# Lightweight Physical Layer Aided Key Agreement and Authentication for the Internet of Things +
## 4 Performance Analysis
### Theoretical Analysis  
To evaluate the proposed authentication scheme, we consider an authentication error probability, which is an incorrect decision probability at the BS and given by the following:  
\[P_{E}=\rho P_{M}+(1-\rho)P_{F} \tag{5}\]  
where \(P_{M}\) and \(P_{F}\) are the miss and false alarm probabilities at the BS, respectively, and \(\rho\) is a weighting factor (\(0\leq\rho\leq 1\)). Note that \(P_{E}=0\) is assumed in the conventional cryptography-based authentication. \(P_{M}\) is the probability that when the legitimate IoT device transmits, the BS decides that the signal is an intrusion signal; \(P_{F}\) is the probability that when the intrusion device transmits, the BS decides that the signal is a legitimate signal. As shown in [8], the distribution of \(f_{\xi|\mathcal{H}_{i}}(x)\) is the Rice distribution as follows:  
\[f_{\xi|\mathcal{H}_{i}}(x)=\frac{x}{\sigma_{i}^{2}}e^{-\frac{x^{2}+\sigma_{i} ^{2}}{2\sigma_{i}^{2}}}I_{0}\left(\frac{xv_{i}}{\sigma_{i}^{2}}\right),\quad x \geq 0\quad\text{and}\quad i=0,1 \tag{6}\]  
\[F_{\xi|\mathcal{H}_{i}}(x)=1-\mathcal{Q}_{1}\left(\frac{v_{i}}{\sigma_{i}^{2} },\frac{x}{\sigma_{i}}\right) \tag{7}\]  
Figure 1: PL-AKA cross-layer authentication protocol [17]. where \(v_{i}=\mathbb{E}[\zeta|\mathcal{H}_{i}]\), \(\sigma_{i}^{2}=\text{Var}[\zeta|\mathcal{H}_{i}]\) and \(\mathcal{Q}_{1}(x,y)\) is the Marcum Q-function respectively [18]. For given target miss and false alarm probabilities denoted by \(P_{M}^{\circ}\) and \(P_{F}^{\circ}\), respectively, the thresholds are determined as follows:  
\[\alpha_{1}=\operatorname*{argmax}_{\alpha}F_{\xi|\mathcal{H}_{1}}(\alpha)\leq P _{M}^{\circ} \tag{8}\]  
\[\alpha_{0}=\operatorname*{argmax}_{\alpha}(1-F_{\xi|\mathcal{H}_{0}}(\alpha)) \leq P_{F}^{\circ} \tag{9}\]  
Then, based on the target miss and false alarm probabilities, the authentication error probability can be obtained in the proposed protocol--that is, we can control the authentication performance with \(P_{M}^{\circ}\) and \(P_{F}^{\circ}\). However, it should be noted that if \(P_{M}^{\circ}\) and \(P_{F}^{\circ}\) are too low, it can induce a large signaling overhead. We compare the proposed protocol with the conventional EPS-AKA protocol in terms of the signaling overhead. As mentioned in the previous subsection, the thresholds (\(\alpha_{1}\) and \(\alpha_{0}\)) are determined with \(P_{M}^{\circ}\) and \(P_{F}^{\circ}\), respectively. Then, the ranges of the three cases (i.e., \"Black\", \"Gray\" and \"White\") are determined as follows:  
\[\Theta= \begin{cases}\text{White}&\text{if}\quad\zeta>\alpha_{0}\\ \text{Gray}&\text{if}\quad\alpha_{1}\leq\zeta\leq\alpha_{0}\\ \text{Black}&\text{if}\quad\zeta<\alpha_{1}\end{cases}  \tag{10}\]  
Note that the lower the target miss and false alarm probabilities are set, the larger the distance between \(\alpha_{1}\) and \(\alpha_{0}\), which determines the range that \"Gray\" becomes. Let \(\lambda\) denote the probability of \"Gray\" (i.e., \(p\) (\(\alpha_{1}\leq\zeta\leq\alpha_{0}\))). Thus, the probability is given by the following:  
\[\lambda=\rho(F_{\xi|\mathcal{H}_{1}}(\alpha_{0})-F_{\xi|\mathcal{H}_{1}}( \alpha_{1}))+(1-\rho)(F_{\xi|\mathcal{H}_{0}}(\alpha_{0})-F_{\xi|\mathcal{H}_ {0}}(\alpha_{1})) \tag{11}\]  
From [19] and Table 1, the signaling overheads of EPS-AKA and PHY-PCRAS-AKA [11] are given as follows:  
\[\Omega_{EPS-AKA}=N(704+608U+528(P-1)), \tag{12}\]  
\[\Omega_{PHY-PCRAS-AKA}=N(\sum_{m=1}^{10}|M_{m}|)+N(P-1)(|M_{1}|+\sum_{q=5}^{10 }|M_{q}|) \tag{13}\]  
\[\Omega_{PHY-PCRAS-AKA}=N(\sum_{m=1}^{4}|M_{m}|+\sum_{n=9}^{10}|M_{n}|)+N(P-1) (|M_{1}|+\sum_{q=9}^{10}|M_{q}|) \tag{14}\]  
where \(N\) and \(U\) are the number of IoT devices and the number of authentication vectors, respectively. In addition, \(P\) denotes the number of authentication trials per IoT device. As shown in Figure 1, the \(M_{9}\) and the \(M_{10}\) messages associated with the cryptography-based authentication are exchanged with a probability of \(\bar{\lambda}=1-(1-\lambda)^{2}\) for the mutual authentication.