Derandomization in Cryptography

Boaz Barak, Shien Jin Ong, Salil P. Vadhan

Abstract: 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 construct: 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 2Ω(n) (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.