## 5 Result Analysis
### Identity Authentication Security Performance Analysis  
In this paper, an identity authentication scheme is designed for blockchain group authentication. Its security is based on the problem of discrete logarithms. The specific analysis is described as follows:  
1. The hash function has a single irreversible feature, which can ensure that the user's identity is not leaked. 2. Due to the intractability of discrete logarithms, even if the attack node intercepts \(v\) and \(a\), [PERSON] cannot find k from Formula (2). \[v=lna^{k}mod\ p\] (2)
3. The attacking node attempts to obtain \(r\), and then intercepts \(y\) and \(e\), so that he can obtain \(k\) from Formula (3); however, finding \(r\) from Formula (1) is also a problem that is difficult to solve for discrete logarithms. Similarly, the verification node cannot find r and then obtain \(k\). \[y=(r-ke)mod\ q\] (3)
4. Replay attack. The scheme is based on the challenge/response method, the authentication process is dynamic and interactive, and the random number sent for each authentication is different. Even if the attacking node obtains all the previously authenticated information, the validating node will not be fooled by outdated information. 5. The verification node sends the message obtained during the transaction verification process to the attacking node. The verification node can only obtain \(v\) and \(y\) and judge whether the node is legal, and knows nothing about the relevant information of the private key \(k\). Therefore, the security of the authentication process is guaranteed. Legitimate users are only selected for authentication, as depicted in Figure 10. As the number of legitimate users increases, the time required for authentication is approximately positively correlated with the time of legitimate users. Therefore, the time cost consumed is huge when the number of authenticated users is very large. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & **Single/Group/Mixed** & **Decentralization** & \multirow{2}{*}{**Time Cost**} & **Certification** \\  & **Authentication** & **/Centralization** & & **Efficiency** \\ \hline Literature [27] & Single & Centralization & O (1) & Poor \\ Literature [28] & Single & Decentralization & O (1) & Poor \\ Literature [14] & Single & Decentralization & O(n\({}^{2}\)) & Poor \\ Literature [29] & Group & Decentralization & O (n\({}^{2}\)) & Median \\ This scheme & Mixed & Decentralization & O (logn) & Strong \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of different authentication schemes. is lower than that of the MHT algorithm when the number of users is less than 200. The advantages of the MHT algorithm are gradually significant when the number of users is more than 200. The Merkel hash tree group authentication method can effectively improve system efficiency when IoT devices are connected on a large scale. Analysis of the internal efficiency of the system. Considering that there are many kinds of Hash algorithms when constructing the MHT, the system efficiency of different algorithms is analyzed in this project. It is assumed in this simulated MHT algorithm that all users in the group are legal members. Figure 12 illustrates the impact of different Hash algorithms on the MHT establishment time. The time cost of all algorithms is positively correlated with the number of users. The time cost of the SHA256 algorithm is lower than that of SHA384 and SHA512 under different numbers of users. The greater the number of users, the greater the gap between the time cost of the SHA256 algorithm and the other two algorithms. This is in accordance with the fact that the complexity of the SHA256 algorithm is lower than the other two algorithms, contributing to a lower time cost. Additionally, the simulation results are consistent with the theory. In other words, the Hash algorithm in the MHT algorithm in this system can employ SHA256 to improve the overall efficiency of the system.