Derandomization in Cryptography

Boaz Barak, Shien Jin Ong, Salil P. Vadhan

Abstract: We give two applications of Nisan–Wigderson‐type (NW‐type) (“noncryptographic”) pseudorandom generators in cryptography. Specifically, assuming the existence of an appropriate NW‐type generator, we construct the following two protocols: (1) a one‐message witness‐indistinguishable proof system for every language in NP, based on any trapdoor permutation. This proof system does not assume a shared random string or any setup assumption, so it is actually an “NP proof system.” (2) a noninteractive bit‐commitment scheme based on any one‐way function. The specific NW‐type generator we need is a hitting set generator fooling nondeterministic circuits. It is known how to construct such a generator if $E = DTIME(2^{O(n)})$ has a function of nondeterministic circuit complexity $2^{\Omega(n)}$. Our witness‐indistinguishable proofs are obtained by using the NW‐type generator to derandomize the ZAPs of Dwork and Naor [Proceedings of the 41st Annual ACM Symposium on Foundations of Computer Science, 2000, pp. 283–293]. To our knowledge, this is the first construction of an NP proof system achieving a secrecy property. Our commitment scheme is obtained by derandomizing the interactive commitment scheme of Naor [J. Cryptology, 4 (1991), pp. 151–158]. Previous constructions of noninteractive commitment schemes were known only under incomparable assumptions.