This paper proposes an id-eCK secure identity-based authenticated key exchange (ID-AKE) scheme, where the id-eCK security implies that a scheme resists against leakage of all combinations of master, static, and ephemeral secret keys except ones trivially break the security. Most existing id-eCK secure ID-AKE schemes require two symmetric pairing operations or a greater number of asymmetric pairing, which is faster than symmetric one, operations to establish a session key. However, our scheme is realized with a single asymmetric pairing operation for each party, and this is an advantage in efficiency. The proposed scheme is based on the ID-AKE scheme by McCullagh and Barreto, which is vulnerable to an active attack. To achieve id-eCK security, we apply the HMQV construction and the NAXOS technique to the McCullagh-Barreto scheme. The id-eCK security is proved under the external Diffie-Hellman for target group assumption and the q-gap-bilinear collision attack assumption.

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.