Searchable encryption is a cryptographic primitive that allows us to perform searches on encrypted data. Searchable encryption schemes require that ciphertexts do not leak information about keywords. However, most of the existing schemes do not achieve the security notion that trapdoors do not leak information about the searched keyword. Shen et al. (TCC 2009) proposed a security notion called full security, which includes both ciphertext privacy and trapdoor privacy, but there are few fully secure constructions. Full security is defined for the secret key setting since it is known that public key schemes cannot achieve the trapdoor privacy in principle. In this paper, we construct a query-bounded fully secure scheme from pseudorandom functions. In addition, we propose three types of efficient (unbounded) fully secure schemes. One of them is based on bilinear groups, and the others are based on lattices. We then analyze the existing constructions. First, we simplify the Cheng et al. scheme (Information Sciences 2023) and prove its security. This scheme had not been proved to be secure. Second, we show that the Li-Boyen pairing-based scheme (IACR CiC 2024) does not achieve the trapdoor privacy, not as claimed.