## 6 Results Discussion  
The proposed framework was implemented using Python, and the smart contract was created using Solidity. The smart contract was deployed on Ethereum's Goerli and Sepolia testnets to evaluate its performance on a near-real-size Blockchain. The Goerli testnet, which uses a proof-of-stake consensus algorithm, is a public network with a comparable Blockchain size to the Ethereum mainnet [29]. All experiments were conducted on a virtual machine running the Linux Ubuntu 18.04.6 LTS operating system, equipped with an AMD Ryzen 7 5800H with Radeon Graphics processor, 8 GB of RAM, and 100 GB of allocated hard disk. The recursive computation of the Chebyshev polynomial is a computing power-intensive process and after a certain degree, the polynomial may take months or years to compute. Instead of recursive computation, the Chebyshev polynomial value was computed using approximation methods. The Chebyshev value returned by an approximation method was not the same as the recursive method but it was close to the actual value. To better understand the time taken by the proposed authentication framework, a time analysis is carried out for various functions such as public key generation, encrypting the _deg_ variable and then decrypting the _deg_ value. The experiments were performed for various digit lengths of _deg_, with randomly generated values, and the time taken by the random number generator is also included in the time analysis. Figure 6 illustrates the time analysis for digit lengths 1 to 3. From the figure, it is evident that as the digit length increases, the time taken to generate a public key, encrypt, and decrypt also increases. For a digit length of 1, which means a polynomial degree in a single digit (i.e., a linear equation), the times taken to generate a public key, encrypt, and decrypt are 0.502 ms, 0.226 ms, and 0.112 ms, respectively. For double-digit polynomial degrees, these times are 0.870 ms, 0.737 ms, and 0.512 ms, respectively. If the degree of the Chebyshev polynomial is increased to 3, the time taken to generate a public key, encrypt, and decrypt is found to be 9.927 ms, 8.993 ms, and 9.752 ms, respectively. Figure 7 shows the time analysis for digit lengths of 4, 5, and 6, showing that as the digit length increases linearly, the time taken increases exponentially. The public key generation time for digit lengths 4 to 6 increases from 63.73 ms to 71,825.43 ms. Similarly, the encryption time rises from 100.88 ms for digit length 4 to 20,090.05 ms for digit length 6. The decryption time follows the same pattern and increases from 62.18 ms for digit length 4 to 68,327.17 ms for digit length 6. No further experiments are carried out for larger digit lengths because of the exponential computation time. Therefore, it is not recommended to use a Chebyshev polynomial degree greater than 6; otherwise, the authentication process will not be completed within a reasonable time. The proposed authentication framework was also tested on Blockchain-related parameters by deploying the smart contract on Ethereum's Goerli testnet. The results are shown in Table 1. Although it is suggested to use a permissioned private Blockchain for the proposed authentication framework where each TA maintains one Blockchain node, the results in Table 1 give an idea of the cost and performance if TAs decide to use Ethereum's mainnet network with real Ethers. The compute units used to deploy the smart contract were 1.4, whereas it was 4.1 for authenticating the group by the deployed smart contract. The median response time for deploying the smart contract was 19 ms, and for authenticating the group, it was 26 ms. The gas price on the network at the time of deployment was 102,483 Wei, whereas it was 101,421 Wei when an experiment of group authentication was carried out. At the time of the experiment, the estimated gas required for deployment was found to be  
Figure 6: Time analysis for the Chebyshev polynomial degree of 1, 2, and 3. Figure 7: Time analysis for the Chebyshev polynomial degrees of 4, 5, and 6.