Capturing One-Way Functions via NP-Hardness of Meta-Complexity

A one-way function is a function that is easy to compute but hard to invert on average. We establish the first characterization of a one-way function by worst-case hardness assumptions, by introducing a natural meta-computational problem whose NP-hardness (and the worst-case hardness of NP) characterizes the existence of a one-way function. Specifically, we generalize the notion of time-bounded conditional Kolmogorov complexity to distributional Kolmogorov complexity, and prove that a one-way function exists if and only if it is NP-hard to approximate the distributional Kolmogorov complexity under randomized polynomial-time reductions and NP is hard in the worst case. We also propose the Meta-Complexity Padding Conjecture, which postulates that distributional Kolmogorov complexity is paddable by an approximation-preserving reduction. Under this conjecture, we prove that the worst-case hardness of an approximate version of the Minimum Circuit Size Problem characterizes the existence of a one-way function.

Our results extend the emerging paradigm of meta-complexity, which suggests that proving NP-hardness of meta-computational problems (i.e., problems that ask to compute complexity) is sufficient to exclude errorless Heuristica and error-prone Pessiland from Impagliazzo's five worlds. The key technical contribution is to conditionally close the gap between errorless and error-prone average-case complexities by combining Nanashima's proof techniques of showing "limits" of black-box reductions (ITCS'21) with non-black-box worst-case-to-average-case reductions of Hirahara (FOCS'18).