# A Blockchain Solution for the Internet of Vehicles with Better Filtering and Adaptive Capabilities
## 6 Theoretical Analysis
### Communication Overhead  
The consensus process of the traditional PBFT has three stages. In the pre-preparation stage, the master node sends messages to other backup nodes. At this time, the number of communications in the network is \(n-1\). In the preparation stage, the backup nodes broadcast the preparation message to each other, and the number of communications is \((n-1)*(n-1)\). In the submission stage, the backup nodes submit messages to each other for further verification, and the number of communications is \((n-1)*(n-1)\). In the reply stage, the backup node replies to the master node, and the number of communications is \(n-1\). Therefore, the communication overhead of PBFT in this process is \(2n*(n-1)\).  
The main stages of the GPBFT algorithm [41] are Pre-prepare, Prepare, and GroupSign. In the Pre-prepare stage, the master node broadcasts messages to other nodes \(n-1\) times. In the Prepare stage, each node broadcasts Prepare messages to other nodes \(n*(n-1)\) times. In the GroupSign stage, after the group administrator and other nodes form a group, the node that receives the Prepare message needs to sign a short group signature to respond to the client, so the number of communications is \(n\). The total communication overhead is \(n^{2}+n-1\).  
In CBFT [42], the consensus process is divided into inter-group and intra-group consensus. Assuming \(n\) nodes, each consensus set has \(m\)\((m\geq 3)\) nodes. Each consensus set has two stages, preparation and submission, which require sending messages for communication. The two are \(n-n/m\), respectively. The number of communications within the consensus group is \(\left(\frac{n}{m}\right)^{2}-n+m\), so the total communication overhead of CBFT is \(\left(\frac{n}{m}\right)^{2}-\frac{3n}{m}+2n\).  
AE-PBFT optimizes the scale of nodes participating in the consensus process, dynamically adjusts the consensus group ratio according to the credit status of the overall nodes, and only requires some trusted nodes to participate in the consensus. Nodes are divided according to the credit values of continuous historical behaviors, eliminating the tedious Commit steps. After multiple verifications, the proportion of the consensus group is stable at \(2/5\) in the actual operating environment, so the communication overhead is approximately \(\left(\frac{2n}{5}-1\right)*\left(\frac{2n}{5}+1\right)\).  
Comparative analysis in Table 5 indicates that AE-PBFT, while maintaining security and stability, incurs lower communication overhead compared to the other three algorithms. Although the maximum fluctuation in the consensus group ratio reaches \(4/5\), it generally remains around \(2/5\) except in extreme cases. The advantage of a smaller consensus group is that it consumes fewer computing resources from the vehicle processors, enables faster response times, and provides greater flexibility over time. Future optimization directions could focus on the development of smaller, more flexible consensus groups and more adaptive dynamic consensus group adjustment algorithms. These improvements are expected to enhance system performance, particularly in terms of reduced energy consumption and lower latency.