Group signature where group manager, members and Open Authority are identity-based

We present the first group signature scheme with provable security and signature size O(λ) bits where the group manager, the group members, and the Open Authority (OA) are all identity-based. We use the security model of Bellare, Shi, and Zhang [3], except to add three identity managers for manager, members, and OA respectively, and we discard the Open Oracle (OO). Our construction uses identity-based signatures summarized in Bellare, Namprempre, and Neven [2] for manager, Boneh and Franklin's IBE [7] for OA, and we extend Bellare et al.[3]'s group signature construction by verifiably encrypt an image of the member public key, instead of the public key itself. The last innovation is crucial in our efficiency; otherwise, Camenisch and Damgard[9]'s verifiable encryption would have to be used resulting in lower efficiency.