# A Robust and Effective Two-Factor Authentication (2 FA) Protocol Based on ECC for Mobile Computing
## 6 Conclusions  
The authentication mechanism has always been an effective way of guaranteeing secure communication in mobile computing. However, existing authentication protocols weaken security in the pursuit of high efficiency. In this study, we designed a robust and effective 2 FA protocol and then fully proved the protocol's security and good performance.  
\begin{table}
\begin{tabular}{c l l} \hline \hline
**Notation** & **Description** & **Time/ms** & **Running Platform** \\ \hline \(T_{c}\) & The computing time for the extended chaotic-map operation & 0.294 & Ubuntu 18.04 with Intel \\ \(T_{m}\) & The computing time for elliptic curve point multiplication & 0.294 & i7-4710 HQ, 2.5 GHz CPU \\ \(T_{s}\) & The computing time for the symmetric cryptography operation & 0.021 & and 8 G memory \\ \(T_{h}\) & The computing time for a one-way hash operation & 0.003 & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Notations with related abbreviations.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Length/Bits** \\ \hline Timestamp & & 16 & \\ User identity & & 160 & \\ Random number & & 128 & \\ Elliptic curve point & & 160 & \\ The output of the hash function & & 160 & \\ The ciphertext of the symmetric encryption/decryption algorithm & & 128 & \\ \hline \hline \end{tabular}
\end{table}
Table 8: Lengths of the safety parameters.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Protocols**} & \multicolumn{3}{c}{**Computation Cost**} & \multicolumn{2}{c}{**Total**} & \multirow{2}{*}{**Message**} \\ \cline{2-2} \cline{4-5}  & **User** & **Service Provider** & & **Total Running** & **Comunication** & **Cost** & **Rounds** \\ \hline [PERSON] et al. (2013) [36] & \(5T_{h}+T_{m}\) & \(5T_{h}+3T_{m}\) & 1.206 ms & 960 bits & 3 \\ [PERSON] et al. (2015) [38] & \(4T_{h}+2T_{c}\) & \(6T_{h}+2T_{c}\) & 1.206 ms & 736 bits & 2 \\ [PERSON] et al. (2016) [35] & \(6T_{h}+3T_{c}\) & \(6T_{h}+3T_{c}\) & 1.8 ms & 1280 bits & 3 \\ [PERSON] et al. (2018) [37] & \(9T_{h}+2T_{c}\) & \(6T_{h}+T_{c}\) & 0.927 ms & 960 bits & 2 \\ [PERSON] et al. (2018) [5] & \(7T_{h}+2T_{m}+T_{s}\) & \(5T_{h}+2T_{m}+T_{s}\) & 1.254 ms & 768 bits & 3 \\
2 FA protocol & \(10T_{h}+3T_{m}\) & \(8T_{h}+3T_{m}\) & 1.818 ms & 1376 bits & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Communication and computation costs in the login and authentication phase.  
The safety analysis results using the ProVerif tool demonstrated that the proposed 2 FA protocol was able to achieve semantic security, satisfy all ten evaluation criteria (\(EC_{1}-EC_{10}\)), and provide mutual authentication for and preserve the security of the session key. By comparing the performance of six state-of-the-art protocols and the presented 2 FA protocol, we showed that the designed 2 FA protocol is more practical. Its design ideas presented in our paper are generic and may be used as guidelines to design AKA protocols. As our ongoing research work, we will focus on a more secure authentication protocol that also considers physical attacks for various situations in mobile computing with IPv6 over low-power wireless personal area networks (6 LoWPAN) [60].  
Validation, methodology, writing--original draft, K.L. and Z.Z.; writing--review and editing, Q.C., G.X. ([PERSON]), C.W. and G.X. ([PERSON]); validation, Y.G. and W.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China under grant no. 62102042 and the National Key Research and Development Program of China under grant no. 2021 YFB3101500.  
Not applicable.  
The authors declare no conflict of interest.