# An Efficient and Secure Certificateless Aggregate Signature Scheme for Vehicular Ad hoc Networks
## 4 Proposed Scheme's Construction  
The proposed scheme consists of the steps listed below, and Table 1 outlines the fundamental symbols utilized in its construction. The construction of the proposed scheme is also illustrated in Figure 3. \begin{table}
\begin{tabular}{l c c} \hline _No_ & _Notation_ & _Descriptions_ \\ \hline
1 & (H\({}_{\text{J}per}\)) & _The hyperelliptic curve of genus 2_ \\ \hline
2 & F\({}^{\text{eild}}\)p & _A finite field of the hyperelliptic curve with order p_ \\ \hline
3 & Dotp & _The private key of DoT_ \\ \hline
4 & Dotp & _The public key of DoT_ \\ \hline
5 & D & _Divisor on hyperelliptic curve_ \\ \hline
6 & H\({}_{\text{01}}\), H\({}_{\text{02}}\), H\({}_{\text{03}}\) & _Hash Function with irreversibility_ \\ \hline \end{tabular}
\end{table} 
Table 1: Symbols used for the construction of the proposed scheme. Figure 2: Network model of the proposed CLAS scheme. \begin{table} \begin{tabular}{l l l} \hline \hline _No_ & _Notation_ & _Descriptions_ \\ \hline \(7\) & \(\mathrm{PB}_{\mathrm{frm}}\) & _Public parameter (param)_ \\ \hline \(8\) & \(\mathrm{U}_{\mathrm{sr}}\) & _Represents the participating user_ \\ \hline \(9\) & \(\mathrm{G}_{\mathrm{usr}}\) & _DoT, the random value selected by user_ \\ \hline _10_ & \(\mathrm{K}_{\mathrm{usr}}\) & _A secret shared key between user and DoT_ \\ \hline _11_ & \(\mathrm{EID}_{\mathrm{usr}}\) & _Encrypted identity of user_ \\ \hline _12_ & \(\mathrm{(F}_{\mathrm{usr}},\mathrm{L}_{\mathrm{usr}})\) & _Public key pair of users_ \\ \hline _13_ & \(\mathrm{(G}_{\mathrm{usr}},\mathrm{P}_{\mathrm{usr}})\) & _Private key pair of users_ \\ \hline _14_ & \(\mathrm{SOBU}\) & _Represent signature generated by OBU_ \\ \hline _15_ & \(\mathrm{P}_{\mathrm{usr}}\) & _Partial Private key of users_ \\ \hline _16_ & \(\mathrm{E}_{\mathrm{K}_{\mathrm{usr}}}\) & _Represents an encryption procedure_ \\ \hline _17_ & \(\mathrm{D}_{\mathrm{K}_{\mathrm{usr}}}\) & _Represents the decryption procedure_ \\ \hline _18_ & \(\mathrm{ID}_{\mathrm{usr}}\) & _Identity of user_ \\ \hline \hline \end{tabular} 
\begin{tabular}{|l l l l l|} \hline \hline \multicolumn{2}{|l|}{**OBU**} & \multicolumn{2}{l|}{**DoT**} & \multicolumn{2}{l|}{**RSU**} \\ \multicolumn{2}{|l|}{\(\bullet\) Select \(G_{\mathit{G}_{\mathit{G}_{\mathit{G}_{\mathit{G}_{\mathit{G}_{\mathit{G}_{ \mathit{G}_{\  
Setup: The DoT can play the role of a TA, and it will be able to run this phase upon receiving the security parameter \(k\), in which it can first choose the hyperelliptic curve (\(H_{\mathit{hyper}}\)) that defines field \(F^{\mathit{ield}}{}_{p}\), a devisor of 80 bis, and then it chooses \(Dot_{p}\) as a private key from \(F^{\mathit{ield}}{}_{p}\) and computes \(Dot_{pb}=Dot_{p}.D\) as a public key. Further, it can choose three hash functions, \(H_{01}\), \(H_{02}\), _and_\(H_{03}\) and the public param \(PB_{\mathit{frm}}=\big{\{}H_{\mathit{hyper}},D,H_{01}\,,\,H_{02},H_{03},F^{ \mathit{ield}}{}_{p},Dot_{pb}\big{\}}\). Partial Private Key Generation (PRPKG): Any participating user \(U_{\mathit{sr}}\) who needs a partial private key can first select \(G_{\mathit{usr}}\) from \(F^{\mathit{ield}}{}_{p}\), compute \(K_{\mathit{usr}}=G_{\mathit{usr}}.Dot_{pb}\), calculate \(EID_{\mathit{usr}}=E_{\mathit{K_{\mathit{usr}}}}. (ID_{\mathit{usr}})\), compute \(F_{\mathit{usr}}=G_{\mathit{usr}}.D\), and send \((EID_{\mathit{usr}},F_{\mathit{usr}})\) to DoT through the internet. Upon receiving \((EID_{\mathit{usr}},F_{\mathit{usr}})\), DoT can compute \(K_{\mathit{usr}}=F_{\mathit{usr}}.Dot_{p}\), recover user identity as \(ID_{\mathit{usr}}=D_{\mathit{K_{\mathit{usr}}}}. (EID_{\mathit{usr}})\), compute \(H_{\mathit{1 usr}}=H_{01}(K_{\mathit{usr}})\), \(H_{\mathit{2 usr}}=H_{02}\Big{(}EID_{\mathit{usr}},F_{\mathit{usr}},Dot_{ pb}\Big{)}\), select \(J_{\mathit{usr}}\) from \(F^{\mathit{ield}}{}_{p}\), compute \(P_{\mathit{usr}}=J_{\mathit{usr}}+Dot_{p}.H_{\mathit{2 usr}}\), calculate \(L_{\mathit{usr}}=J_{\mathit{usr}}.D\), encrypt \(PPK_{\mathit{usr}}=E_{\mathit{K_{\mathit{usr}}}}.