## 2 Related Work  
We performed a thorough literature assessment of key searchable encryption (KSE) and verifiable key-aggregate searchable encryption. The concept of searchable encryption (SE), a search method that permits the searching of encrypted data while concurrently protecting individual privacy, was first formally described by [PERSON] et al. [13] with reference to key searchable encryption (KSE). Searchable symmetric encryption and searchable asymmetric encryption are the two main components of SE. However, the seeking of authorization for a particular dataset one of the disadvantages of these methods. In [14], researchers presented searchable encryption using key aggregation as a solution to this problem. For this approach to allow for search capabilities over a set of files, a \(dg_{i}\) is required give a user an aggregate key. Then, in order to perform searches on the shared dataset, the user has to send a trapdoor. However, this strategy is susceptible to both \(k\) assumption and attacks on the chosen \(k\). [PERSON] et al. [15] successfully addressed the identified vulnerabilities, especially those pertaining to guessing other users' private keys. Nevertheless, searching across multi-generator data is not supported by these systems. To overcome this limitation, [PERSON] et al. [16] proposed an efficient key-aggregate scheme that satisfies all key-aggregate efficiency requirements, allowing a data owner to enforce dynamic updates in user access rights much more efficiently than existing methods. Additionally, some schemes [17, 18] have been proposed with the aim of aggregating multi-generator keys. The authors of [17] developed a technique that enables users to utilize a query trapdoor to search through the document sets of multiple generators. This technique significantly enhances the query performance of current key searchable encryption (KSE) schemes. Later, [PERSON] et al. [18] succeeded in achieving a multi-generator setting, making sure that the aggregation key's size stayed constant, irrespective of the quantity of records. Similarly, [PERSON] et al. [14] proposed a method to produce a constant-size key. Although this approach successfully addresses the issue of key leakage during data exchange in the authentication process, it has a significant flaw. This flaw allows unauthorized individuals to access any arbitrary set of files stored in the fog server and forge the authentication key. Subsequently, [PERSON] et al. [19] developed a key aggregation searchable encryption system with auxiliary input, which may improve resistance to key disclosure. The previously described efforts accomplished aggregated keys. [PERSON] et al. [20] introduced a novel decentralized ciphertext policy attribute-based encryption scheme to address the single point of failure and the limited research focusing on the privacy concerns of user identity during the key generation process. However, they failed to consider the potential for vehicles to inadvertently disclose sensitive \(k\) information. Consequently, we propose an efficient and secure method for aggregated multi-key management. In the field of verifiable searchable key-aggregate encryption, [PERSON] et al. [21] proposed an effective technique to ensure the integrity of search results. They introduced appropriate confirmation flags generated by the \(dg_{i}\) with the assistance of a server. However, if the assistance server is compromised, the security of the system is at risk. To address this issue, [PERSON] et al. [7] suggested an efficient verifiable searchable encryption method utilizing aggregate keys. Additionally, [PERSON] et al. [19] employed a Bloom filter to validate the search results. [PERSON] et al. [22] introduced a certifiateless verifiable public \(k\)-searchable encryption scheme to mitigate the problem of a semi-trusted third party providing incorrect search results, thereby conserving computational and bandwidth resources. Nevertheless, due to its high false-positive rate, the Bloom filter may yield inaccurate verified results.