## 5 Performance and Security Analysis
### Security Analysis  
In what follows, the resilience of AAD against sensor compromises, PDoS attack and FEDoS attack will be described, respectively. **Resilience Against Sensor Compromises. ** Let \(\bar{d}_{j}^{\bar{r}}\) be the counterfeit data the adversary constructs to substitute \(d_{j}^{\bar{r}}\). For the adversary, to successfully deceive the sink into accepting the bogus \(\hat{d}_{j}^{\bar{r}}\), it needs to compromise all the sensors in \(\bar{S}_{j}^{\bar{r}}\) and \(\mathcal{E}_{\bar{S}_{j}^{r}}\), where \(\mathcal{E}_{\bar{S}_{j}^{r}}\) denotes the set of all the sensors on the route from \(s_{j}\) to the sensors in \(\bar{S}_{j}^{\bar{r}}\), before the target round \(\bar{r}\). In other words,  
\[\bar{S}_{j}^{\bar{r}}\cup\mathcal{E}_{\bar{S}_{j}^{r}}\subseteq\{C_{1}, \ldots,C_{\bar{r}}\} \tag{2}\]  
Figure 8: The network topology in the example. Compromising the sensors in \(\bar{S}_{j}^{\bar{r}}\) is used to substitute the authentication tag corresponding to \(\hat{d}_{j}^{\bar{r}}\) while compromising the sensors in \(\mathcal{E}_{\bar{S}_{j}^{\bar{r}}}\) aims to know the keys stored in the sensors in \(\mathcal{E}_{\bar{S}_{j}^{\bar{r}}}\) so that the legitimacy of the bogus authentication tag can also be counterfeited. To evaluate the resilience of AAD against the sensor compromises, we have to estimate the number of sensors involved in the authentication procedure of one datum in AAD. Let \(|N_{\sigma}|\) be the number of sensors that is exactly \(\sigma\)-hop away from a specific sensor in a randomly deployed network. Before starting to estimate the number of sensors involved in the authentication procedure of one datum in AAD, we first need to know how to calculate \(|N_{\sigma}|,1\leq\sigma\leq\lambda\). Obviously, \(|N_{\sigma}|\) is dependent on the size of the sensing region, the number \(n\) of sensors, and the communication range \(R\) of each sensor. Although \(|N_{1}|=\pi R^{2}\frac{n}{A}-1\) can be easily obtained, there is no explicit formula for the expression of \(N_{\sigma}\) for \(\sigma\geq 2\). An approximation of \(|N_{\sigma}|\) is calculated in [21]. Although the calculation of \(N_{1}\) is straightforward and the calculation of \(N_{2}\) is not difficult, calculating \(N_{3}\) is quite difficult. Considering two sensors \(a\) and \(b\) with communication range being \(r\). There could be the case that \(b\) lies in the communication disk centered at \(a\) with the radius \(3r\) but \(a\) cannot reach \(b\)_via_ the exactly \(3\)-hop communication. Even if \(b\) which lies in the communication disk centered at \(a\) with the radius \(3r\) can reach \(a\), we have to make sure whether they can communicate with each other _via_\(3\)-hop communication or they can communicate with each other _via_\(2\) or \(1\)-hop communication. Simultaneously considering all these kinds of possibilities is an extremely difficult task. To the best of our knowledge, no exact formula has been proposed to deal with the calculation of \(N_{\sigma}\). As [21] indicates, \(N_{\sigma}\) is well approximated by using their proposed approximation formula. In our AAD scheme, for the energy conservation, \(\lambda\) is usually not too large, which means that we do not need to consider \(N_{\sigma}\) for large \(\sigma\). Thus, although the approximation formula of \(N_{\sigma}\) gradually deviates from the true \(N_{\sigma}\) as \(\sigma\) increases, it is still sufficient for our use. Therefore, in the following discussion, we refer to [21] for the derivation of \(|N_{\sigma}|\), \(2\leq\sigma\leq\lambda\). For \(1\leq\sigma\leq\lambda\), the probability of randomly selecting a sensor whose hop distance to the sensor \(s_{j}\) is exactly \(\sigma\) can be expressed as \(\frac{|N_{\sigma}|}{n}\).