The study of security and privacy in vehicular ad hoc networks (VANETs) has become a hot topic that is wide open to discussion. As the quintessence of this aspect, authentication schemes deployed in VANETs play a substantial role in providing secure communication among vehicles and the surrounding infrastructures. Many researchers have proposed a variety of schemes related to information verification and computation efficiency in VANETs. In 2018, Kazemi et al. proposed an evaluation and improvement work towards Azees et al.'s efficient anonymous authentication with conditional privacy-preserving (EAAP) scheme for VANETs. They claimed that the EAAP suffered from replaying attacks, impersonation attacks, modification attacks, and cannot provide unlinkability. However, we also found out if Kazemi et al.'s scheme suffered from the unlinkability issue that leads to a forgery attack. An adversary can link two or more messages sent by the same user by applying Euclid's algorithm and derives the user's authentication key. To remedy the issue, in this paper, we proposed an improvement by encrypting the message using a shared secret key between sender and receiver and apply a Nonce in the final message to guarantee the unlinkability between disseminated messages.