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 is contained in P\poly iff NEXP=MA; this can be interpreted to say that no derandomization of MA 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 iff EE=BPE, where EE is the double-exponential time class and BPE is the exponential-time analogue of BPP.

This is joint work with Russell Impagliazzo and Avi Wigderson.