This paper revisits the concept of searchable encryption, focusing on public-key encryption with keyword search (PEKS) and its consistency properties. The authors identify and address gaps in the existing notion of perfect consistency, defining computational and statistical relaxations. They show that the Boneh et al. (BDOP) PEKS scheme is computationally consistent but not statistically consistent. A new PEKS scheme, PEKS-STAT, is introduced, which is statistically consistent and secure under the random oracle model. The paper also explores the connection between PEKS and anonymous identity-based encryption (IBE), proposing a transform from IBE to PEKS that guarantees consistency. Additionally, the authors discuss extensions to anonymous hierarchical identity-based encryption, public-key encryption with temporary keyword search, and identity-based encryption with keyword search. The paper provides formal definitions, proofs, and experimental results to support these findings.This paper revisits the concept of searchable encryption, focusing on public-key encryption with keyword search (PEKS) and its consistency properties. The authors identify and address gaps in the existing notion of perfect consistency, defining computational and statistical relaxations. They show that the Boneh et al. (BDOP) PEKS scheme is computationally consistent but not statistically consistent. A new PEKS scheme, PEKS-STAT, is introduced, which is statistically consistent and secure under the random oracle model. The paper also explores the connection between PEKS and anonymous identity-based encryption (IBE), proposing a transform from IBE to PEKS that guarantees consistency. Additionally, the authors discuss extensions to anonymous hierarchical identity-based encryption, public-key encryption with temporary keyword search, and identity-based encryption with keyword search. The paper provides formal definitions, proofs, and experimental results to support these findings.