Public-key encryption with keyword search (PEKS) is a cryptographic primitive that allows us to search encrypted data for those of including particular keywords without decrypting them. PEKS is expected to be used for enhancing security of cloud storages. It is known that PEKS can be constructed from anonymous identity-based encryption (IBE), anonymous attribute-based encryption (ABE) and so on. It is believed that it is difficult to construct PEKS schemes that can specify a flexible search condition such as logical disjunctions and logical conjunctions from weaker cryptographic tools than ABE. However, this intuition has not been rigorously justified. In this paper, we formally prove it by constructing key-policy ABE from PEKS for monotone boolean formulas.