We identify and fill some gaps with regard to consistency (the extent to which false positives are produced) for public-key encryption with keyword search (PEKS), providing new definitions, consistency proofs for existing schemes, and a new scheme with stronger consistency properties than previous ones. We provide a transform of an anonymous IBE scheme to a secure PEKS scheme that, unlike the previous one, guarantees consistency. Finally we define and implement three extensions of the basic notions considered here, namely anonymous HIBE, public-key encryption with temporary keyword search, and identity-based publickey encryption with keyword search. Keywords. Foundations, Random-Oracle Model, Anonymity, IdentityBased Encryption, Searchable Encryption.