Cryptography from sublinear-time average-case hardness of time-bounded Kolmogorov complexity

Let MKtP[s] be the set of strings x such that Kt(x) ≤ s(|x|), where Kt(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: (i) the existence of some function T ∈ F such that T-hard one-way functions (OWF) exists (with non-uniform security); (ii) the existence of some function T ∈ F such that MKtP[T−1] is mildly average-case hard with respect to sublinear-time non-uniform algorithms (with running-time nδ for some 0<δ<1). For instance, existence of subexponentially-hard (resp. quasi-poly-nomially-hard) OWFs is equivalent to mild average-case hardness of MKtP[poly logn] (resp. MKtP[2O(√logn))]) w.r.t. sublinear-time non-uniform algorithms. We additionally note that if we want to deduce T-hard OWFs where security holds w.r.t. uniform T-time probabilistic attackers (i.e., uniformly-secure OWFs), it suffices to assume sublinear time hardness of MKtP w.r.t. uniform probabilistic sublinear-time attackers. We complement this result by proving lower bounds that come surprisingly close to what is required to unconditionally deduce the existence of (uniformly-secure) OWFs: MKtP[polylogn] is worst-case hard w.r.t. uniform probabilistic sublinear-time algorithms, and MKtP[n−logn] is mildly average-case hard for all O(t(n)/n3)-time deterministic algorithms.