The Internet of Things (IoT) is a technological paradigm that has gained significant momentum the last decade and, among other features, enables the development of intelligent and interoperable device networks. In this respect, it has triggered the creation and evolution of vehicular ad-hoc networks (VANETs), which are initially implemented in order to guarantee the safety of drivers and the avoidance of traffic accidents. The drawback is that this fast evolution comes with serious concerns in terms of the privacy of users, while the population of attackers or entities that try to eavesdrop and intercept information has significantly increased. This imposes a serious risk for drivers moving across a Smart City. The research presented in this paper aims to evaluate privacy protection mechanisms in VANET environments, based on the efficiency and security level they ensure, considering the fact that VANETs provide limited resources to users/drivers. Moreover, the usage of elliptic curve cryptography in reduced resources environments is discussed. Finally, this paper compares the performance of three cryptographic algorithms, elliptic curve cryptography (ECC), hyperelliptic curve cryptography genus 2 (HECC-2) and HECC genus 3 (HECC-3), employed for an efficient authentication and safe message transmission mechanism in VANETs, aimed at reaching conclusions related to the implementation of each cryptographic scheme in this specific application area. The evaluation results indicate that ECC supersedes HECC-2 and HECC-3 in most metrics. However, HECC-2 and HECC-3 demonstrate better responses than ECC does in selected energy metrics. Overall, it is observed that HECC algorithms are not yet mature enough to compete with ECC. This is due to the fact that the research community has not sufficiently progressed toward the optimization of HECC, and moreover, HECC builds on quite complex mathematics. There are indications, however, that once HECC curves are indeed optimized, HECC will outperform ECC in speed as well as in other metrics, sinceHECC-2 and HECC-3 use a significantly smaller key size with the same level of security as that of ECC.