## 2 Related Work  
[PERSON] et al. [5] initially proposed a PEKS scheme based on bilinear mapping. However, subsequent studies have shown that the scheme has serious security problems, such as the need for secure channels to transmit trapdoors [23], keyword guessing attacks [7], and other problems. To address these problems, several researchers have proposed improvement schemes. [PERSON] et al. [23] designed an SCF-PEKS scheme that transmits keyword trapdoors without a secure channel. [PERSON] et al. [24] introduced the concept of registered keywords and required the sender to register the keyword with the receiver before generating the keyword ciphertext. [PERSON] et al. [25] introduced the concept of trapdoor indistinguishability and proposed a public key searchable encryption scheme (dPEKS) with a designated tester. Recently, [PERSON] et al. [26] proposed a hierarchical PEKS scheme (dDHHEKS) with decodable encryption for designated testers to satisfy security against external keyword guessing attacks. However, due to the powerful attack capabilities of cloud servers, the above schemes cannot address internal KGANs from malicious servers. Therefore, researchers have started to investigate PEKS schemes that can resist internal KGANs. [PERSON] et al. [9] constructed a public key encryption with fuzzy keyword search scheme (PEFKS), where each keyword corresponds to an exact keyword search trapdoor and a fuzzy keyword search trapdoor, which is effective against internal KGA but has a high communication overhead. [PERSON] et al. [10] proposed a server-assisted scheme (SA-PEKS). To solve the problem of internal KGA, some researchers have introduced the concept of public key authenticated encryption for keyword search (PAEKS). In 2021, [PERSON] et al. [27] proposed a new public key authenticated encryption with keyword search scheme that achieves both multi-ciphertext and multi-trapdoor indistinguishability. [PERSON] et al. [28] introduced an improved ciphertext (CI security) model for PAEKS to guarantee the indistinguishability of multiple cipher-keywords in a multi-user environment. [PERSON] et al. [11] proposed a certificateless public key authentication encryption with keyword search scheme (CLPAEKS), which is free of certificates and key management, while solving internal keyword attacks.  
In a system, an effective mechanism or policy is needed to ensure correct operation [29]. Verifiable keyword search is a technique used to ensure the accuracy of retrieval results and detect the dishonest or malicious behavior of cloud servers. Several researchers have proposed different verifiable keyword search schemes, such as [PERSON] et al.'s verifiable attribute-based keyword search (VABKS) [30] and [PERSON] et al.'s efficient verifiable connected keyword search (VCRS) for encrypted cloud data [31]. [PERSON] et al. [32] also developed a verifiable keyword search scheme with fine-grained authorization control using reversible Bloom lookup tables and Merkle hash trees. However, most of these schemes are vulnerable to keyword guessing attacks (KGA). To address this issue, [PERSON] et al. [33] created a basic verifiable search framework (VSEF) that solves both verifiable search and internal KGA problems.