Non-interactive zero-knowledge proofs play an essential role in many cryptographic protocols. We suggest several NIZK proof systems based on prime order groups with a bilinear map. We obtain linear size proofs for relations among group elements without going through an expensive reduction to an NP-complete language such as Circuit Satisfiability. Security of all our constructions is based on the decisional linear assumption.
The NIZK proof system is quite general and has many applications such as digital signatures, verifiable encryption and group signatures. We focus on the latter and get the first group signature scheme satisfying the strong security definition of Bellare, Shi and Zhang  in the standard model without random oracles where each group signature consists only of a constant number of group elements.
We also suggest a simulation-sound NIZK proof of knowledge, which is much more efficient than previous constructions in the literature.
Caveat: The constants are large, and therefore our schemes are not practical. Nonetheless, we find it very interesting for the first time to have NIZK proofs and group signatures that except for a constant factor are optimal without using the random oracle model to argue security.