# Grouping Sensors for the Key Distribution of Implicit Certificates in Wireless Sensor Networks
## 4 Results  
In this study, we used a tree topology to simulate a WSN environment. The simulated network has a tree structure in which simulated sensor nodes are connected like the branches of a tree. The tree structure network has three branches, and 1000 nodes were implanted randomly in the network with an increment of 100. Each sensor node was assumed to transmit one data. The sensor nodes were grouped into 10, 15, 20, 25, and 50 using TDG, BUG, and ABG. The aggregators were the reference of node grouping because power volume and memory space are critical resources for sensor nodes. The aggregator proportion was set from 10% to 50%. Space consumption can be evaluated from the number of group key parameters needed to be stored in the node. Table 1 lists the network parameter setting. Encryption and decryption follow the same estimation rules. Figure 9 shows the encryption and decryption procedures and how to estimate the number of encryption and decryption operations. Data are transmitted from one sensor node (Node\({}_{1}\)) to the target node (Aggregator). The data may be transmitted between the nodes from different groups. The data transmitted from Node\({}_{1}\) to the aggregator passes through the node from three different groups (Figure 10), and they need three encryptions and two decryptions. We infer that one node transmits data to the aggregator through nodes from different \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Values** \\ \hline Topology & Tree topology \\ \hline Deployment way & Random deployment \\ \hline The number of sensor nodes & 100–1000 nodes (100-node increases) \\ \hline The number of aggregators & 10–50\% (10\% increases each time) \\ \hline MN\_G & 10, 15, 20, 25, and 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Network parameter setting. groups, and \(t+1\) encryptions and \(t\) decryption are needed. The node from the same bottom level (Node\({}_{5-7}\)) must be aggregated as one aggregator (Aggregator\({}_{2}\)) using the aggregate function, and the data are encrypted as one data. The aggregator (Aggregator\({}_{2}\)) transmits the data to the previous level (depth = 2), repeating the aggregation procedure until the data are transmitted to the top level (depth = 1). Thus, one encryption is needed. The aggregator needs to decrypt each child non-aggregator node (green node). If one aggregator has \(y\) child non-aggregator nodes, it needs \(y\) decryption. Figure 10: Data receival by the aggregator. Figure 9: Data transmission to the aggregator. In this experiment, we simulated wireless sensor networks using C programming language by randomly deploying 100-1000 nodes in a tree structure with 100-node increments in each simulation, and the degree of the tree was set to three. The aggregator nodes accounted for 10-50% of all network nodes. Based on the algorithm used for calculating the number of encryption and decryption operations in the simulations, we inferred that as the number of groups between transmitting and aggregator nodes increases, the number of encryption and decryption operations required for data transmission by a given node increases. We assumed that each node would transmit a piece of data, and the maximum group size was set to 10, 15, 20, 25, and 50 individuals. The experimental parameters included the number of encryption and decryption operations required for the entire network to transmit data to a base station and the average number of group key parameters that the nodes need to store. After reading data, nodes perform encryption before transmitting the data to the upper layer. Notably, there exist two cases that require additional decryption and encryption operations during the transmission. The first case pertains to data transmission from one group to another: when the connecting node receives the data, it is decrypted using the current group key and then encrypted using the next group key until all data are transmitted to the base station.