A Cooperative Jamming Game in Wireless Networks Under Uncertainty

Abstract

Considered is a multi-channel wireless network for secret communication that uses the signal-to-interference-plus-noise ratio (SINR) as the performance measure. An eavesdropper can intercept encoded messages through a degraded channel of each legitimate transmitter-receiver communication pair. A friendly interferer, on the other hand, may send cooperative jamming signals to enhance the secrecy performance of the whole network. Besides, the state information of the eavesdropping channel may not be known completely. The transmitters and the friendly interferer have to cooperatively decide on the optimal jamming power allocation strategy that balances the secrecy performance with the cost of employing intentional interference, while the eavesdropper tries to maximize her eavesdropping capacity. To solve this problem, we propose and analyze a non-zero sum game between the network defender and the eavesdropper who can only attack a limited number of channels. We show that the Nash equilibrium strategies for the players are of threshold type. We present an algorithm to find the equilibrium strategy pair. Numerical examples demonstrate the equilibrium and contrast it to baseline strategies.

This material is based upon work supported by the National Science Foundation (Grant No.1901721).

Appendices

Appendix A: Proof of Theorem 3

Note that

$$\begin{aligned} \mathbb {E}\left[ C_{E_i}(J_i)\right] = \frac{\partial v_E(\varvec{J},\varvec{y})}{\partial y_i}. \end{aligned}$$

Given $y^o_m=1$ and $y^o_i = 0, \forall i \ne m$, Theorem 2 implies

$$\begin{aligned} y^o_i T_i \sum _{k=1}^{K_i} \frac{\mathbb {E}\left[ A_i \right] \beta ^k_i p^k_i}{{\sigma _i}^2} - \gamma = - \gamma < w_D, \forall i \ne m, \end{aligned}$$

which leads to $\tilde{J}^o_m(\tilde{w}_D, \varvec{y^o})=0, \forall i \ne m$.

Assume $(\varvec{\tilde{J}^o}(\tilde{w}_D),\varvec{y})$ is a pair of NE strategies. It is required that

$$\begin{aligned} w_A \ge \mathbb {E}\left[ C_{E_i}(0)\right] , \forall i \ne m, \end{aligned}$$

by KKT conditions (8). Also, since $y^o_m > 0$, then

$$\begin{aligned} w_A = \mathbb {E}\left[ C_{E_m}(J^o_m(\tilde{w}_D))\right] \le \mathbb {E}\left[ C_{E_m}(0\right) ], \end{aligned}$$

by KKT conditions (8). Thus, it must be true that

$$\begin{aligned} \mathbb {E}\left[ C_{E_m}(0\right) ] \ge \mathbb {E}\left[ C_{E_i}(0\right) ], \forall i \ne m, \end{aligned}$$

$$\begin{aligned} \mathbb {E}\left[ C_{E_m}(J^o_m(\tilde{w}_D))\right] \ge \mathbb {E}\left[ C_{E_i}(0)\right] , \forall i \ne m, \end{aligned}$$

for the assumption to be true.    $\square$

Appendix B: Proof of Theorem 4

Let $w_A$ be the Lagrange multiplier for the eavesdropper’s optimization problem (6). Note that

$$\begin{aligned} R_i(J_i) = \frac{\partial v_E(\varvec{J},\varvec{y})}{\partial y_i}, \end{aligned}$$

$$\begin{aligned} R_i(0) = \frac{\mathbb {E}\left[ A_i \right] T_i}{\sigma _i}, \end{aligned}$$

and $R_i(J_i)$ is decreasing w.r.t. $J_i \ge 0$. Let $\varvec{J^*}$ be the cooperative jamming strategy in the NE. Note that

$$\begin{aligned} R_i(J^*_i) \le w_A, \forall i=1,...,N, \end{aligned}$$

as required by KKT condition (9). Thus, if $R_i(0) > w_A$, then $J^*_i > 0$. Let $w_D$ be the Lagrange multiplier for the defender’s optimization problem (5) and $\varvec{y^*}$ be the attack strategy in the NE. By KKT condition (7), if $J^*_i > 0$, then

$$\begin{aligned} y^*_i T_i \sum _{k=1}^{K_i} \frac{\mathbb {E}\left[ A_i \right] \beta ^k_i p^k_i}{(\sigma _i + \beta ^k_i J^*_i)^2} - \gamma = w_D \ge 0, \end{aligned}$$

thus giving $y^*_i > 0$. But, if $y^*_i > 0$, then

$$\begin{aligned} R_i(J^*_i) = w_A, \end{aligned}$$

by KKT condition (9). In summary, if $R_i(0) > w_A$, then $J^*_i$ is the unique root of the equation,

$$\begin{aligned} R_i(x) = w_A. \end{aligned}$$

If $R_i(0) \le w_A$, then

$$\begin{aligned} w_A \ge R_i(0)> R_i(J_i), \forall J_i > 0, \end{aligned}$$

which leads to $J^*_i=0$ since $R_i(J^*_i) < w_A$ if $J^*_i>0$. Thus, let us define to show that the NE cooperative jamming strategy is dependent on $w_A$.

Note that $H_i(w_A,w_D)$ is the unique root of the equation,

$$\begin{aligned} \frac{\partial v_D(\varvec{J},\varvec{y^*})}{\partial J_i} \bigg |_{\varvec{J}=\varvec{J^*}} = w_D, \end{aligned}$$

w.r.t. $y^*_i$.

If $R_i(0) > w_A$, then $J^*_i > 0$, which leads to $y^*_i = H_i(w_A,w_D)$ by KKT condition (7).

If $R_i(0) < w_A$, then $R_i(J^*_i) = R_i(0) < w_A$ since $J^*_i=0$. It follows that $y^*_i=0$ by KKT condition (9).

If $R_i(0) = w_A$, then $J^*_i=0$, but it is possible to have $y^*_i>0$ in a NE as long as

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial v_D(\varvec{J},\varvec{y^*})}{\partial J_i} \big |_{\varvec{J}=\varvec{J^*}} \le w_D, \text {by KKT condition (7)},\\ y^*_i \ge 0,\\ y^*_i + \sum \limits _{j \in I(w_A)} y^*_j \le 1. \end{array}\right. } \end{aligned}$$

Thus, $\varvec{y}(w_A,w_D)$ in (12) is correctly defined to satisfy KKT conditions (7) and (9). Note that it is possible to have

• $y_i(w_A,w_D) = H_i(w_A,w_D) < 1 - \sum \limits _{j \in I(w_A)} y_j(w_A,w_D)$, or

• $\sum \limits _{j \in I(w_A)} y_j(w_A,w_D) > 1$.

So the constraint $\sum _{i=1}^N y_i(w_A,w_D)=1$ is not guaranteed by (12).

Finally, it is required that

$$\begin{aligned} w_D(J-\sum _{i =1}^{N} J_i(w_A))=0. \end{aligned}$$

Thus, if $\sum _{i \in I(w_A)} J_i(w_A) < J$, then $w_D=0$. In this case, the only requirement missing for $(\varvec{J}(w_A), \varvec{y}(w_A,w_D))$ to be a pair of NE strategies is to satisfy

$$\begin{aligned} \sum _{i=1}^N y_i(w_A, w_D)=1. \end{aligned}$$

Now look at the case when $\sum _{i \in I(w_A)} J_i(w_A) = J$. Following (11) and (12) when $\sum _{i \in I(w_A)} J_i(w_A) = J$, it is guaranteed that $\sum _{i=1}^N y_i(w_A, w_D)=1$. However, it is possible that $w_D<0$, so the only requirement for $(\varvec{J}(w_A), \varvec{y}(w_A,w_D))$ to be a pair of NE strategies is to satisfy $w_D \ge 0$.

In summary, following (10), (11) and (12), if

• 1. $w_D \ge 0$, and

• 2. $\sum _{i=1}^N y_i(w_A,w_D)=1$,

then $(w_D,w_A)$ is a proper pair of Lagrange multipliers for the Nash equilibrium, $(\varvec{J}(w_A), \varvec{y}(w_A,w_D))$.    $\square$