A Cookbook for Black-Box Separations and a Recipe for UOWHFs

Abstract

We present a new framework for proving fully black-box separations and lower bounds. We prove a general theorem that facilitates the proofs of fully black-box lower bounds from a one-way function (OWF). Loosely speaking, our theorem says that in order to prove that a fully black-box construction does not securely construct a cryptographic primitive Q (e.g., a pseudo-random generator or a universal one-way hash function) from a OWF, it is enough to come up with a large enough set of functions $\mathcal{F}$ and a parameterized oracle (i.e., an oracle that is defined for every f ε{0,1} n → {0,1} n ) such that $\mathcal{O}_{f}$ breaks the security of the construction when instantiated with f and the oracle satisfies two local properties. Our main application of the theorem is a lower bound of Ω(n/log(n)) on the number of calls made by any fully black-box construction of a universal one-way hash function (UOWHF) from a general one-way function. The bound holds even when the OWF is regular, in which case it matches to a recent construction of Barhum and Maurer [4].