Pseudorandomness and average-case complexity via uniform reductions

Impagliazzo and Wigderson (1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. We obtain an optimal worst-case to average-case connection for EXP: if EXP BPTIME(( )), EXP has problems that are cannot be solved on a fraction 1/2 1/'( ) of the inputs by BPTIME('( )) algorithms, for ' = /sup 1/. We exhibit a PSPACE-complete downward self-reducible and random self-reducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson (1998), which used a a P-complete problem with these properties. We argue that the results in Impagliazzo and Wigderson (1998) and in this paper cannot be proved via "black-box" uniform reductions.