## 6 Computation Overhead Analysis  
To evaluate the computation overhead of the proposed protocol, we implemented it using a simplified C-V2X protocol simulator and a bilinear pairing cryptography library called MCL [36]. MCL implements functions for the elliptic curve cryptography that supports optimal pairing over (BN) curves. We chose a BN curve with an embedding degree \(k=12\), which supports a 128-bit security level over a prime field of size 256 bits [37].  
To compare the computation overhead, we also implemented the six previous methods [19; 20; 21; 22; 23; 24] that were analyzed in Section 4. We tested all methods under the same experimental environment for a fair comparison. Table 4 shows the average execution time of the primary cryptographic operations of the BN256 curve in our simulator. The simulations are conducted in a hardware platform employing an Intel Core I7-4770 processor with a 3.6 GHz clock, Linux gcc.5.4.0, and a main memory of 4 GB.  
Figure 12: Comparison of communication cost for the proposed authentication methods and the compared security protocols for the multi-zone scenario.  
In the following, we analyze the signature generation time and signature verification time of the six previous methods--(1) [PERSON] et al. [19], (2) [PERSON] et al. [20], (3) [PERSON] et al. [21], (4) [PERSON] et al. [22], (5) [PERSON] et al. [23], and (6) [PERSON] et al. [24], as well as the proposed MAPP for the single-zone and multi-zone scenarios. We analyze the computation overhead for signature generation and signature verification. In [PERSON] et al.'s protocol [19], the signature generation per message requires two scalar multiplication and two addition in the group \(G_{1}\), which results in a computation overhead of \(2\text{TM}_{\text{G1}}+2\text{TA}_{\text{G1}}=0.9594\) ms. Its signature verification per message requires three pairings, one scalar multiplication in \(G_{1}\), and one hash map, which results in a computation overhead of \(3\text{TP}+\text{TM}_{\text{G1}}+\text{TM}_{\text{2P}}=3\times 2.446+0.479+0.135=7.8305\) ms.  
Similarly, we analyzed the signature generation time and verification time of the other previous methods [20; 21; 22; 23; 24]. The analysis results are summarized by the formulas in Table 5.  
Finally, we analyzed the signature generation time and verification time for the three proposed MAPP protocol.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation** & **Definition** & **Time in (ms)** \\ \hline \multirow{4}{*}{Bilinear pairing} & The time needed to perform & \\  & one bilinear pairing of & \\  & elements from group \(G_{1}\) and & \\  & group \(G_{2}\) to group \(G_{T}\) & \\ \hline \multirow{2}{*}{Addition in \(G_{1}\)} & The time of addition of two & \\  & points inside group \(G_{1}\) & TAG1 = 0.007 \\ \hline \multirow{2}{*}{Multiplication in \(G_{1}\)} & Scalar multiplication of a & \\  & point inside group \(G_{1}\)and a & \\  & random integer & \\ \hline \multirow{2}{*}{Addition in \(G_{2}\)} & The time of addition of two & \\  & points inside group \(G_{2}\) & TAG2 = 0.013 \\ \hline \multirow{2}{*}{Multiplication in \(G_{2}\)} & Scalar multiplication of a & \\  & point inside group \(G_{2}\) and a & \\  & random integer & \\ \hline \multirow{2}{*}{HashAndMap to \(G_{1}\) or \(G_{2}\)} & The time of hashing message & \\  & using sha-256 then map the & \\  & hashed result to a point in the & \\  & group \(G_{1}\) or group \(G_{2}\) & \\ \hline \multirow{2}{*}{Hashing Operation} & the time defined for one hash & \\  & function operation using & \\  & SHA-256 algorithm & \\ \hline \multirow{2}{*}{Addition in \(F_{p}\)} & The time of addition of two & \\  & points over the finite filed \(F_{p}\) & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Average execution time of the BN254 pairing functions used in our simulation.