We propose short group signature (GS) schemes which are provably secure without random oracles. Our basic scheme is about 14 times shorter than the Boyen-Waters GS scheme at Eurocrypt 2006, and 42% shorter than the recent GS schemes due to Ateniese et al. The security proofs are provided in the Universally Composable model, which allows the proofs of security valid not only when our scheme is executed in isolation, but also in composition with other secure cryptographic primitives. We also present several new computational assumptions and justify them in the generic group model. These assumptions are useful in the design of high-level protocols and may be of independent interest.