Boaz Barak, Shien Jin
We give two applications of Nisan-Wigderson-type
(“non-cryptographic”) pseudorandom generators in cryptography.
Specifically, assuming the existence of an appropriate NW-type generator, we
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 = TIME(2^O(n)) has a function of nondeterministic circuit complexity (Miltersen and Vinodchandran, FOCS 99). Our witness-indistinguishable proofs are obtained by using the NW-type generator to derandomize the ZAPs of Dwork and Naor (FOCS 00). 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, 1991). Previous constructions of noninteractive commitment schemes were only known under incomparable assumptions.