Revision #1 Authors: Yanyi Liu, Rafael Pass

Accepted on: 1st May 2021 00:57

Downloads: 174

Keywords:

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.

TR21-059 Authors: Yanyi Liu, Rafael Pass

Publication: 25th April 2021 13:26

Downloads: 408

Keywords:

We present the first natural $\NP$-complete problem whose average-case hardness w.r.t. the uniform distribution over instances implies the existence of one-way functions (OWF). In fact, we prove that the existence of OWFs is \emph{equivalent} to mild average-case hardness of this $\NP$-complete problem. The problem, which originated in the 1960s, is the \emph{Conditional Time-Bounded Kolmogorov Complexity Problem}: let $K^t(x \mid z)$ 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 $K^t(x \mid z)< k$, 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 observe that the result of Liu-Pass (FOCS'20) extends 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.