## 4 Proposed Scheme
### Informal Analysis  
In this subsection, security goals should be satisfied in our work for 5G-enabled vehicular networks as below:  
* Satisfying Pseudonym Identity: The vehicle's original identity \(OID_{v_{i}}\) is hidden with an inter-pseudonym identity \(IPID_{vi}\) by the TA at the enrollment phase of our work. Before broadcasting the message to vehicle-to-everything (V2X) communication, the vehicle computes a public pseudonym identity \(PPID_{v_{i}}=IPID_{v_{i}}\oplus h(A_{v_{i}}||T_{v_{i}})\), where \(T_{v_{i}}\) is the latest timestamp and then sends the message-tuple \(\{PPID_{vi},\,M_{i},\,T_{vi},\,\sigma_{vi}\}\). Therefore, our work is safe, and the adversary does not have the capability to attack the scheme to obtain the vehicle's identity from the message. * Satisfying Authentication and Integrity: In our work, the recipient verifies the node authentication and message integrity by checking through \(\mathcal{T}_{PPID_{v_{i}}SM^{-}_{vi}}(Pr_{vi})\stackrel{{? }}{{=}} \mathcal{T}_{PPID_{v_{i}}}(\sigma_{vi})\). After completing the verification process, the vehicle then accepts the safety-related message \(M_{i}\) included of the message-tuple \(\{PPID_{vi},\,M_{i},\,T_{vi},\,\sigma_{vi}\}\). Therefore, our work is achieving requirements of authentication and integrity in 5G-enabled vehicular networks. * Satisfying Traceability: The original identity of vehicle \(OID_{v_{i}}\) is hid in an inter-pseudonym identity \(IPID_{v_{i}}\), where \(IPID_{v_{i}}=h(OID_{v_{i}}||A_{v_{i}}||SVP_{v_{i}})\). Due to the \(IPID_{vi}\) being hid in \(PPID_{vi}\) of the message-tuple \(\{PPID_{vi},\,M_{i},\,T_{vi},\,\sigma_{vi}\}\), the attacker does not have the ability to disclose it. The TA is only able to retrieve the original identity by computing \(IPID_{v_{i}}=PPID_{v_{i}}\oplus h(A_{v_{i}}||T_{v_{i}})\), where \(A_{v_{i}}\) is stored on the TA. Therefore, our work is satisfying traceability requirement. * Satisfying Unlinkability: In our work, the vehicle generates public pseudonym identity \(PPID_{v_{i}}=IPID_{v_{i}}\oplus h(A_{v_{i}}||T_{v_{i}})\) by the chaotic map based a hash function \(h\), and the adversary does not have the ability to determine two \(PPID_{v_{i}}\) that are issued from the same enrolled vehicle. Hence, our work satisfies the unlinkability requirement in 5G-enabled vehicular networks. * Resisting Side-Channel Attack: The assumption of the existing authentication scheme is that a TPD is not compromised by an adversary; thereby, the secret key of the system (TA) is preloaded and saved in the enrolled vehicle. Nevertheless, the adversary could conduct malicious activities by launching an attack on a side channel to acquire the sensitive data preserved on TPD. Then, the adversary impersonates a legal vehicle and sends fake messages to collapse the system of vehicular networks. In our work, before the short valid period \(SVP_{v_{i}}\) is to expire, the inter-pseudonym identity \(IPID_{v_{i}}\) is regularly renewed to be new \(IPID_{v_{i}}^{new}\) to resist side-channel attack. Therefore, our work is safe from the side-channel attack, and the adversary does not have the ability to attack the scheme to acquire the sensitive information saved on the TPD of enrolled vehicles in 5G-enabled vehicular networks. * Resisting Replay Attack: The message-tuple \(\{PPID_{vi},\,M_{i},\,T_{vi},\,\sigma_{vi}\}\) sent by the enrolled vehicle includes a timestamp \(T_{vi}\); the recipient has the ability to check if the message is replayed by verifying the timestamp \(T_{vi}\). Therefore, our work is safe for a replay attack, and the adversary does not have the ability to attack the scheme to replay the message in 5G-enabled vehicular networks. * Resisting Modify Attack: The message-tuple \(\{PPID_{yi},\,M_{i},\,T_{vi},\,\sigma_{vi}\}\) sent by the enrolled vehicle includes a signature \(\sigma_{vi}\); the recipient has the ability to check if the message is modify by verifying the \(\sigma_{vi}\), as shown in Equation (3).