Concurrent zero knowledge with logarithmic round-complexity

We show that every language in NP has a (black-box) concurrent zero-knowledge proof system using O˜(log n) rounds of interaction. The number of rounds in our protocol is optimal, in the sense that any language outside BPP requires at least Ω˜(log n) rounds of interaction in order to be proved in black-box concurrent zero-knowledge. The zero-knowledge property of our main protocol is proved under the assumption that there exists a collection of claw free functions. Assuming only the existence of one-way functions, we show the existence of O˜(log n)-round concurrent zero-knowledge arguments for all languages in NP.