On One-Way Functions, the Worst-Case Hardness of Time-Bounded Kolmogorov Complexity, and Computational Depth

Whether one-way functions (OWF) exist is arguably the most important problem in Cryptography, and beyond. While lots of candidate constructions of one-way functions are known, and recently also problems whose average-case hardness characterize the existence of OWFs have been demonstrated, the question of whether there exists some worst-case hard problem that characterizes the existence of one-way functions has remained open since their introduction in 1976. In this work, we present the first “OWF-complete” promise problem—a promise problem whose worst-case hardness w.r.t. BPP (resp. P/poly) is equivalent to the existence of OWFs secure against PPT (resp. nuPPT) algorithms. The problem is a variant of the Minimum Time-bounded Kolmogorov Complexity problem (MK^tP[s] with a threshold s), where we condition on instances having small “computational depth”. We furthermore show that depending on the choice of the threshold s, this problem characterizes either “standard” (polynomially-hard) OWFs, or quasi polynomially- or subexponentially-hard OWFs. Additionally, when the threshold is sufficiently small (e.g., 2^O(logn) or polylogn) then sublinear hardness of this problem suffices to characterize quasi-poly-nomial/sub-exponential OWFs. While our constructions are black-box, our analysis is non-black box; we additionally demonstrate that fully black-box constructions of OWF from the worst-case hardness of this problem are impossible. We finally show that, under Rudich’s conjecture, and standard derandomization assumptions, our problem is not inside coAM; as such, it yields the first candidate problem believed to be outside of AM∩coAM, or even SZK, whose worst case hardness implies the existence of OWFs.