Cryptography in NC ⁰ (extended abstract)

We study the parallel time-complexity of basic crypto-graphic primitives such as one-way functions (OWFs) and pseudorandom generators (PRGs). Specifically, we study the possibility of computing instances of these primitives by NC ⁰ circuits, in which each output bit depends on a constant number of input bits. Despite previous efforts in this direction, there has been no significant theoretical evidence supporting this possibility, which was posed as an open question in several previous works. We essentially settle this question by providing over-whelming positive evidence for the possibility of cryptography in NC ⁰. Our main result is that every "moderately easy" OWF (resp., PRG), say computable in NC ¹, can be compiled into a corresponding OWF (resp., low-stretch PRG) in NC ₄ ⁰, i. e. whose output bits each depend on at most 4 input bits. The existence of OWF and PRG in NC ¹ is a relatively mild assumption, implied by most number-theoretic or algebraic intractability assumptions commonly used in cryptography. Hence, the existence of OWF and PRG in NC ⁰ follows from a variety of standard assumptions. A similar compiler can also be obtained for other cryptographic primitives such as one-way permutations, encryption, commitment, and collision-resistant hashing. The above results leave a small gap between the possibility of cryptography in NC ₄ ⁰ and the known impossibility of implementing even OWF in NC ₂ ⁰. We partially close this gap by providing evidence for the existence of OWF in NC ₃ ⁰. Finally, our techniques can also be applied to obtain unconditionally provable constructions of non-cryptographic PRGs. In particular, we obtain ε-biased generators in NC ₃ ⁰, resolving an open question posed by Mossel et al. [25], as well as a PRG for logspace in NC ⁰. Our results make use of the machinery of randomizing polynomials [19], which was originally motivated by questions in the domain of information-theoretic secure multi-party computation.