# Lightweight Physical Layer Aided Key Agreement and Authentication for the Internet of Things +
## 4 Performance Analysis
### Theoretical Analysis
Then, the average signaling overhead of the proposed protocol and signaling overhead at MME are given by the following:  
\[\mathbb{E}[\Omega_{PL-AKA}]=N(\sum_{m=1}^{8}|M_{m}|+\bar{\lambda}\sum_{n=9}^ {10}|M_{n}|)+N(P-1)(|M_{1}|+\sum_{p=5}^{8}|M_{p}|+\bar{\lambda}\sum_{q=9}^{1 0}|M_{q}|) \tag{15}\]  
\[\mathbb{E}[\Omega_{PL-AKA}{}_{MME}]=N(\sum_{m=1}^{4}|M_{m}|+\bar{\lambda}\sum _{n=9}^{10}|M_{n}|)+N(P-1)(|M_{1}|+\bar{\lambda}\sum_{q=9}^{10}|M_{q}|) \tag{16}\]  
where \(M_{i}\) is the \(i\)th message in the proposed protocol. Based on the related parameters in Table 1, \(|M_{1}|=|M2|=176\), \(|M_{3}|=608U+L\), \(|M_{4}|=|M_{5}|=|M_{6}|=|M_{7}|=|M_{8}|=L\), \(|M_{9}|=288 We present the simulation results to see the authentication and signaling overhead performance of the proposed protocol. For simulations, we assume the [PERSON] fading channel model and generate random phases, where \(H(f_{l})\sim\mathcal{CN}(0,\sigma_{h}^{2})\) and define SNR as \(10\text{log}_{10}(\frac{E_{b}}{\sigma^{2}})\). Figure 2 depicts the probability density functions of \(\zeta\) for legitimate and intrusion signals in which \(P_{M}^{\circ}=P_{F}^{\circ}=10^{-6}\), \(L=64\), and \(\text{SNR}=5\text{ dB}\). As shown in Figure 2, the distribution of \(\zeta|\mathcal{H}_{1}\) from the legitimate device is sufficiently distinguishable from that of \(\zeta|\mathcal{H}_{0}\), the intrusion one. In addition, the ranges of the three cases (\"Black\", \"Gray\" and \"White\") are determined by two thresholds (\(\alpha_{1}\) and \(\alpha_{0}\)) with \(P_{M}^{\circ}\) and \(P_{F}^{\circ}\). Here, \"Gray\" plays a role as a guard interval to prevent a wrong authentication decision of a BS caused by noise and interference. From Figure 2, it seems obvious that the probability that \(\zeta\) is included in \"Gray\" is negligibly low, compared to \"Black\" and \"White\". It implies that the signaling overhead induced by cryptography-based authentication (i.e., \(M_{9}\) and \(M_{10}\)) is insignificant. Figure 3 shows the simulation results for the signaling overhead over different target false alarm probabilities to compare the proposed PL-AKA with conventional EPS-AKA and physical layer phase challenge response authentication AKA (PHY-PCRAS-AKA) [11], where \(L=64\), \(P=30\), \(U=20\), \(N=200\), \(P_{M}^{\circ}=10^{-6}\) and \(\text{SNR}=5\text{ dB}\). In the simulation, the signaling overhead is divided into two types: (i) total signaling overhead, and (ii) signaling overhead at MME. As shown in Figure 3, although \(P_{M}^{\circ}=10^{-10}\), the proposed PL-AKA has a small total signaling overhead, compared to the conventional methods. In particular, the PHY-PCRAS-AKA needs a larger signaling overhead than EPS-AKA because it simply cascades both layer authentication methods for the enhancement of security as described in (13) and (14). Furthermore, while the signaling overhead at MME is the same as the total signaling overhead in the conventional EPS-AKA, in the proposed PL-AKA, the signaling overhead at MME is significantly smaller than the total signaling overhead because the BS performs a preemptive authentication instead of the MME in the proposed PL-AKA protocol. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Symbol** & **Descriptions** & **Bits** \\ \hline IMSI & International mobile subscriber identity & 128 \\ SR & Service request & 8 \\ LAI & Location area identity & 40 \\ RES & Response & 64 \\ XRES & Expected response & 64 \\ RAND & Random challenge & 128 \\ AUTN & Authentication token & 160 \\ \(\kappa\) & Secret key for PHY-CRAM & \(L\) \\ AV & Authentication vector & \(608+L\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Related parameters.