## 7 Conclusions and Future Work  
The management of trust in Internet of Things (IoT) devices controlled by different entities is an important aspect of security in IoT networks. Presently, digital certificates based on public key cryptography are utilized for secure communication, but the associated cost is substantial, particularly if every IoT device is issued a digital certificate from a certificate authority. In this research paper, a group authentication framework based on the Chebyshev polynomial and Blockchain technology is proposed. The proposed framework is capable of authenticating IoT devices situated on different entities and can facilitate secure data communication between that groups. The proposed framework was implemented in Python to understand the time analysis of the Chebyshev polynomial and was then deployed on Ethereum's Goerli testnet using a Solidity-based smart contract. From the results, it was found that the public key generation time, encryption, and decryption time increase as the degree of the Chebyshev polynomial increases. It was also found that a Chebyshev polynomial degree of four digits can be reasonably used in the proposed framework. The public key generation time, encryption time, and decryption time of 63.73 milliseconds, 100.18 milliseconds, and 62.18 milliseconds are achieved with a four-digit long Chebyshev polynomial degree. If a three-digit length polynomial degree is used, the public key generation time, encryption time, and decryption time reduce to 9.927 milliseconds, 8.993 milliseconds, and 9.752 milliseconds, respectively. It is evident \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Use Cases** & **Group Authentication** \\
**Parameters** & **Deployment** & **Transaction** \\ \hline Average Compute Units & 1.4 & 4.1 \\ Median Response (ms) & 19 & 26 \\ _eth\_gasPrice_ & 102,483 & 101,421 \\ _eth\_estimateGas_ & 476,029 & 23,587 \\ _eth\_getBlockByNumber_: Base Fe Per Gas & 91,861 & 115,261 \\ _eth\_getBlockByNumber_:Gas Used & 28,623,706 & 19,079,703 \\ Max Fe Per Gas & — & 1,500,230,522 \\ Max Priority Fe Per Gas & — & 1,500,000,000 \\ _eth\_getTransactionRecip_: Cumulative Gas Used & 16,403,924 & — \\ _eth\_getTransactionRecip_: Effective Gas Price & 102,483 & — \\ _eth\_getTransactionRecip_: Gas Used & 476,029 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance evaluation results of the deployed blockchain. from the obtained results that a polynomial degree of digit length beyond six digits is not advised because of long key generation, encryption, and decryption time. The Blockchain results show that compute units of 1.4 and 4.1 are used for the deployment of a smart contract and one transaction on a smart contract. The median response time for deployment and transaction was 19 milliseconds and 26 milliseconds, respectively. The group authentication framework is also analyzed on various parameters of Blockchain, such as effective gas used, estimated gas, the base fee per gas, and gas used, etc. One major issue in Blockchain-based solutions is scalability. In the future, the proposed framework could be tested for scalability. In this research, the framework was tested on the Goerli network, which is a large Blockchain and represents almost the same scalability parameter as the real Blockchain. However, there is a need to develop ideas to reduce the response time of the Blockchain for real-time group authentication. Another future direction is to modify the proposed framework to reduce the burden of Chebyshev polynomial computation over the smart contract. Smart contracts are not designed to perform large complex polynomial approximations, so the proposed framework can be modified to offload that task onto a trusted authority. It will be interesting to analyze the impact on various performance parameters with the modified mechanism. Conceptualization, R.S., H.T. and S.S.; methodology, R.S. ; software, S.S.; validation, H.T. and S.S.; formal analysis, R.S., H.T. and S.S.; investigation, S.S.; resources, S.S.; data curation, R.S. and H.T.