NON-BLACK-BOX WORST-CASE TO AVERAGE-CASE REDUCTIONS WITHIN NP

There are significant obstacles to establishing an equivalence between the worst-case and average-case hardness of NP. Several results suggest that black-box worst-case to averagecase reductions are not likely to be used for reducing any worst-case problem outside coNP/poly to a distributional NP problem. This paper overcomes the barrier. We present the first non-blackbox worst-case to average-case reduction from a problem conjectured to be outside coNP/poly to a distributional NP problem. Specifically, we consider the minimum time-bounded Kolmogorov complexity problem (MINKT) and prove that there exists a zero-error randomized polynomial-time algorithm approximating the minimum time-bounded Kolmogorov complexity k within an additive error (sic)(root k) if its average-case version admits an errorless heuristic polynomial-time algorithm. We observe that the approximation version of MINKT is Random 3SAT-hard, and more generally it is harder than avoiding any polynomial-time computable hitting set generator that extends its seed of length n by (sic)(root n), which provides strong evidence that the approximation problem is outside coNP/poly and thus our reductions are non-black-box. Our reduction can be derandomized at the cost of the quality of the approximation. We also show that, given a truth table of size 2(n), approximating the minimum circuit size within a factor of 2((1-epsilon)n) is in \sansB NP for some constant epsilon > 0 iff its averagecase version is easy. Our results can be seen as a new approach for excluding Heuristica. In particular, proving NP-hardness of the approximation versions of MINKT or the minimum circuit size problem is sufficient for establishing an equivalence between the worst-case and average-case hardness of NP.