## 6 Performance Analysis
### Security Performance
In conclusion, we have that all phantom locations lie in the two regions including \(E_{1}E_{s}E_{2}\) and \(E_{3}E_{s}E_{4}\) with \(4\theta\) range, where \(\theta=arccos((h-1)\)/h)\) as shown in Figure 3.  
(2) SLP and SLP-E against one Attacker  
Different from existing work, SLP and SLP-E can enhance the source location protection by using widely distributed phantom locations, so even if the attacker can obtain our privacy protocols, it is difficult for him to localize the source by tracing multiple phantom locations. This is because the  
Figure 3: The phantom locations distribution. phantom locations are scattered in SLP and SLP-E. Thus, it takes a long time for the attacker to trace multiple phantom locations and hence the source. In a real situation, once the source detects an event, it will send several messages to the base station over a period of time, so the attacker can only trace messages hop by hop during that period of time. Since the phantom locations are scattered, SLP and SLP-E can extend the time that the attacker takes to locate the phantom locations and hence improve the source location's security. According to the _h_-directed routing phase introduced in Section 4, the more random the directed paths are, the wider distribution the phantom locations have and consequently the more difficult it is for the attacker to trace the source. Therefore, we measure the security performance of SLP by the number of random directed paths. **Definition 2**. : _The random directed path is defined as the path from the source to the random location by h-directed routing. Specifically, the number of random directed paths is used to measure the degree of geographical dispersion of phantom locations. This is because the phantom locations are more widely scattered as the number of random directed paths grows. As shown in Figure 3, any phantom location will be located at the circumference of the circle with center s and radius h. As a result, compared to previous work [4, 9], the number of random directed paths for both SLP and SLP-E increases by \(\Phi=1-4\theta/2J=1-2\arccos((h-1)/h)\). Furthermore, Table 3 shows the changes of \(\Phi\) under varying h, where h \(\geq\) 2. When h = 2, \(\Phi\) increases by at least 33.33% compared to phantom single-path. Table 3 also shows that \(\Phi\) increases as h increases. In particular, \(\Phi\) increases to 88.36% when h = 60. We further obtain the mean value of \(\Phi\) when h grows from 2 to 60, i.e., \(1-\frac{2\sum\limits_{h=2}^{\infty}\arccos(1-\frac{1}{h})}{\pi}=79.83\%\). In conclusion, both SLP and SLP-E provide a stronger protection for source location because they increase the number of random directed paths significantly when compared with previous work [4, 9]._  
Theorem 2 indicates that for an attacker with a wider field of vision, SLP-E can avoid the inefficient paths completely and hence further enhance the source location protection. **Theorem 2**. : _By marking sensors in the unsafe area and bypassing these marked sensors during the shortest path routing process, the inefficient path can be avoided completely for SLP-E._  
Proof of Theorem 2.: In SLP-E, a sensor that transmits the packet for the first time during the shortest path routing process, say \(u\), is out of the unsafe area because \(h>r\). Node \(u\) forwards the packet to a neighbor that belongs to V\({}_{0}\)\(\cap\)\(u\)_set_parent_. The packet forwarding process will be repeated until the packet reaches the base station. Since the packet is only transmitted by the sensors outside the unsafe area, no inefficient paths will be generated. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **h = 2** & **h = 20** & **h = 30** & **h = 40** & **h = 50** & **h = 60** \\ \hline
**SLP** & 33.33\% & 79.78\% & 83.52\% & 85.73\% & 87.15\% & 88.36\% \\ \hline \end{tabular}
\end{table} 
Table 3: The percentage of random directed paths increase. (3) SLP and SLP-E against Several Attackers  
Assume that there are several attackers and only one source.