On One-way Functions and Kolmogorov Complexity Yanyi Liu Rafael Pass∗ Cornell University Cornell Tech
[email protected] [email protected] September 24, 2020 Abstract We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial t(n) ≥ (1 + ")n; " > 0, the following are equivalent: • One-way functions exists (which in turn is equivalent to the existence of secure private-key encryption schemes, digital signatures, pseudorandom generators, pseudorandom functions, commitment schemes, and more); • t-time bounded Kolmogorov Complexity, Kt, is mildly hard-on-average (i.e., there exists a t 1 polynomial p(n) > 0 such that no PPT algorithm can compute K , for more than a 1 − p(n) fraction of n-bit strings). In doing so, we present the first natural, and well-studied, computational problem characterizing the feasibility of the central private-key primitives and protocols in Cryptography. ∗Supported in part by NSF Award SATC-1704788, NSF Award RI-1703846, AFOSR Award FA9550-18-1-0267, and a JP Morgan Faculty Award. This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via 2019-19-020700006. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein. 1 Introduction We prove the equivalence of two fundamental problems in the theory of computing: (a) the exis- tence of one-way functions, and (b) mild average-case hardness of the time-bounded Kolmogorov Complexity problem.