Inner product functional encryption (IPFE) is a subclass of functional encryption (FE), whose function class is limited to inner product. We construct an efficient private-key IPFE scheme with full-hiding security, where confidentiality is assured for not only encrypted data but also functions associated with secret keys. Recently, Datta et al. presented such a scheme in PKC 2016 and this is the only scheme that achieves full-hiding security. Our scheme has an advantage over their scheme for the two aspects. More efficient: keys and ciphertexts of our scheme are almost half the size of those of their scheme. Weaker assumption: our scheme is secure under the k-linear (k-Lin) assumption, while their scheme is secure under a stronger assumption, namely, the symmetric external Diffie-Hellman (SXDH) assumption. It is well-known that the k-Lin assumption is equivalent to the SXDH assumption when k=1 and becomes weak as k increases.

Subspace membership encryption (SME), a generalization of inner product encryption (IPE), was recently formalized by Boneh, Raghunathan, and Segev in Asiacrypt 2013. The main motivation for SME was that traditional predicate encryptions did not yield function privacy, a security notion introduced by Boneh et al. in Crypto 2013 that captures the privacy of the predicate associated to the secret key. Although they gave a generic construction of SME based on any IPE, we show that their construction of SME for small attribute space was incorrect and provide an attack that breaks the attribute hiding security, a baseline security notion for predicate encryptions that captures the privacy of the attribute associated with the ciphertext. Then, we propose a generalized construction of SME and prove that the attribute hiding security can not be achieved even in the newly defined setting. Finally, we further extend our generalized construction of SME and propose a SME that achieves the attribute hiding property even when the attribute space is small. In exchange our proposed scheme does not yield function privacy and the construction is rather inefficient. Although we did not succeed in constructing a SME both yielding function privacy and attribute hiding security, ours is the first attribute hiding SME scheme whose attribute space is polynomial in the security parameter, and we formalized a richer framework for constructing SMEs and discovered a trade-off like relationship between the two security notions.