There is significant evidence from cryptography that there exist NP-complete problems that are hard in the average case (meaning that e.g. $AvgP nsupseteq DistNP$). Namely, we have candidate one-way functions and candidate pseudorandom generators.
Is there similar evidence for an average case analogue of the separation between EXP and NEXP?
EDIT: I have the outline of the argument implying the above separation but the conclusion seems too strong so I suspect there is an error somewhere. The separation between $EXP$ and $NEXP$ corresponds to the separation between $P$ and $NP cap TALLY$ via unary encoding. Given a language $L in TALLY$ we can construct the language $L’:=lbrace x | 1^{|x|} in L rbrace$. However, it seems that the existence of a heuristic algorithm for $L’$ wrt the uniform distribution implies that $L in BPP$. Hence, $BPP nsupseteq NP cap SPARSE$ implies $HeurP nsupseteq DistNP$. However, if such a simple average-case to worst-case reduction existed I would probably have encountered it in the literature by now, so I must have got something wrong. What is going on?
