## 4 Proposed Method
### Proposed Method: AAD
\begin{table}
\begin{tabular}{|c||c|} \hline Notation & Description \\ \hline \hline \(s_{j}\) & The sensor \(j\) \\ \hline \(r\) & The \(r\)-th round \\ \hline \(d^{r}_{j}\) & The data sensed by \(s_{j}\) at round \(r\) \\ \hline \(k^{r}_{j}\) & The key used by \(s_{j}\) at round \(r\) \\ \hline \(E_{k^{r}_{j}}(d^{r}_{j})\) & The encryption of \(d^{r}_{j}\) with key \(k^{r}_{j}\) \\ \hline \(z^{r}_{j}=h_{k^{r}_{j}}(d^{r}_{j})\) & The hash value of \(d^{r}_{j}\) with key \(k^{r}_{j}\) \\ \hline \(S^{r}_{j}\) & A random subset of sensors selected by \(s_{j}\) at round \(r\) \\ \hline \(\widetilde{t}^{r}_{j}\) & The number of co-authenticators selected by \(s_{j}\) at round \(r\) \\ \hline \end{tabular}
\end{table} 
Table 1: Notation table. the sink can be aware of the \(\lambda\)-hop proximity relationship of each sensor. Now, before the sensors start to sense data, the mobile sink starts its first itinerary over the network. For the contact of each sensor \(s_{j}\), the sink stores a Bloom filter, \(\mathfrak{B}_{j}\), containing the \(\lambda\)-hop proximity information of \(s_{j}\), in \(s_{j}\). \(\mathfrak{B}_{j}\) can be constructed by the sink according to the topology of the network graph. Specifically, for each \(s_{i}\) that has \(\lambda\)-hop proximity relationship with \(s_{j}\), the sink embeds \(s_{j}||s_{i}\) into \(\mathfrak{B}_{j}\). Note that although the number of \(\lambda\)-hop neighbors of \(s_{j}\) could be different for each \(s_{j}\), the size of Bloom filter used is chosen to be the same. **Sensing Phase. ** The description of sensing phase is shown in Figure 5. The sensing phase is executed by each sensor at each round. The sensor \(s_{j}\) constructs the corresponding MAC, \(z_{j}^{r}=h_{k_{j}^{r}}(d_{j}^{r})\), after it has the sensed data \(d_{j}^{r}\) at round \(r\). If the sensed data and corresponding MAC are simply stored in its local memory, once the sensor is compromised by the adversary, all the security materials will be exposed to the adversary, and \(d_{j}^{r}\) and \(z_{j}^{r}\) can be arbitrarily generated so that the security breach will occur. Thus, in the sensing phase, the sensed data and its corresponding MACs of a sensor will be distributed over some sensors randomly chosen from its \(\lambda\)-hop neighbors. Specifically, a subset \(S_{j}^{r}\) of \(t\) sensors is first randomly sampled from \(\{s_{1},\ldots,s_{n}\}\) (line 3 in Figure 5). \(SELECT-DISTINCT(t,n,j,k_{j}^{r})\) is a function randomly generating \(t\) sensor IDs from \(\{s_{1},\ldots,s_{n}\}\setminus\{s_{j}\}\). It can be implemented by using the hash function whose output is of length \(\lceil\log n\rceil\) bits. Then, a subset \(\bar{S}_{j}^{r}\) of \(S_{j}^{r}\) is constructed by choosing the sensors in \(S_{j}^{r}\) whose hop distance to \(s_{j}\) is within the pre-determined threshold \(\lambda\). Here, each sensor \(s_{j}\) uses the Bloom filter \(\mathfrak{B}_{j}\) constructed in post-deployment phase to check the hop distances to the randomly selected sensors in \(\bar{S}_{j}^{r}\). To construct \(\bar{S}_{j}^{r}\), what each sensor \(s_{j}\) needs to do is to check whether the \(s_{j}\) itself has \(\lambda\)-hop proximity relationship with the generated sensors in \(S_{j}^{r}\). For example, for \(s_{i}\in S_{j}^{r}\), the sensor issues \(s_{j}||s_{i}\) to query \(\mathfrak{B}_{j}\). We can know that \(s_{i}\in\bar{S}_{j}^{r}\) if the query result is positive and \(s_{i}\
otin\bar{S}_{j}^{r}\) otherwise (line 4 in Figure 5). Afterwards, the MAC, \(z_{j}^{r}\), is transmitted to the sensors in \(\bar{S}_{j}^{r}\). Note that \(\bar{S}_{j}^{r}=\bar{t}_{j}^{r}\) may vary as \(j\) and \(r\) vary. In general, \(\bar{t}_{j}^{r}\) is not a constant. Nonetheless, for ease of the explanation, \(\bar{t}_{j}^{r}\) is described as if it is a constant. In the following, the term \"authentication tag of \(s_{j}\)\" is used to denote \(z_{j}^{r}\) and the term \"co-authenticator of \(s_{j}\) at round \(r\)\" is used to denote the sensors in \(\bar{S}_{j}^{r}\).