## 5 Communication Overhead Analysis  
In this section, we compare the proposed MAPP protocol and the certificateless bilinear pairing methods [19; 20; 21; 22; 23; 24]. To calculate the communication cost, we analyze the message structure of the MAPP protocol and the previous methods, based on the bilinear pairing [19; 20; 21; 22; 23; 24]. For the security overhead calculations, we exclude the size of the traffic message payload, since it is common for all methods. In our implementation, we define the elliptic curve equation \(E\) over a finite field \(F_{p}\), which is given by Equation (14). \[y^{2}=x^{3}+b\ mod\ p \tag{14}\]  
We employ the Barreto-Naehrig (BN256) curves that offer asymmetric bilinear pairing (e.g., \(G_{1}\
eq G_{2}\)) [34]. For BN256, we choose \(r=256\) bits to give finite field points \(2^{256}\) and \(b\) a random number \(\
eq 0\) to be a non-singular curve, which means that the curve has no cusps or self-intersections. The BN curve chooses \(b=2\) to satisfy the non-singular condition and give optimum security. \(G_{1}\) indicates the cyclic additive subgroup defined over \(F_{p}\),\(G_{2}\), denoting a cyclic additive subgroup defined over \(F_{p2}\), and \(G_{T}\) represents the cyclic multiplicative subgroup defined over \(F_{p12}\). \(G_{1}\), \(G_{2}\), and \(G_{T}\) are defined with order \(r\). \(G_{1}\) and \(G_{2}\) elements are represented in a compressed form by the values of the x-coordinate instead of representing them by (x, y, z), which reduces their sizes to 32 and 64 bytes, respectively [35]. Table 3 summarizes the BN256 curve's parameters and element size in the bilinear groups. \begin{table}
\begin{tabular}{c c} \hline \hline Pairing Curve equation & E: \(y^{2}=x^{3}+b\) \\ \hline Size of elements in the finite field \(F_{p}\) & \(\left\|F_{p}\right\|=32\) bytes \\ \hline \(b\) & Integer number over \(F_{p}\) where \(b=2\) \\ \hline Size of elements in the elliptic curve group \(G_{1}\) & \(\left\|G_{1}\right\|=32\) bytes \\ \hline Size of elements in the elliptic curve group \(G_{2}\) & \(\left\|G_{2}\right\|=2\)\(\left\|G_{1}\right\|=64\) bytes \\ \hline Size of elements in the result mapping group \(G_{T}\) & \(\left\|G_{T}\right\|=12\)\(\left\|G_{1}\right\|=384\) bytes \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The BN256 bilinear pairing curve parameters. In the following, we analyze the overhead message of 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. (1) [PERSON] et al. (19):  
The structure of the transmitted message is expressed by Equation (15):  
\[m=\{M_{i},ID_{i},\mathit{v}pk_{i},t_{i},\sigma_{i}\} \tag{15}\]  
where \(M_{i}\) is the message payload, \(ID_{i}\) represents the vehicle pseudo-identity, \(\mathit{v}pk_{i}\) represents the vehicle public key, \(t_{i}\) represents the time stamp, and \(\sigma_{i}\) represents the signature. \(ID_{i}\) consists of two parts \(\{ID_{i}^{1},ID_{i}^{2}\}\), where \(ID_{i}^{1}\in G_{2}\), \(ID_{i}^{2}\in F_{p}\), _and \(\mathit{v}pk_{i}\), \(\sigma_{i}\in G_{2}\)_. The total communication overhead for the message of Equation (15) is \(|ID_{i}|+|\mathit{v}pk_{i}|+|t_{i}|+|\sigma_{i}|=64+32+64+4+64=228\) bytes. (2) [PERSON] et al. (20):  
The structure of the transmitted message in (20) is also represented by Equation (15) and introduces a total communication overhead of 228 bytes. (3) [PERSON] et al. (21):  
The structure of the transmitted message is expressed by Equation (16):  
\[m=\big{\{}M_{i},PS_{j},PS_{1j},P_{i},t_{i},\sigma_{ijk}\big{\}} \tag{16}\]  
where \(\mathrm{M_{i}}\) is the message payload, while the vehicle's pseudo-identity is represented by two parts \(\{\mathrm{PS_{j}},\mathrm{PS_{ij}}\}\)\(\in G_{2}\). \(P_{i}\) represents the public key \(\in G_{2}\), and \(t_{i}\) represents the time stamp. \(\sigma_{ijk}\) represents the signature over the message and consists of two parts \(\{\mathrm{U_{i}},\mathrm{V_{ijk}}\}\)\(\in G_{2}\).