**On One-way Functions from NP-Complete Problems**

*Yanyi Liu and Rafael Pass*

**Abstract: **We present the first natural NP-complete problem whose average-case hardness w.r.t. the uniform distribution over instances is equivalent to the existence of one-way functions (OWFs). The problem, which originated in the 1960s, is the Conditional Time-Bounded Kolmogorov Complexity Problem: let $K^t(x | z)$ be the length of the shortest ``program'' that, given the ``auxiliary input'' $z$, outputs the string $x$ within time $t(|x|)$, and let
McKTP$[t,\zeta]$ be the set of strings $(x,z,k)$ where $|z| = \zeta(|x|)$, $|k| = \log |x|$ and $K^t(x | z)< k$, where, for our purposes, a ``program'' is defined as a RAM machine.

Our main results shows that for every polynomial $t(n)\geq n^2$, there exists some polynomial $\zeta$ such that McKTP$[t,\zeta]$ is NP-complete. We additionally extend the result of Liu-Pass (FOCS'20) to show that for every polynomial $t(n)\geq 1.1n$, and every polynomial $\zeta(\cdot)$, mild average-case hardness of McKTP$[t,\zeta]$ is equivalent to the existence of OWFs.

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

**Original Publication**** (with minor differences): **On https://eccc.weizmann.ac.il/

**Date: **received 19 Apr 2021, last revised 30 Apr 2021

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

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

**Version: **20210430:222411 (All versions of this report)

**Short URL: **ia.cr/2021/513

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