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Impagliazzo and Wigderson (FOCS 2002) recently gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely, that EXP does not equal 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.In this paper: We obtain an optimal worst-case to average-case connection for EXP: if EXP is not contained in BPTIME(t(n)), then EXP has problems that are cannot be solved on a fraction 1/2 +1/t'(n) of the inputs by BPTIME(t'(n)) algorithms, for t'=t^{Omega(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, which used a a #P-complete problem with these properties. We argue that the results of Impagliazzo and Wigderson and of this paper cannot be proved via "black-box" uniform reductions.