The paper revisits searchable encryption with keyword search (PEKS), focusing on consistency properties and their relation to anonymous identity-based encryption (IBE). It identifies gaps in the existing notion of perfect consistency for PEKS and proposes computational and statistical relaxations. The Boneh et al. (BDOP-PEKS) scheme is shown to be computationally consistent but not statistically consistent. A new statistically consistent PEKS scheme is introduced, and a transformation from anonymous IBE to secure PEKS is proposed, ensuring consistency. The paper also explores three extensions: anonymous hierarchical IBE, public-key encryption with temporary keyword search (PETKS), and identity-based encryption with keyword search (IBEKS). It discusses the implications of these results for practical applications, such as secure email routing and encrypted audit logs. The paper defines new consistency notions, including perfect, statistical, and computational consistency, and shows how they apply to PEKS schemes. It also addresses the connection between PEKS and IBE, demonstrating how anonymous IBE can be used to construct secure PEKS schemes. The paper concludes with a discussion of the implications of these results for future research and practical implementations of searchable encryption.The paper revisits searchable encryption with keyword search (PEKS), focusing on consistency properties and their relation to anonymous identity-based encryption (IBE). It identifies gaps in the existing notion of perfect consistency for PEKS and proposes computational and statistical relaxations. The Boneh et al. (BDOP-PEKS) scheme is shown to be computationally consistent but not statistically consistent. A new statistically consistent PEKS scheme is introduced, and a transformation from anonymous IBE to secure PEKS is proposed, ensuring consistency. The paper also explores three extensions: anonymous hierarchical IBE, public-key encryption with temporary keyword search (PETKS), and identity-based encryption with keyword search (IBEKS). It discusses the implications of these results for practical applications, such as secure email routing and encrypted audit logs. The paper defines new consistency notions, including perfect, statistical, and computational consistency, and shows how they apply to PEKS schemes. It also addresses the connection between PEKS and IBE, demonstrating how anonymous IBE can be used to construct secure PEKS schemes. The paper concludes with a discussion of the implications of these results for future research and practical implementations of searchable encryption.