## 5 Analyses
### Security and Privacy-Preservation Analyses  
This part discusses the security and privacy-preservation features satisfied by the proposed scheme, specifically this is in respect to anonymity (identity privacy), message authentication, data integrity, traceability, unlinkability and resistance to attacks. * Anonymity: In the proposed scheme the vehicle's identification \(ID_{i}\) is not the real identification \(RID_{i}\), but rather a pseudo-identity as offered by the TRA for purposes of achieving conditional privacy of the vehicle in VANETs. The only way for an adversary or any malicious party to obtain the real identity it by computing \(RID_{i}=ID_{i}\oplus H_{1}(\beta.PID_{1}||T_{i}||T_{pub})\). Without knownledge of the TRA's master private key \(\beta\), no other party can know the vehicle's real identity \(RID_{i}\), since it requires \(\beta\) to calculate \(H_{1}(\beta.PID_{1}||T_{i}||T_{pub})\). This manipulation is infeasible for an adversary to achieve since the extraction of \(\beta\) from \(T_{pub}=\beta.P\), involves an intractable ECDL problem. Therefore, these claims ascertain the satisfaction of user identity privacy-preservation. * Message Integrity and Authentication: By virtue of signing a message before broadcasting, the legitimate user's authenticity is verified. Based on the ECDLP assumption the authenticity and integrity of the message \((ID_{i},Q_{ID_{i}},vpk_{i},M_{i},t_{i},\sigma_{i})\) is upheld by verifying the computation \(S_{i}.P=h_{i}.R_{i}+vpk_{i}+Q_{ID_{i}}+h_{i,0}.P_{pub}\). Since \(h_{i}=H_{3}(M_{i}||D_{i}||Q_{ID_{i}}||vpk_{i}||R_{i}||t_{i})\) and \(h_{i,0}=H_{2}(ID_{i}||Q_{ID_{i}})\), no malicious party can forge \(\sigma_{i}=(R_{i},S_{i})\) which achieves the message integrity and authentication of which needs knowledge of full private key \(sk_{i}=x_{i}+psk_{i}\) in its formulation. * Traceability: Although the vehicle is identified by a pseudonym, in necessary circumstances the real identity of a particular vehicle can be mapped back from the pseudonym. For instance, the pseudo-identity of a vehicle is \(ID_{i}=(PID_{1}||PID_{2}||T_{i})\) and the TRA can revoke the real identity by calculating \(PID_{2}=RID_{i}\oplus H_{1}(\beta.PID_{1}||T_{i}||T_{pub})\). As such, once a vehicle is flagged as questionable the TRA is able to trace its true identity and thereby carrying out whatever necessary procedures to curb any kind of malpractice. Once this is done the TRA records the real identity \(RID_{i}\) on the revocation list of the system and as a result the vehicle cannot use its corresponding pseudo-identity \(ID_{i}\). * Unlinkability: The message transmitted \((ID_{i},Q_{ID_{i}},vpk_{i},M_{i},t_{i},\sigma_{i})\) from a vehicle \(V_{i}\) to others has the component \(PID_{1}=k_{i}P\), where \(k_{i}\in Z_{q}^{*}\) is random, that is randomly generated for any particular message transmitted. Since the \(PID_{1}\) is also a component for pseudo-identity generation, it means the randomness in \(PID_{1}\) results in the randomness of the publicized pseudo-identity \(ID_{i}\), hence, any two individual captures of the pseudo-identity \(ID_{i}\) for \(V_{i}\) stills seem random and unrelated to the real identity \(RID_{i}\), in the eyes of eavesdroppers. So by virtue of the identification being anonymous and distinct any captured signatures cannot be linked to previously captured identity nor to a particular true signer. Thus, any communication is seen as random and new in the plying eyes of an adversary and has no any relationship to previous communications for an eavesdropper to learn any useful information from such communication. * Resistance to Attacks: At this point we will present a demonstration of how the proposed ECLAS scheme can resist the main common attacks such as--replay attack, modification attack, impersonation attack, and stolen verifier attack. * Replay Attack Resilience: In the message \((ID_{i},Q_{ID_{i}},\mathit{vpk}_{i},M_{i},t_{i},\sigma_{i})\) the \(t_{i}\) in the message helps in checking replay attacks.