Concurrent zero knowledge with logarithmic round-complexity

We show that every language in NP has a (black-box) concurrent zero-knowledge proof system using Õ(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 Ω̃(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 Õ(log n)-round concurrent zero-knowledge arguments for all languages in NP.