## 1 Introduction  
The Vehicular Ad Hoc Network (VANET), as a branch of Intelligent Transportation Systems (ITS), is a dynamic wireless network that seamlessly connects vehicles and roadside units (RSUs) to provide a wide range of traffic-related and entertainment applications. The VANET is designed to enhance traffic safety and efficiency by increasing the awareness of vehicles about their surrounding traffic [1]. The key aspect of VANET applications lies in communication, enabling vehicles to exchange information and interact with other vehicles (V2V), road infrastructure (V2I), and everything else (V2X) [2, 3, 4, 5]. This communication facilitates enhanced awareness of surrounding traffic conditions, thereby significantly improving traffic safety and efficiency [1, 6, 7]. Figure 1 illustrates the VANET scenario, wherein it is formed by a short-haul wireless network connecting RSUs, vehicles, and a backbone network. RSUs are established by trusted authorities for dedicated short-range communication (DSRC). Vehicles are equipped with On-board Units (OBUs) that possess unique IDs, primarily utilized for authentication purposes by the authority. This authentication process enables vehicles to establish safe connections with one another, either directly or by leveraging RSUs for re-transmission. In VANETs, vehicles broadcast safety messages to report accidents or congestion. Additionally, to support various functionalities within the VANET, vehicles are required to periodically broadcast beacons, known as Cooperative Awareness Messages (CAMs), to apprise others of their state information [8]. Despite the enhanced traffic safety and efficiency brought by periodically broadcasting beacons, the lack of security protection of vehicles in VANETs suffers from the problem of low data trustworthiness [9; 10]. There is growing concern about the potential threat to confidentiality, particularly regarding privacy issues [11; 12]. Due to the inherent openness of wireless communication in VANETs [13], attackers can intercept and eavesdrop on CAM beacons and exploit the state information they contain, such as vehicle ID, speed, GPS position, timestamp, and acceleration, to uncover a vehicle's trajectory. Consequently, this could result in severe consequences, such as unauthorized vehicle tracking, potential misuse of personal information, and even the facilitation of targeted attacks on individual vehicles or groups of vehicles. Therefore, VANETs have gained considerable research attention in addressing privacy concerns arising from beacon broadcasting while preserving the core functionality of the VANET [14]. Using pseudonyms instead of real IDs in authentication processes is  
Figure 1: The diagram of the VANET scenario. widely recognized as an effective measure to safegu vehicle privacy [15]. The European standard ETSI TS 102 941 [16] exemplifies the responsibility of the authority in issuing pseudonym certificates, which should be updated within a time frame of five minutes [17]. Despite the implementation of the pseudonym, the threat of pseudonym-linking attacks persists, enabling attackers to associate previous and new pseudonyms [18]. These attacks can take on a syntactic level, involving correlation of previous and current pseudonyms when a vehicle changes its pseudonym individually, or a semantic level, wherein the vehicle behavior and trajectory are estimated based on the state information of broadcast beacons to unveil identity and undermine pseudonym changes [19]. To combat pseudonym-linking attacks, various mix-zone-related approaches have been proposed [19; 20; 21]. As shown in Figure 1, vehicles collectively replace their pseudonyms upon leaving the mix zone by communicating with the mix-zone Roadside Unit (mRSU) [20]. Anonymity algorithms like k-anonymity [22] are employed to make the pseudonyms used within the mix zone unlinkable.