In search of an easy witness: exponential time vs. probabilistic polynomial time

Restricting the search space {0, 1}ⁿ to the set of truth tables of “easy” Boolean functions on log n variables, as well as using some known hardness-randomness tradeoffs, we establish a number of results relating the complexity of exponential-time and probabilistic polynomial-time complexity classes. In particular, we show that NEXP⊂P/poly⇔NEXP=MA; this can be interpreted to say that no derandomization of MA (and, hence, of promise-BPP) is possible unless NEXP contains a hard Boolean function. We also prove several downward closure results for ZPP, RP, BPP, and MA; e.g., we show EXP=BPP⇔EE=BPE, where EE is the double-exponential time class and BPE is the exponential-time analogue of BPP