2 1 0.4 0 300 0.2 0 200 100 User 1 Price p 0 150 User 1 Price p 100 50 200 2 100 50 100 2 150 300 User 2 Price p 1 0 User 2 Price p 1 Fig. 4. Us versus prices of the two users. Fig. 6. U2 versus prices of the two users. not concave. Next we п¬Ѓx the source and friendly jammer powers to Ps = Pi = 0.02. In Figure 3, we show how much power the source buys from the jammer as a function of the requested price. We can see that the power is reduced if the price goes high. At some point, the source would stop buying the power. So there is a tradeoп¬Ђ for setting the price, i.e., if the price too high, the source would buy less power or even would buy nothing. For the two-jammer case, N = 2, we set up the following locations of the nodes. Malicious node is located at (50,90), friendly jammer 1 is located at (50,50), and friendly jammer 2 is located at (50,75). In Figure 4, Figure 5, and Figure 6, we show the sourceвЂ™s utility Us , the utility of jammer 1, U1 , and the utility of jammer 2, U2 , as function of both usersвЂ™ price, respectively. We can see that the source would buy service from only one of the friendly jammers. If the friendly jammer asks too low price, the jammerвЂ™s utility is very low. On the other hand, if the jammer asks too high price, it risks the situation in which the source would buy the service from the other friendly jammer. There is an optimal price for each friendly jammer to ask, and the source would always select the one that can provide the best performance improvement. V. Conclusions 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 eп¬Ђectively improve secrecy capacity. In this paper, we investigate the interaction between the source and friendly jammers using game theory for having a distributed solution. The source pays the friendly jammers to interfere the malicious eavesdropper so as to increase the secrecy capacity. The friendly jammers charge the source with a price for the jamming. To analyze the game outcome, we investigate the Stackelberg game and construct the distributed algorithm. Some properties such as equilibrium and convergence are analyzed. From the simulation results, we conclude the following: First, there 8 is a tradeoп¬Ђ for the price: if the price is too low, the proп¬Ѓt is low; if the price is too high, the source would not buy or buy from the other jammers. Second, for the multiple jammer case, the source would buy service from only one jammer. Overall, the proposed game theoretical scheme can achieve a good performance with distributed implementation. References [1] A. D. Wyner, вЂњThe wire-tap channel,вЂќ Bell System Technical Journal, vol. 54, no. 8, pp. 1355 вЂ“ 1387, 1975. [2] S. K. Leung-Yan-Cheong and M. E. Hellman, вЂњThe Gaussian wiretap channel,вЂќ IEEE Transactions on Information Theory, vol. 24, no. 4, pp. 451 вЂ“ 456, July 1978. [3] I. CsiszВґ ar and J. 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