## 6 Simulation Results
### Communication Overhead
#### 6.2.2 Communication Overhead of Network  
The algorithm has a message overhead of 4 bytes per data packet as opposed to two MAC of 4 bytes each in DAA. Let \(\alpha\) represent the number of data packets generated by legitimate nodes, and \(\beta\) represent the number of false data packets injected by up to \(T\) compromised nodes. Let \(H\) denote the average number of hops that a data packet travels in the network, and \(H_{f}\) denote the average number of hops between two consecutive aggregators. Let \(L_{d}\) denote the length of the data packet in FESA, then the data packet length in DAA is \(L_{d}+6\). Let \(D_{\text{FESA}}\), \(D_{\text{DAA}}\) and \(D_{\text{SDA-PH}}\) denote the amount of data transmitted over a sensor network using FESA, DAA and SDA-PH, respectively. Therefore, \(D_{\text{FESA}}\), \(D_{\text{DAA}}\) and \(D_{\text{SDA-PH}}\) can be expressed as follows:  
\[D_{\text{FESA}}=L_{d}\times\left[(\alpha\times H)+\left(\beta\times H_{f} \right)\right]+T\times L_{d}\times(\alpha+\beta)+T\times\frac{4}{T+1}\times( \alpha+\beta)\text{ bytes} \tag{8}\]  
\[D_{\text{DAA}}=(L_{d}+6)\times\left[(\alpha\times H)+\left(\beta\times H_{f} \right)\right]+T\times(L_{d}+6)\times(\alpha+\beta)+T\times\frac{4}{T+1}\times( \alpha+\beta)\text{ bytes} \tag{9}\]  
\[D_{\text{SDA-PH}}=(L_{d}-8)\times H\times(\alpha+\beta)\text{ bytes} \tag{10}\]  
Although the value of \(T\) is nothing to do with the packet size, it can affect the number MAC transmissions between aggregators and monitoring nodes during data aggregation. Substituting \(H=100\) and \(L_{d}=38\) in Equation (8), Figure 6 illustrates the effect of monitoring nodes to the total data transmission in FESA and DAA. As the number of monitoring nodes increases, the total data transmission in the network also increases. From Figure 6, we can see that DAA has a much more data transmission than FESA mainly because the data packets forwarded in DAA which contain both plaintext and ciphertext are larger than FESA. Figure 5: The performance in aggregator under DAA, SDA-PH and our scheme. Although it is more complex using FHE, the performance of our scheme is also better than other two schemes. (**a**) The average delay of aggregator; (**b**) The average delivery ratios of the aggregator; (**c**) The average throughput of the aggregator. In (**b**), when there are less than seven neighboring nodes forward data to an aggregator at the same time, the average data delivery ratios of the aggregator are close to 1. Compared to no data aggregation protocols, the schemes using aggregators to aggregate receiving data can effectively reduce the data redundancy. Figure 6: The impact of monitoring nodes on the communication overhead for FESA and DAA. Figure 7: The entire network performance under DAA, SDA-PH and our scheme. Although using FHE increases the communication overhead, the performance of our scheme is still approaching to the other two schemes. **(a)** The average delay of the entire network; **(b)** The average delivery ratios of the entire network; **(c)** The average throughput of the entire network. In **(b)**, when there are less than six aggregators forward data to the base station at the same time, the data delivery ratios are close to 1. Therefore, in secure data aggregation protocols, the number of aggregators in the network affects the performance of the whole network. In this simulation scenario, we assume no false data exists. Figure 7 illustrates the performance of these three protocols under different number of aggregators, including transmission delay, delivery radios and throughput. The comparison result for average delays of the aggregator is SDA-PH \(<\) FESA \(<\) DAA. Since FESA increases part of the communication overhead using FHE, the average delivery ratios and the average throughputs of these three protocols are approximate. Figure 8 shows the total data transmission _versus \(\beta\)/\(\alpha\)_ for FESA, DAA and SDA-HP.