## 6 Performance Analysis
### Security Performance  
In this section, we will analyze the security performance of the existing typical source location protection protocols [3, 4, 13] firstly and then our proposed protocols respectively. (1) The Existing Typical Source Location Protection Protocols  
The security performance of the protocols described in [3, 4, 13] improves as the generated phantom locations are more widely distributed. However, Lemma 1 proves that the phantom locations generated by [3, 4, 13] gather in two regions. As a result, the attacker is likely to be led to the source as he can reach to one of the two regions easily. **Lemma 1**. : _Suppose \(\mathit{Hop}_{u,bs}\) and \(\mathit{Hop}_{v,bs}\) are the shortest distance from sensor \(u\) and \(v\) to the base station respectively. The absolute difference between \(\mathit{Hop}_{u,bs}\) and \(\mathit{Hop}_{v,bs}\) is no more than 1, where \(v\) is \(u\)'s neighbor._  
**Proof of Lemma 1. ** According to the broadcast scheme initialized by the base station in [3, 4, 9], for each \(v\in u\)._neighbor_, we have that \(\mathit{Hop}_{v,u}=1\), where \(u\)._neighbor_ denotes the set of all neighbors of \(u\). Apparently \(\mathit{Hop}_{v,bs}-\mathit{Hop}_{u,bs}\leq 1\) if \(\mathit{Hop}_{u,bs}=\mathit{Hop}_{v,bs}\). If \(\mathit{Hop}_{u,bs}>\mathit{Hop}_{v,bs}\), there is a path from \(u\) to base station through \(v\), namely \(R_{u,bs}\). \(R_{u,bs}\) consists of two parts: the shortest path from \(u\) to \(v\) and the shortest \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(r\)/\(H\) & 1/2 & 1/5 & 1/10 & 1/15 & 1/20 \\ \hline \(f_{avg}\) & 28 & 8 & 3 & 2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The increased amount of communication cost on average. path from \(v\) to base station. Thus, we have \(|R_{u,bs}|=1\)\(+\)\(Hop_{v,bs}\) and \(|R_{u,bs}|\geq Hop_{u,bs}\), where \(|R_{u,bs}|\) is the length of path \(R_{u,bs}\) measured by the number of hops. Finally, we have \(1\)\(+\)\(Hop_{v,bs}\)\(\geq\)\(Hop_{u,b}\) and \(Hop_{u,bs}-\)\(Hop_{v,bs}\)\(\leq\)\(1\). Similarly, we can prove that \(Hop_{v,b}-\)\(Hop_{u,bs}\)\(\leq\)\(1\) if \(Hop_{u,bs}\)\(<\)\(Hop_{v,bs}\). In summary, we have \(|Hop_{v,bs}-\)\(Hop_{u,bs}|\)\(\leq\)\(1\). **Theorem 1**. : _If the packet is forwarded to a sensor randomly chosen from the set determined by the source (i.e., parent set or child set of the current sensor), all phantom locations will gather in two regions after h-hops of transmissions._  
Proof of Theorem 1.: Once a source determines the parent set as its forwarding set, the source \(s\) will send its packet to a sensor randomly chosen from its parent set. Similarly, the sensor responsible for the packet transmission will also forward the packet to a sensor randomly from its parent set. After \(h\) transmissions, the packet will reach the phantom location \(p\), where the shortest distance between \(p\) and \(s\) measured by hops \(Hop_{p,s}\)\(\leq\)\(h\). As is shown in Figure 3, \(p\) lies in the region \(E_{i}E_{s}E_{2}\). According to Lemma 1, we have \(Hop_{p,bs}=H-h\), where \(H\) denotes the shortest distance from the source to base station measured by the number of hops. Given that the center point of the base station and the radius of the \(i\)-th circle \(i\)*\(R\) (\(I\) = 1,2,3...), a sensor, say \(u\), lies in the area between the \(k\)-th and the \((k-1)\)-th circle if \(Hop_{u,bs}=k\) (\(k\) = 2,3...), where \(R\) denotes the transmission range of a sensor. Since any sensor within the transmission radius of the base station is one hop away from base station, given any phantom location, say \(p\), it is located in the area between the \((H-h)\)-th and \((H-h-1)\)-th circle. Similarly, we can prove that \(p\) lies in the region \(E_{i}E_{s}E_{i}\) if \(s\) determines its child set as its forwarding set.