Strategic Defense Against Deceptive Civilian GPS Spoofing of Unmanned Aerial Vehicles

Abstract

The Global Positioning System (GPS) is commonly used in civilian Unmanned Aerial Vehicles (UAVs) to provide geolocation and time information for navigation. However, GPS is vulnerable to many intentional threats such as the GPS signal spoofing, where an attacker can deceive a GPS receiver by broadcasting incorrect GPS signals. Defense against such attacks is critical to ensure the reliability and security of UAVs. In this work, we propose a signaling game framework in which the GPS receiver can strategically infer the true location when the attacker attempts to mislead it with a fraudulent and purposefully crafted signal. We characterize the necessary and sufficient conditions of perfect Bayesian equilibrium (PBE) of the game and observe that the equilibrium has a PLASH structure, i.e., pooling in low types and separating in high types. This structure enables the development of a game-theoretic security mechanism to defend against the civil GPS signal spoofing for civilian UAVs. Our results show that in the separating part of the PLASH PBE, the civilian UAV can infer its true position under the spoofing attack while in the pooling portion of the PLASH PBE, the corresponding equilibrium strategy allows the civilian UAV to rationally decide the position that minimizes the deviation from its true position. Numerical experiments are used to corroborate our results and observations.

A Appendix

1.1 A.1 Appendix A: Proof of Lemma 1

Proof

Since we require $\frac{d \alpha _z(t_z)}{d t_z}\ge 0$, the strategy $\alpha _z(t_z)$ in the separating portion must satisfy $\alpha _z(t_z)> s^*_z(t_z) = \frac{t_z}{1+\rho }$. Suppose that $\alpha _z$ is constant on some interval $\varPhi \subseteq (t^s_z,t^l_z))$, then there exists some type $t_z\in \varPhi$ such that S can send a signal $s_z(t_z+\delta )$ with $\delta >0$ indicating a slightly higher type $t_z+\delta \in \varPhi$ without inducing the additional deception cost, which contradicts the hypothesis of separating equilibrium in Lemma 1; therefore, $\alpha _z$ is strictly increasing on $(t^s_z, t^l_z)$; thus, $\alpha _z \in (t^m_z, t^M_z)$ for any $t_z \in (t^s_z, t^l_z)$.

The incentive compatibility of SPBE requires that for any $t_z\in (t^s_z, t^l_z)$, $\alpha _z(t_z) \in \arg \min _{s_z} \ C^{S,z}(t_z, t_z, s_z).$ (8) is obtained by differentiating $C^{S,z}(t_z, t_z, s_z)$, which can be done only if $\alpha _z(t_z)$ is differentiable. In order to prove that $\alpha _z(t_z)$ on $(t^s_z, t^l_z)$, we first prove that $\alpha _z(t_z)>\arg \min _s C^{D,z}$ and $\alpha _z(t_z)$ is continuous for all $t_z\in (t^s_z, t^l_z)$.

We prove $\alpha _z(t_z) >s^*_z = \frac{t_z}{1+\rho }$ for all $t_z\in (t^s_z, t^l_z)$ in two steps as follows.

Step 1: Suppose $\alpha _z(\bar{\tau }_z) = s^*_z(\bar{\tau }_z)=\frac{\bar{\tau }_z}{1+\rho }$ for some $\bar{\tau }_z\in (t^s_z, t^l_z)$. Then, $C^{D,z}_2(t_z, \alpha _z(\bar{\tau }_z)) = 0$. Let $\delta >0$ be a position constant with small enough $| \delta |$. Let $U(\delta )$ be the expected change in the cost for type $\bar{\tau }_z - \delta \in (t^s_z, t^l_z)$ by changing from $\alpha _z(\bar{\tau }_z-\delta )$ to $\alpha _z(\bar{\tau }_z)$. Then,

$$ \begin{aligned} U(\delta ) =&C^{S,z}(\bar{\tau }_z,\bar{\tau }_z-\delta , s^*_z(\bar{\tau }_z)) -C^{S,z}(\bar{\tau }_z-\delta ,\bar{\tau }_z-\delta ,\alpha _z(\bar{\tau }_z-\delta ))\\ =&\big [ C^{A,z}(\bar{\tau }_z, \bar{\tau }_z-\delta ) -C^{A,z}(\bar{\tau }_z-\delta , \bar{\tau }_z-\delta ) \big ] \\&+ k_1\big [ C^{D,z}(\bar{\tau }_z-\delta , s^*_z(\bar{\tau }_z))-C^{D,z}(\bar{\tau }_z-\delta , \alpha _z(\bar{\tau }_z-\delta ))\big ]. \end{aligned} $$

Since $C^{A,z}(\bar{\tau }_z, \bar{\tau }_z-\delta )<C^{A,z}(\bar{\tau }_z-\delta , \bar{\tau }_z-\delta )$ and $C^{D,z}(\bar{\tau }_z-\delta , s^*_z(\bar{\tau }_z-\delta ))\le C^{D,z}(\bar{\tau }_z-\delta , \alpha _z(\bar{\tau }_z-\delta ))$, $U(\delta )<0$, which implies that S strictly prefers to use the strategy $\alpha _z(\bar{\tau }_z)$ when the type is $\bar{\tau }_z -\delta$; this means that S uses the strategy $\alpha _z(\bar{\tau }_z)$ for both type $\bar{\tau }_z -\delta$ and type $\bar{\tau }_z$, which contradicts the hypothesis of SPBE for $\bar{\tau }_z$. Thus, $\alpha _z(\bar{\tau }_z)\ne s^*_z(\bar{\tau }_z)$.

Step 2: Suppose there exists a $\hat{\tau }_z\in (t^s_z, t^l_z)$ such that $\alpha _z(\hat{\tau }_z)<s^*_z(\hat{\tau }_z)<\hat{\tau }_z$. From (8), we have $\frac{d \alpha _z(\hat{\tau }_z)}{d \hat{\tau }_z}<0$. Thus, the strict monotonicity of $\alpha _z(t_z)$ gives that $\alpha _z(\hat{\tau }_z-\delta )>\alpha _z(\hat{\tau }_z)$ for all $\delta >0$. Then for small enough $\delta >0$, we have $C^{D,z}(\hat{\tau }_z-\delta , \alpha _z(\hat{\tau }_z))<C^{D,z}(\hat{\tau }_z-\delta ,\alpha _z(\hat{\tau }_z-\delta ))$. Also, we have $C^{A,z}(\hat{\tau }_z, \hat{\tau }_z-\delta )< C^{A,z}(\hat{\tau }_z-\delta , \hat{\tau }_z-\delta )$. As a result, $C^{S,z}(\hat{\tau }_z, \hat{\tau }_z-\delta , \alpha _z(\hat{\tau }_z)) < C^{S,z}(\hat{\tau }_z-\delta , \hat{\tau }_z-\delta , \alpha _z(\hat{\tau }_z-\delta ))$. Therefore, S prefers to use the same strategy $\alpha _z(\hat{\tau }_z)$ for $\hat{\tau }_z-\delta$ as for $\hat{\tau }_z$, which contradicts the hypothesis of SPBE for $\hat{\tau }_z$. Thus, Step 1 and 2 yield that $\alpha _z(t_z)> s^*_z(t_z)$.

Now we prove the continuity of $\alpha _z(t_z)$ on $t_z\in (t^s_z, t^l_z)$. Suppose that there exists a discontinuity point at some $\textit{t}_z\in (t^s_z, t^l_z)$. Let $\alpha _z(\textit{t}_z)> \lim _{t_z\rightarrow \textit{t}_z^-}=\hat{\alpha }_z$. Then,

$$ \begin{aligned}&\lim _{\delta \rightarrow 0+} \big [C^{A,z}(\textit{t}_z-\delta , \alpha _z(\textit{t}_z-\delta ))-C^{A,z}(\textit{t}_z-\delta , \alpha _z(\textit{t}_z)) \big ]=0. \end{aligned} $$

Since $\alpha _z$ is strictly increasing and $s^*_z(\textit{t}_z)\le \hat{\alpha }_z<\alpha _z(\textit{t}_z)$, we also have

$$ \begin{aligned}&\lim _{\delta \rightarrow 0} \big [C^{D,z}(\textit{t}_z-\delta , \alpha _z(\textit{t}_z-\delta ) -C^{D,z}(\textit{t}_z-\delta , \alpha _z(\textit{t}_z)) \big ] = C^{D,z}(\textit{t}_z, \hat{\alpha }_z) -C^{D,z}(\textit{t}_z, \alpha _z(\textit{t}_z))<0. \end{aligned} $$

Therefore, the cost of $\alpha _z(\textit{t}_z-\delta )$ is less than $\alpha _z(\textit{t}_z)$; thus, S prefers to use the same strategy $\alpha _z(\textit{t}_z-\delta )$ for $\textit{t}_z$ as for $\textit{t}_z-\delta$ for small enough $\delta >0$, which contradicts the hypothesis of SPBE. Similar proof for the case $\alpha _z(\textit{t}_z)< \lim _{t_z\rightarrow \textit{t}_z^+}=\hat{\alpha }_z$ can show that S prefers to use the same strategy $\alpha _z(\textit{t}_z+\delta )$ for $\textit{t}_z$ as for $\textit{t}_z+\delta$ for small enough $\delta >0$, contradicting the SPBE. Therefore, $\alpha _z(t_z)$ is continuous on $(t^s_z, t^l_z)$.

Based on the same argument of the Proposition 2 in the Appendix of Mailath’s work in [14] (also see the proof of [9]), $\alpha _z$ is differentiable. Therefore, Lemma 1 is proved.

1.2 A.2 Appendix B: Proof of Theorem 1

In this part, we prove that there exists a unique solution on $[\hat{t}, t^M_z]$ to (8) with initial condition $\alpha ^*_z(t^M_z) = t^M_z = \hat{s}_z(t^M_z)$ and $\frac{d \alpha ^*_z(t_z)}{d t_z}>0$.

Proof

Step 1: Local uniqueness and existence

Let $B_z(t_z, s_z)$ be the inverse initial value problem and let $\eta _z(s_z)$ be the solution of $B_z(t_z, s_z)$. Then,

$$\begin{aligned} \begin{aligned} \eta '_z = B_z(\eta _z, s_z)&= -\frac{C^{S,z}(\eta _z, \eta _z, s_z)_3}{C^{S,z}(\eta _z, \eta _z, s_z)_1} \text {, with } \eta _z(s^*_z(t^M_z)) = t^M_z. \end{aligned} \end{aligned}$$

(16)

From the definition of $C^{S,z}$, $B_z$ is Lipschitz continuous on $T\times T$. Then, from the existence and uniqueness theorems [11], we can find some $\delta >0$ such that $\hat{s}_z(t_z)-\delta \ge s^*_z(t_z) = \frac{t_z}{1+\rho }$ and there exists a unique solution $\hat{\eta }_z$ to (16) on $[\hat{s}_z(t^M_z)-\delta , \hat{s}_z(t^M_z))$, and $\hat{\eta _z}$ is continuously differentiable on $[\hat{s}_z(t^M_z)-\delta , \hat{s}_z(t^M_z))$. From the definition of $\hat{s}_z(t^M_z)$, we have $B_z(t^M_z, \hat{s}_z(t^M_z))>0$, $B_z(t^M_z, s^*_z(t^M_z))=0$ and $\hat{s}^{-1}_z(t^M_z) = \frac{1}{\hat{s}_z(\hat{s}^{-1}_z(t^M_z) )}>0$; $\delta$ can be small enough such that $s_z<\hat{s}_z(\hat{\eta }_z(s_z))$ for all $s_z\in (\hat{s}_z(t^M_z)-\delta , \hat{s}_z(t^M_z)))$; and thus $\hat{\eta }'_z(s_z)>0$. Let $\hat{\alpha }_z = \hat{\eta }^{-1}_z$ be a solution to 8 on $(\breve{t}_z, t^M_z]$ for some $\breve{t}_z <t^M_z$ with $\frac{d \hat{\alpha }_z}{d t_z}>0$. Since the solution $\hat{\eta }_z$ to the inverse initial value problem is locally unique, the solution to the initial value problem (8) is locally unique.

Step 2: Suppose $\hat{\alpha }_z$ is the a solution to (8) with initial condition $\alpha ^*_z(t^M_z) = t^M_z = \hat{s}_z(t^M_z)$ and $\frac{d \alpha ^*_z(t_z)}{d t_z}>0$, on $(t'_z, t^M_z]$. Let $\bar{\alpha }_z = \lim _{t_z\rightarrow t'_z} \hat{\alpha }_z$. As been proved above, $\hat{\alpha }_z > s^*_z(t_z)$ for all $(t'_z, t^M_z]$, and $\bar{\alpha }_z\ge s^*_z(t_z)$. Suppose $\bar{\alpha }_z = s^*_z(t_z)$. Then, $C^{S,z}_3 = 0$, which yields $\lim _{t_z\rightarrow t'_z} = \infty$. Let $\zeta = \sup _{t_z\in [t'_z, t^M_z]} (s^*_z(t_z))' = \frac{1}{1+\rho }<\infty$. Since $\hat{\alpha }'_z(t^M_z)>0$ exists, there exists a $t^{''}_z>t'_z$ such that $\alpha _z(t^{''}_z)>\zeta$ for all $t_z\in [t'_z, t^{''}_z]$. Let $\epsilon > 0$ such that $\hat{\alpha }_z(t^{''}_z)>s^*_z(t^{''}_z)+\epsilon$. Since $\bar{\alpha }_z = \lim _{t_z\rightarrow t'_z} \hat{\alpha }_z$, it follows

$$ \begin{aligned} \bar{\alpha }_z&= \hat{\alpha }_z(t^{''}_z)+ \lim _{t_z\rightarrow t'_z} \int _{t_z}^{t^{''}_z} \alpha '_z(\tau )d\tau> s^*_z(t^{''}_z)+\epsilon + \int _{t'_z}^{t^{''}_z} \alpha '_z(\tau )d\tau \\&\quad > s^*_z(t^{''}_z)+ \int _{t'_z}^{t^{''}_z} \big (s^*_z(\tau )\big )' d\tau +\epsilon = s^*_z(t'_z) + \epsilon , \end{aligned} $$

which contradicts that $\bar{\alpha }_z = s^*_z(t_z)$. Therefore, we have $\bar{\alpha }_z > s^*_z(t_z)$.

If the solution $\hat{\alpha }_z(t_z)$ is well defined on $(t'_z, t^M_z]$ with $\lim _{t_z\rightarrow t'_z} \hat{\alpha }_z(t_z)>t^M_z$, then $-\frac{C^{S,z}_1}{C^{S,z}_3}$ is Lipschitz continuous and bounded in a neighborhood of $( \bar{\alpha }_z, t'_z)$. According to the existence and uniqueness theorems, there exists a unique differentiable solution $\hat{\alpha }_z$ to (8) on $( t'_z- \epsilon ', t^M_z]$ for some $\epsilon '>0$ with $\lim _{t_z\rightarrow t'_z- \epsilon '} \hat{\alpha }_z(t_z) > s^*_z(t'_z- \epsilon ')$ for $t'_z \in (t^m_z, t^M_z)$.

Clearly, $\hat{t}_z = \sup \{\hat{\tau }_z: \hat{\alpha }_z \text { is well defined on} \,(\hat{\tau }_z, t^M_z]\}$, and setting $\hat{\alpha }_z(\hat{t}_z )= t^m_z$ finishes the proof of Theorem 1.