Public-key encryption with keyword search (PEKS) is a cryptographic primitive that allows us to search for particular keywords over ciphertexts without recovering plaintexts. By using PEKS in cloud services, users can outsource their data in encrypted form without sacrificing search functionality. Concerning PEKS that can specify logical disjunctions and logical conjunctions as a search condition, it is known that such PEKS can be (generically) constructed from anonymous attribute-based encryption (ABE). However, it is not clear whether it is possible to construct this types of PEKS without using ABE which may require large computational/communication costs and strong mathematical assumptions. In this paper, we show that ABE is crucial for constructing PEKS with the above functionality. More specifically, we give a generic construction of anonymous key-policy ABE from PEKS whose search condition is specified by logical disjunctions and logical conjunctions. Our result implies such PEKS always requires large computational/communication costs and strong mathematical assumptions corresponding to those of ABE.