Let MK t P[s] be the set of strings x such that K t (x) ≤ s(|x|), where K t (x) denotes the t-bounded Kolmogorov complexity of the truthtable described by x. Our main theorem shows that for an appropriate notion of mild average-case hardness, for every ε > 0, polynomial t(n) ≥ (1 + ε)n, and every "nice" class F of super-polynomial functions, the following are equivalent: * A preliminary version of this paper will appear in the proceedings of STOC'21. This is the full version.