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URN: urn:nbn:de:0030-drops-165981
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On One-Way Functions from NP-Complete Problems

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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 McK^tP[ζ] be the set of strings (x,z,k) where |z| = ζ(|x|), |k| = log |x| and K^t(x∣z) < k, where, for our purposes, a "program" is defined as a RAM machine.
Our main result shows that for every polynomial t(n) ≥ n², there exists some polynomial ζ such that McK^tP[ζ] is NP-complete. We additionally extend the result of Liu-Pass (FOCS'20) to show that for every polynomial t(n) ≥ 1.1n, and every polynomial ζ(⋅), mild average-case hardness of McK^tP[ζ] is equivalent to the existence of OWFs. Taken together, these results provide the following crisp characterization of what is required to base OWFs on NP ⊈ BPP:
There exists concrete polynomials t,ζ such that "Basing OWFs on NP ⊈ BPP" is equivalent to providing a "worst-case to (mild) average-case reduction for McK^tP[ζ]".
In other words, the "holy-grail" of Cryptography (i.e., basing OWFs on NP ⊈ BPP) is equivalent to a basic question in algorithmic information theory.
As an independent contribution, we show that our NP-completeness result can be used to shed new light on the feasibility of the polynomial-time bounded symmetry of information assertion (Kolmogorov'68).

BibTeX - Entry

@InProceedings{liu_et_al:LIPIcs.CCC.2022.36,
  author =	{Liu, Yanyi and Pass, Rafael},
  title =	{{On One-Way Functions from NP-Complete Problems}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{36:1--36:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16598},
  URN =		{urn:nbn:de:0030-drops-165981},
  doi =		{10.4230/LIPIcs.CCC.2022.36},
  annote =	{Keywords: One-way Functions, NP-Completeness, Kolmogorov Complexity}
}

Keywords: One-way Functions, NP-Completeness, Kolmogorov Complexity
Seminar: 37th Computational Complexity Conference (CCC 2022)
Issue date: 2022
Date of publication: 11.07.2022


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