On One-way Functions and Kolmogorov Complexity

TL;DR

This paper establishes an equivalence between the existence of one-way functions and the average-case hardness of time-bounded Kolmogorov complexity, linking fundamental cryptographic primitives to a natural computational problem.

Contribution

It proves the equivalence of one-way functions and the average-case hardness of time-bounded Kolmogorov complexity, providing a natural problem characterizing core cryptographic primitives.

Findings

One-way functions exist if and only if $K^t$ is mildly hard-on-average.

First natural problem characterizing private-key cryptographic primitives.

Links fundamental cryptographic assumptions to Kolmogorov complexity.

Abstract

We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial $t(n)\geq (1+\varepsilon)n, \varepsilon>0$, 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 PPT algorithm can compute $K^t$, for more than a $1-\frac{1}{p(n)}$ 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.