## 7 Conclusions  
In this paper, the Chebyshev polynomial-based scheme for resisting side-channel attacks in 5G-enabled vehicular networks could achieve both important properties in terms of chaotic and semi-group. Compared with other methods, our work can thwart the side-channel attack by periodically renewing the sensitive data preserved on the TPD on the enrolled node's OBU. Our work is also proven to be secure against the model of Dolev-Yao attacks in the AVISPA simulator. Security analysis shows that our work archives all of the design goals with regard to privacy (pseudonym identity and unlikability) and security (authentication, integrity, and traceability) in 5G-enabled vehicular networks. Nevertheless, the resistance to side-channel attacks could not be achieved in the relevant works. Ultimately, since our work does not employ BPC and ECC operations, the efficiency performance of our work is the lowest compared to the four related schemes. Thus, our work has better efficiency with regard to costs of computation and communication.  
In future work, we will carry out the simulation experiment of the proposed scheme by running a traffic simulator (SUMO) and a network simulator (OMNeT++) to provide more performance details in terms of varying the speed and numbers of UE. Moreover, we plan to explore achieving the privacy and security factors in a more dynamic scenario containing fog computing technology over 5G communication.  
Conceptualization, writing--review and editing, M.A.A.-S.; writing--original draft preparation, investigation, supervision, S.M.; funding acquisition, software, visualization, B.A.M.; methodology, funding acquisition, resources, Z.G.A.-M.; project administration, funding acquisition, software, A.Q.; funding acquisition, investigation, resources, A.J.A.; data curation, software, visualization, G.A.; visualization, methodology, visualization, supervision, A.A.S.; and investigation, methodology, validation, K.A. All authors have read and agreed to the published version of the manuscript. This research has been funded by the Scientific Research Deanship at the University of Ha'il, Saudi Arabia, through project number RG-21098.  
Not applicable.  
We would like to acknowledge the Scientific Research Deanship at the University of Ha'il, Saudi Arabia, for funding this research.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Schemes** & **Message Tuple** & **Size of Tuple (Bits)** & **Improvement** \\ \hline [PERSON] et al. (2015) & \(\{ID_{1}^{i},ID_{2}^{i},\sigma_{i},T_{i}\}\) & \(1024+160+160+32=1376\) & 74.42\% \\ [PERSON] et al. (2016) & \(\{ID_{1}^{i},ID_{2}^{i},\sigma_{i},T_{i}\}\) & \(1024+160+160+32=1376\) & 74.42\% \\ [PERSON] et al. (2018) & \(\{AID_{i,1},AID_{i,2},T_{i},R_{i},\sigma_{i}\}\) & \(320+160+32+320+160=992\) & 64.52\% \\ [PERSON] et al. (2019) & \(\{PID_{wi},T_{i},T_{wi},h_{ki},R_{i},\sigma_{i}\}\) & \(320+32+32+160+320+160=1024\) & 65.63\% \\ [PERSON] et al. (2011) & \(\{PID_{j},DT_{ij},\sigma_{j},D_{j},T_{j}\}\) & \(320+320+160+160+32=992\) & 64.52\% \\ Our work & \(\{PPID_{wi},T_{wi},\sigma_{wi}\}\) & \(160+32+160=352\) & \\ \hline \end{tabular}
\end{table}
Table 6: Comparison of cost of communication.