Existing semantically secure public-key searchable encryption schemes take search time linear with the total number of the ciphertexts. This makes retrieval from large-scale databases prohibitive. To alleviate this problem, this paper proposes searchable public-key ciphertexts with hidden structures (SPCHS) for keyword search as fast as possible without sacrificing semantic security of the encrypted keywords. In SPCHS, all keyword-searchable ciphertexts are structured by hidden relations, and with the search trapdoor corresponding to a keyword, the minimum information of the relations is disclosed to a search algorithm as the guidance to find all matching ciphertexts efficiently. We construct an SPCHS scheme from scratch in which the ciphertexts have a hidden star-like structure. We prove our scheme to be semantically secure in the random oracle (RO) model. The search complexity of our scheme is dependent on the actual number of the ciphertexts containing the queried keyword, rather than the number of all ciphertexts. Finally, we present a generic SPCHS construction from anonymous identity-based encryption and collision-free full-identity malleable identity-based key encapsulation mechanism (IBKEM) with anonymity. We illustrate two collision-free full-identity malleable IBKEM instances, which are semantically secure and anonymous, respectively, in the RO and standard models. The latter instance enables us to construct an SPCHS scheme with semantic security in the standard model.