**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 $t(n)\geq (1+\varepsilon)n, \varepsilon>0$, 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.

**Category / Keywords: **foundations / one-way functions, Kolmogorov complexity, average-case hardness

**Date: **received 14 Apr 2020, last revised 24 Sep 2020

**Contact author: **yl2866 at cornell edu, rafael at cs cornell edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20200924:062756 (All versions of this report)

**Short URL: **ia.cr/2020/423

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