Game theoretic approach of eavesdropping attack in millimeter-wave-based WPANs with directional antennas

Abstract

In this paper, we propose a game theoretic framework to analyze a passive eavesdropping attack in millimeter-wave-based wireless personal area networks. As the interaction between defenders and attackers in network security issues can be considered as a game, devices can be considered as players. Usually a player has insufficient information of other players such as strategies and payoffs especially in static situation. We assume that each player selects its strategy based on its belief about the information on opponents and the belief is known as a prior probability. The Bayesian game model is a relevant framework for this situation. It is assumed that each device is equipped with a directional antenna and knows the channel state. In the analysis, the exposure region (ER) that is determined by the antennas directions of players is considered and the ratio of ER for the total area of a network is calculated by using the probability density function of distance between players. The optimal strategies of two players, a transmitter and an eavesdropper, are derived in terms of Bayesian Nash equilibriums and payoffs for the players are computed for pure and mixed strategies in static situation. The analysis shows how the effects of using a directional antenna and the prior probability of transmitters belief are involved in the Nash equilibriums. Numerical results show the equilibriums and two players payoffs. Those values depend on several parameters such as beamwidth and prior probability of belief as well as its opponents strategies. It also shows the effect of directional antennas in the eavesdropping attack; the use of directional antenna seems to increase the transmitters payoff, while it seems to decrease the eavesdroppers payoff. The obtained results will provide the criteria for selecting appropriate strategy to transmitters when an eavesdropper exists in the network.

Appendices

Appendix 1: Derivation of $P_{\mathrm{ER}}$

Let X be a random variable of the distance between two DEVs in the $L \times L$ room. Then, it is known that the probability density function (PDF) of X, f(x),  is given as follows [45]:

$$\begin{aligned} f(x) = \left\{ {\begin{array}{ll} \frac{{2x}}{{L^2 }}\left( {\frac{{x^2 }}{{L^2 }} - 4\frac{x}{L} + \pi } \right) , &{}{\hbox { if }}0 \le x \le L \\ \frac{{2x}}{{L^2 }}\left\{ {4\sqrt{\frac{{x^2 }}{{L^2 }} - 1} - \left( {\frac{{x^2 }}{{L^2 }} + 2 - \pi } \right) - 4\tan ^{ - 1} \sqrt{\frac{{x^2 }}{{L^2 }} - 1} } \right\} ,&{}{\hbox { if }}L < x \le \sqrt{2} L. \\ \end{array}} \right. \end{aligned}$$

(16)

The value of $P_{\mathrm{ER}}$ depends on antenna equipment of a DEV. We differentiate $P_{\mathrm{ER}}$ either $P_\mathrm{{{ER,d}}}$ or $P_\mathrm{{{ER,o}}},$ when the DEV is equipped with a directional and an omni-directional antenna, respectively. Then, they are given by

$$\begin{aligned} P_{\mathrm{{ER,d}} }= & {} \left[ {\left( {\frac{\theta }{{2\pi }}} \right) ^2 \int _0^{r_{1} } {f(x)dx} + \left( {\frac{\theta }{{2\pi }}} \right) \left( {1 - \frac{\theta }{{2\pi }}} \right) \sum \limits _{i = 2}^3 {\int _0^{r_{i} } {f(x)dx} } + \left( {1 - \frac{\theta }{{2\pi }}} \right) ^2 \int _0^{r_{4} } {f(x)dx} } \right] \nonumber \\&- \,\left[ \begin{array}{l} \left( {\frac{\theta }{{2\pi }}} \right) ^3 \left( {1 - \frac{\theta }{{2\pi }}} \right) \left\{ {\prod \limits _{i = 1,2} {\int _0^{r_{i} } {f(x)dx} } + \prod \limits _{i = 1,3} {\int _0^{r_{i} } {f(x)dx} } } \right\} \\ + \left( {\frac{\theta }{{2\pi }}} \right) ^2 \left( {1 - \frac{\theta }{{2\pi }}} \right) ^2 \left\{ {\prod \limits _{i = 1,4} {\int _0^{r_{i} } {f(x)dx} } + \prod \limits _{i = 2,3} {\int _0^{r_{i} } {f(x)dx} } } \right\} \\ + \left( {\frac{\theta }{{2\pi }}} \right) \left( {1 - \frac{\theta }{{2\pi }}} \right) ^3 \left\{ {\prod \limits _{i = 2,4} {\int _0^{r_{i} } {f(x)dx} } + \prod \limits _{i = 3,4} {\int _0^{r_{i} } {f(x)dx} } } \right\} \\ \end{array} \right] \nonumber \\&+ \,\left[ \begin{array}{l} \left( {\frac{\theta }{{2\pi }}} \right) ^4 \left( {1 - \frac{\theta }{{2\pi }}} \right) ^2 \prod \limits _{i = 1,2,3} {\int _0^{r_{i} } {f(x)dx} } \\ + \left( {\frac{\theta }{{2\pi }}} \right) ^3 \left( {1 - \frac{\theta }{{2\pi }}} \right) ^3 \left\{ {\prod \limits _{i = 1,2,4} {\int _0^{r_{i} } {f(x)dx} + \prod \limits _{i = 1,3,4} {\int _0^{r_{i} } {f(x)dx} } } } \right\} \\ + \left( {\frac{\theta }{{2\pi }}} \right) ^2 \left( {1 - \frac{\theta }{{2\pi }}} \right) ^4 \prod \limits _{i = 2,3,4} {\int _0^{r_{i} } {f(x)dx} } \\ \end{array} \right] \nonumber \\&- \,\left( {\frac{\theta }{{2\pi }}} \right) ^4 \left( {1 - \frac{\theta }{{2\pi }}} \right) ^4 \prod \limits _{i = 1}^4 {\int _0^{r_{i} } {f(x)dx} } \end{aligned}$$

(17)

$$\begin{aligned} {\hbox { and }} P_\mathrm{{ER,o}}=\int _0^\mathrm{{min}(r_\mathrm{{ER,o}},\sqrt{2}L)}f(x)dx, \end{aligned}$$

(18)

respectively, where $r_i$ in Eq. (17) is ER radius given by $r_i=\mathrm{{min}}(r_{{\mathrm{ER}},i},\sqrt{2}L).$ The ER of a DEV depends on the antenna directions of the DEV and other DEV that emits the signal and there are four different ER radii $\{ r_{{\mathrm{ER}},i}\}_{i=1}^{4}$ which are given by

$$\begin{aligned}&r_\mathrm{{{ER,1}}}=(\kappa G_\mathrm{{{TM}}}G_\mathrm{{{RM}}}P_\mathrm{{{T}}}/P_\mathrm{{{R}}})^{1/n}, r_\mathrm{{{ER,2}}}=(\kappa G_\mathrm{{{TS}}}G_\mathrm{{{RM}}}P_\mathrm{{{T}}}/P_\mathrm{{{R}}})^{1/n}, \nonumber \\&r_\mathrm{{{ER,3}}}=(\kappa G_\mathrm{{{TM}}}G_\mathrm{{{RS}}}P_\mathrm{{{T}}}/P_\mathrm{{{R}}})^{1/n},{\hbox { and }} r_\mathrm{{{ER,4}}}=(\kappa G_\mathrm{{{TS}}}G_\mathrm{{{RS}}}P_\mathrm{{{T}}}/P_\mathrm{{{R}}})^{1/n}, \end{aligned}$$

(19)

where $G_\mathrm{{{TM}}}(G_\mathrm{{{TS}}})$ and $G_\mathrm{{{RM}}}(G_\mathrm{{{RS}}})$ are the antenna gains of the main lobe (side lobe) of a transmitter and a receiver, respectively. $P_\mathrm{{{R}}}$ is the receiver sensitivity corresponding to the base data rate of 25.8 Mbps, which is $-70$ dB in this case. Equation (18) is the radius of ER when an omni-directional antenna is equipped on a DEV, which is given by

$$\begin{aligned} r_\mathrm{{{ER,o}}}=(\kappa G_\mathrm{{{0}}}^{2}P_\mathrm{{{T}}}/P_\mathrm{{{R}}})^{1/n}, \end{aligned}$$

(20)

where $G_\mathrm{{{0}}}$ is the main lobe antenna gain obtained for $\theta =2 \pi .$ During the derivation of Eqs. (19)–(20), the path loss model of IEEE 802.15.3c were used. For the detailed derivation of the ER radii, see [39].

Appendix 2: Proof of Theorem 2

• (a) For the strategies, the expected payoffs of tx and ev are given as follows:

  $$\begin{aligned} \left\{ \begin{array}{ll} E_{\mathrm{e}}(s_{\mathrm{e,1}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) = P_{\mathrm{ER}}p_{\mathrm{e}}( w_{\mathrm{e}} p_{\mathrm{ch}} p_{\mathrm{tx}}- c_{\mathrm{e}}), \\ E_{\mathrm{e}}(s_\mathrm{{{e,2}}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) = w_{\mathrm{e}}p_{\mathrm{ch}}p_{\mathrm{tx}} P_{\mathrm{ER}} p_{\mathrm{e}}- c_{\mathrm{e}}, \\ E_{\mathrm{e}}(s_{\mathrm{e,1}}^{\mathrm{m}},s_\mathrm{{{t,2}}}^{\mathrm{m}}) = -c_{\mathrm{e}}(1-p_{\mathrm{ch}} p_{\mathrm{tx}}) P_{\mathrm{ER}} p_{\mathrm{e}}, \\ E_{\mathrm{e}}(s_\mathrm{{{e,2}}}^{\mathrm{m}},s_\mathrm{{{t,2}}}^{\mathrm{m}}) = -c_{\mathrm{e}}, {\hbox { }} E_{\mathrm{e}}(s_{\mathrm{e,3}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) = E_{\mathrm{e}}(s_{\mathrm{e,3}}^{\mathrm{m}},s_\mathrm{{{t,2}}}^{\mathrm{m}}) = 0, \end{array} \right. \end{aligned}$$

  and

  $$\begin{aligned} \left\{ \begin{array}{l l} E_{\mathrm{t}}(s_{\mathrm{e,1}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) = p_{\mathrm{ch}}(w_{\mathrm{t}}-c_{\mathrm{t}}-2 w_{\mathrm{t}}p_{\mathrm{on}}P_{\mathrm{ER}} p_{\mathrm{e}}), \\ E_{\mathrm{t}}(s_\mathrm{{{e,2}}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) = p_{\mathrm{ch}}(w_{\mathrm{t}}- c_{\mathrm{t}}-2 w_{\mathrm{t}}P_{\mathrm{ER}} p_{\mathrm{e}}), \\ E_{\mathrm{t}}(s_{\mathrm{e,3}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) = p_{\mathrm{ch}} (w_{\mathrm{t}}- c_{\mathrm{t}}), \\ E_{\mathrm{t}}(s_{\mathrm{e,1}}^{\mathrm{m}},s_\mathrm{{{t,2}}}^{\mathrm{m}}) = E_{\mathrm{t}}(s_\mathrm{{{e,2}}}^{\mathrm{m}},s_\mathrm{{{t,2}}}^{\mathrm{m}}) = E_{\mathrm{t}}(s_{\mathrm{e,3}}^{\mathrm{m}},s_\mathrm{{{t,2}}}^{\mathrm{m}})=0, \end{array} \right. \end{aligned}$$

  (21)

  respectively. To find BNEs, we consider the following cases:

  • (i) For tx,  if tx selects $s_{\mathrm{t,1}}^{\mathrm{m}},$ev must select the strategy which gives maximum the expected payoff. $E_{\mathrm{e}}(s_{\mathrm{e,1}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) \ge E_{\mathrm{e}}(s_\mathrm{{{e,2}}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}})$ holds always, while

    $$\begin{aligned} \left\{ \begin{array}{l l} E_{\mathrm{e}}(s_{\mathrm{e,1}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}) \ge E_{\mathrm{e}}(s_{\mathrm{e,3}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}), {\hbox { if }} w_{\mathrm{e}} \ge c_{\mathrm{e}}/p_{\mathrm{ch}} p_{\mathrm{tx}} \\ E_{\mathrm{e}}(s_{\mathrm{e,1}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}})< E_{\mathrm{e}}(s_{\mathrm{e,3}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}}), {\hbox { if }} w_{\mathrm{e}} < c_{\mathrm{e}}/p_{\mathrm{ch}} p_{\mathrm{tx}} \end{array} \right. \end{aligned}$$

    (22)

    holds. Therefore, ev selects $s_{\mathrm{e,1}}^{\mathrm{m}}$ if $w_{\mathrm{e}} \ge c_{\mathrm{e}}/p_{\mathrm{ch}} p_{\mathrm{tx}},$ while it selects $s_{\mathrm{e,3}}^{\mathrm{m}}$ if $w_{\mathrm{e}} < c_{\mathrm{e}}/p_{\mathrm{ch}} p_{\mathrm{tx}}.$ If tx selects $s_\mathrm{{{t,2}}}^{\mathrm{p}},$ev must select $s_{\mathrm{e,3}}^{\mathrm{p}}.$ Therefore, if $w_{\mathrm{e}} < c_{\mathrm{e}}/p_{\mathrm{ch}} p_{\mathrm{tx}},$$s_{\mathrm{e,3}}^{\mathrm{m}}$ is the dominant strategy, while if $w_{\mathrm{e}} \ge c_{\mathrm{e}}/p_{\mathrm{ch}} p_{\mathrm{tx}},$ there is no dominant strategy for ev.

  • (ii) For ev,  if ev selects $s_{\mathrm{e,1}}^{\mathrm{m}},$tx selects $s_{\mathrm{t,1}}^{\mathrm{m}}$ if $w_{\mathrm{t}} \ge c_{\mathrm{t}}/(1-2p_{\mathrm{on}}P_{\mathrm{ER}} p_{\mathrm{e}}).$ Otherwise, it selects $s_\mathrm{{{t,2}}}^{\mathrm{m}}.$ If ev selects $s_\mathrm{{{e,2}}}^{\mathrm{m}},$tx selects $s_\mathrm{{{t,2}}}^{\mathrm{m}}$ since $w_{\mathrm{t}} < c_{\mathrm{t}}/(1-2P_{\mathrm{ER}} p_{\mathrm{e}}).$ Finally, if ev selects $s_{\mathrm{e,3}}^{\mathrm{m}},$tx selects $s_{\mathrm{t,1}}^{\mathrm{m}}$ always since $w_{\mathrm{t}} > c_{\mathrm{t}}.$ Therefore, there is no dominant strategy for tx if $c_{\mathrm{t}}/(1-2p_{\mathrm{on}}P_{\mathrm{ER}}p_{\mathrm{e}}) \le w_{\mathrm{t}} < c_{\mathrm{t}}/(1-2P_{\mathrm{ER}} p_{\mathrm{e}}).$$s_{\mathrm{t,1}}^{\mathrm{p}}$ is the dominant strategy if $w_{\mathrm{t}} \ge c_{\mathrm{t}}/(1-2P_{\mathrm{ER}} p_{\mathrm{e}}).$

    By (i)-(ii), we conclude that $[s_{\mathrm{e,1}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}} ]$ is a BNE if $c_{\mathrm{t}}/(1-2p_{\mathrm{on}}P_{\mathrm{ER}}p_{\mathrm{e}}) \le w_{\mathrm{t}} < c_{\mathrm{t}}/(1-2P_{\mathrm{ER}} p_{\mathrm{e}})$ and $w_{\mathrm{e}} \ge c_{\mathrm{e}}/p_{\mathrm{ch}}p_{\mathrm{tx}},$$[s_{\mathrm{e,3}}^{\mathrm{m}},s_{\mathrm{t,1}}^{\mathrm{m}} ]$ is a BNE if $w_{\mathrm{t}} < c_{\mathrm{t}}/(1-2P_{\mathrm{ER}} p_{\mathrm{e}})$ and $w_{\mathrm{e}} < c_{\mathrm{e}}/p_{\mathrm{ch}}p_{\mathrm{tx}}.$

  • (b) Therefore, no BNE exists if $w_{\mathrm{t}} < c_{\mathrm{t}}/(1-2p_{\mathrm{on}}P_{\mathrm{ER}}p_{\mathrm{e}})$ and $w_{\mathrm{e}} \ge c_{\mathrm{e}}/p_{\mathrm{ch}}p_{\mathrm{tx}}.$

Appendix 3: Proof of Theorem 3

Since $E_{\mathrm{t}}$ and $E_{\mathrm{e}}$ are computed as

$$\begin{aligned} E_{\mathrm{t}}= \sum _{a \in A} E_{\mathrm{t}}(\mathrm{payoff}|a)P(a)=p_{\mathrm{tx}}p_{\mathrm{ch}}\{(w_{\mathrm{t}}-c_{\mathrm{t}})-2w_{\mathrm{t}}p_{\mathrm{on}}P_{\mathrm{ER}}p_{\mathrm{e}}\} \end{aligned}$$

(23)

and

$$\begin{aligned} E_{\mathrm{e}}= \sum _{a \in A} E_{\mathrm{e}}(\mathrm{payoff}|a)P(a)=p_{\mathrm{on}}(w_{\mathrm{e}}p_{\mathrm{tx}}p_{\mathrm{ch}}P_{\mathrm{ER}}p_{\mathrm{e}}-c_{\mathrm{e}}) \end{aligned}$$

(24)

respectively, the results follow. Here A is the set of environments of a tx-ev pair where the players’ payoffs can be considered, which is given as Eq. (4) in Sect. 3.