Provably secure non-interactive key distribution based on pairings

We define a security notion for non-interactive key distribution protocols. We identify an apparently hard computational problem related to pairings, the Bilinear Diffie-Hellman problem (BDH). After extending Sakai, Ohgishi, and Kasahara's pairing based protocol to a slightly more general setting, we show that breaking the system is polynomially equivalent to solving BDH in the random oracle model and thus establish a security proof.