Paper 2020/423

On One-way Functions and Kolmogorov Complexity

Yanyi Liu and Rafael Pass

Abstract

We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial $t(n)\ge (1+\epsilon )n,\epsilon >0$, 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, $K^t$, is mildly hard-on-average (i.e., there exists a polynomial $p(n)>0$ such that no $\text{PPT}$ algorithm can compute $K^t$, for more than a $1-\frac{1}{p(n)}$ fraction of $$-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.