Predicate encryption is a new paradigm of public key encryption that enables searches on encrypted data. Using the predicate encryption, we can search keywords or attributes on encrypted data without decrypting the ciphertexts. In predicate encryption, a ciphertext is associated with attributes and a token corresponds to a predicate. The token that corresponds to a predicate $f$ can decrypt the ciphertext associated with attributes $x$ if and only if $f(x)=1$. Hidden vector encryption (HVE) is a special kind of predicate encryption. In this thesis, we consider the efficiency, the generality, and the security of HVE schemes. The results of this thesis are described as follows. The first results of this thesis are efficient HVE schemes where the token consists of just four group elements and the decryption only requires four bilinear map computations, independent of the number of attributes in the ciphertext. The construction uses composite order bilinear groups and is selectively secure under the well-known assumptions. The second results are efficient HVE schemes that are secure under any kind of pairing types. To achieve our goals, we proposed a general framework that converts HVE schemes from composite order bilinear groups to prime order bilinear groups. Using the framework, we convert the previous HVE schemes from composite order bilinear groups to prime order bilinear groups. The third results are fully secure HVE schemes with short tokens. Previous HVE schemes were proven to be secure only in the selective security model where the capabilities of the adversaries are severely restricted. Using the dual system encryption techniques, we construct fully secure HVE schemes with match revealing property in composite order groups.