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

Abstract

Up to know, the known derandomization methods have been derived assuming average-case hardness conditions. In this paper we instead present the first worst-case hardness conditions sufficient to obtain P = BPP. Our conditions refer to the worst-case circuit complexity of Boolean operators computable in time exponential in the input size. Such results are achieved by a new method that departs significantly from the usual known methods based on pseudo-random generators. Our method also gives a worst-case hardness condition for the circuit complexity of Boolean operators computable in NC (with respect to their output size) to obtain NC = BPNC.