## 5 Summary Comparisons: Functionality and Performance
### Storage, Communication, and Computation Cost Comparisons  
In order to provide a comprehensive evaluation of storage and communication overheads, Table 4 provides reasonable reference lengths for all components.  
Simultaneously, to ascertain computational costs during the login and verification phases, we executed the RSA algorithm with a key length of 1024 bits on a 12 th-generation intel core i7-12700 H with 16 G memory; we report that the elapsed time with respect to 1024-bit RSA modular exponentiation is 0.63 ms. For other cryptographic functions, based on the results from [10; 11; 12; 28; 29], the time required for the SHA-1 hash function is 0.00069 ms [28], the PUF function requires 0.43 ms [29], and symmetric encryption/decryption and the bio-hash function demand 0.1303 ms and 0.01 ms, respectively [11]. ECC point multiplication requires 0.0018 ms [12], and the fuzzy extractor function and bilinear pairing require 2.226 ms and 5.811 ms, respectively [10].  
Then, we provide Table 5, which presents a comparative analysis covering the storage, communication, and computational overheads consumed in all compared schemes.  
In our protocol, our total storage overhead is the smallest at 1376 bits, and the storage overhead of [PERSON] et al. [10] is the largest at 3952 bits. The storage overhead of each scheme  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbols** & **Bits** & **Symbols** & **Bits** \\ \hline Module n\({}_{0}\) & 32 & ECC point p & 160 \\ Counter c & 32 & Hash value h & 160 \\ Threshold value t & 16 & Secret key value x & 160 \\ Timestamp T & 32 & Random/once r & 160 \\ User’s/BASN’s identity & 128 & Symmetric ciphertext size enc & 256 \\ BASN’s identity set \(\text{S}_{\text{M}}/\text{S}_{\text{M}}^{\text{U}_{i}}\) & 32 & Public reproduction parameter \(\pi_{i}\) & 128 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The lengths of all terms involved in storage and communication costs.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multirow{2}{*}{**Schemes**} & \multirow{2}{*}{**Ref.**} & \multicolumn{3}{c}{**Storage Cost: Bits**} & \multicolumn{3}{c}{**Communication Cost: Bit**} & \multicolumn{3}{c}{**Computation Cost: ms**} \\ \cline{2-11}  & & \(\mathbf{U_{i}}\) & \(\mathbf{GWN}\) & \(\mathbf{BASN}\) & \(\mathbf{U_{i}}\) & \(\mathbf{GWN}\) & \(\mathbf{BASN}\) & \(\mathbf{U_{i}}\) & \(\mathbf{GWN}\) & \(\mathbf{BASN}\) \\ \hline [PERSON] et al. & [10] & 1328 & 288 & 2336 & 1056 & 800 & 1280 & 10.27 & 0.005 & 5.81 \\ \hline [PERSON] et al. & [11] & 1057 & 322 & 128 & 1408 & 1856 & 352 & 0.16 & 0.14 & 0.003 \\ \hline [PERSON] et al. & [12] & 576 & 288 & 1024 & 1888 & 3488 & 2144 & 0.03 & 0.03 & 0.90 \\ \hline Ours & — & 864 & 352 & 160 & 1920 & 3616 & 1664 & 3.28 & 0.20 & 0.64 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Storage, communication, and computation costs in the login and authentication phase.  
increases in the order of 1376 bits, 1507 bits, 1888 bits, and 3952 bits. Moreover, our proposal has obvious advantages in terms of storage overheads. As for communication overheads, the user, GWN, and BASN costs are 1920 bits, 3616 bits, and 1664 bits, respectively, with corresponding computation times of 3.28 ms, 0.20 ms, and 0.64 ms. It is evident that our solution boasts the lowest cumulative storage overhead compared to [10; 11; 12]. Simultaneously, the consumed times of the user and BASN in our scheme are 3.28 ms and 0.64 ms, respectively, which can reduce the user's and BASN's computation cost by 68.1% and 83.8% compared to the scheme reported in [10].  
In summary, our proposal outperforms others in terms of optimal security, superior storage and communication efficiency, and competitively efficient computational overheads. Other schemes, to varying degrees, require improvements in terms of security, communication overheads, or computational overheads.