We give direct constructions of pseudorandom function (PRF) families based on conjectured hard lattice problems and learning problems. Our constructions are asymptotically efficient and highly parallelizable in a practical sense, i.e., they can be computed by simple, relatively small low-depth arithmetic or boolean circuits (e.g., in NC 1 or even TC 0 ). In addition, they are the first low-depth PRFs that have no known attack by efficient quantum algorithms. Central to our results is a new "derandomization" technique for the learning with errors (LWE) problem which, in effect, generates the error terms deterministically.

We show that every language in N P has a (black-box) concurrent zero-knowledge proof system usingÕ(log n) rounds of interaction. The number of rounds in our protocol is optimal, in the sense that any language outside BPP requires at leastΩ(log n) rounds of interaction in order to be proved in black-box concurrent zero-knowledge. The zeroknowledge 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 ofÕ(log n)-round concurrent zero-knowledge arguments for all languages in N P.

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