## 5 Performance and Security Analysis
### Security Analysis
On the other hand, with \(k\bar{r}\leq n\), there are \(\binom{n}{\bar{r}}\) ways to compromise the sensors in the networks and at least \(\sum_{\sigma=1}^{\lambda}\frac{|N_{\sigma}|}{n}\sigma\) specific sensors should be included in the set \(\{C_{1},\ldots,C_{\bar{r}}\}\) of compromised sensors. As a result, the upper bound of \(P_{ADV}\) defined in Equation (5) holds. Here, it should be noted that instead of compromising random sensors over the network, the adversary that is aware of the application of AAD on the network may intentionally compromise the sensors in the nearby region of \(s_{j}\) so as to increase its success probability \(P_{ADV}\). Nevertheless, this strategy is infeasible for the adversary since the target sensor \(s_{j}\) and the target round \(\bar{r}\) are known only after the end of the target round. Thus, without the prior knowledge of the target sensor, in general, what the adversary can do is only to randomly compromise the sensors over the network. Define \(P_{AAD}\) as the _survival probability_, the probability that the data \(d_{j}^{\bar{r}}\) remain unforged can be calculated as:  
\[P_{AAD}=1-P_{ADV} \tag{6}\]  
Because of the upper bound of \(P_{ADV}\) in Equation (5), the lower bound of \(P_{AAD}\) can be represented as:  
\[\left\{ \begin{array}{ll}1-\frac{\binom{n-L(\lambda,n)}{\binom{n}{\bar{r}}} }{\binom{n}{\bar{r}}}&\text{if }k\bar{r}\leq n\\ 0&\text{if }k\bar{r}>n\end{array} \right. \tag{7}\]  
\(P_{AAD}\)'s in different settings are shown in Figure 9. Note that the dotted line represents the lower bound of \(P_{AAD}\) in the setting of \(n=100\), \(k=10\), and \(|N_{1}|=10\), irrespective of the parameter \(t\) used in AAD. Although the dashed line and solid line act as the upper bound of \(P_{AAD}\) in the setting of \(n=100\), \(k=10\), and \(|N_{1}|=10\). They, however, are pretty accurate approximations of \(P_{AAD}\) in a relatively dense network. We can observe that the curve of survival probabilities generated by the use of AAD with the parameters \(\bar{t}_{j}^{r}=3\) and \(\lambda=5\) is close to that in [4] with \(15\) co-authenticators used in their proposed protocol. Therefore, we can also claim that, when the similar level of security is required, the communication overhead incurred by AAD is lower than that incurred by prior works. **Resilience Against PDoS attacks. ** In general, the adversary is always able to launch PDoS attacks. In other words, PDoS attacks can only be mitigated, but not eliminated. The evaluation method similar to the ones used in [22, 23, 24, 25] is conducted here to demonstrate the superiority of AAD over CoMAC and ExCo in terms of the resilience against PDoS attacks. Recall that the resilience of CoMAC and ExCo are the same. Let \(L_{AAD}\) and \(L_{ExCo}\) be the bit-length of the authentication material of AAD and ExCo, respectively. Assume that a MAC is of length \(128\) bits and an encrypted data is also of length \(128\) bits. In AAD, the field representing the counter \(C\) is assumed to be of length \(8\) bits because the setting of \(\lambda\leq 2^{8}-1\) is sufficient in most cases. The field representing the aggregated prime number \(P\) is assumed to have \(32\) bits. Assume that in total \(n\) sensors are deployed. Consequently, \(L_{AAD}\) can be calculated as \(128+128+32+8+32+2\sqrt{n}=328+2\sqrt{n}\) bits. On the other hand, since the authentication material only contains the authentication tag, the size of \(L_{ExCo}\) is \(128+2\sqrt{n}\) bits. For a flat network whose \(n\) sensors are uniformly distributed, the average hop distance between arbitrary two sensors is \(\sqrt{n}\)[26]. In the following evaluation, it is assumed that the compromised sensor sends bogus message to a random sensor, instead of sending to the most distant sensors. Also note that such evaluation is advantageous to the calculation of ExCo's energy consumption but is disadvantageous to the calculation of ours. Nevertheless, such evaluation will imply the lower bound of the energy saving of our AAD over ExCo.