Non-Black-Box Worst-Case to Average-Case Reductions Within [math]

SIAM Journal on Computing, Volume 52, Issue 6, Page FOCS18-349-FOCS18-382, December 2023.
Abstract. There are significant obstacles to establishing an equivalence between the worst-case and average-case hardness of [math]. Several results suggest that black-box worst-case to average-case reductions are not likely to be used for reducing any worst-case problem outside [math] to a distributional [math] problem. This paper overcomes the barrier. We present the first non-black-box worst-case to average-case reduction from a problem conjectured to be outside [math] to a distributional [math] 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 [math] within an additive error [math] 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 [math] by [math], which provides strong evidence that the approximation problem is outside [math] 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 [math], approximating the minimum circuit size within a factor of [math] is in [math] for some constant [math] iff its average-case version is easy. Our results can be seen as a new approach for excluding Heuristica. In particular, proving [math]-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 [math].