Worst-case hardness suffices for derandomization: a new method for hardness-randomness trade-offs

Abstract

Up to now, the known derandomization methods for BPP have been derived assuming the existence of an EXP function that has a “hard” average-case circuit complexity. In this paper we instead present the first construction of a de-randomization method for BPP that relies on the existence of an EXP function that is hard only in the worst-case.The construction is based on a new method that departs significantly from the usual known methods based on pseudo-random generators. Indeed, we prove new particular bounds on the circuit complexity of partial Boolean functions which are then used to derive efficient constructions of hitting set generators. As recently proved, such generators can derandomize any BPP-algorithm.Our method is efficiently parallelizable and hence yields also a hardness condition on the worst-case circuit complexity of Boolean operators running in NC which is sufficient to obtain NC = BPNC.