License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.45
URN: urn:nbn:de:0030-drops-156411
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15641/
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Chen, Lijie ; Hirahara, Shuichi ; Vafa, Neekon

Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions

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Abstract

What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness of NP or PH:
- NTIME[n] cannot be solved in quasi-linear time on average if UP ̸ ⊆ DTIME[2^{Õ(√n)}].
- Σ₂TIME[n] cannot be solved in quasi-linear time on average if Σ_kSAT cannot be solved in time 2^{Õ(√n)} for some constant k. Previously, it was not known if even average-case hardness of Σ₃SAT implies the average-case hardness of Σ₂TIME[n].
- Under the Exponential-Time Hypothesis (ETH), there is no average-case n^{1+ε}-time algorithm for NTIME[n] whose running time can be estimated in time n^{1+ε} for some constant ε > 0.
Our results are given by generalizing the non-black-box worst-case-to-average-case connections presented by Hirahara (STOC 2021) to the settings of fine-grained complexity. To do so, we construct quite efficient complexity-theoretic pseudorandom generators under the assumption that the nondeterministic linear time is easy on average, which may be of independent interest.

BibTeX - Entry

@InProceedings{chen_et_al:LIPIcs.ITCS.2022.45,
  author =	{Chen, Lijie and Hirahara, Shuichi and Vafa, Neekon},
  title =	{{Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{45:1--45:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15641},
  URN =		{urn:nbn:de:0030-drops-156411},
  doi =		{10.4230/LIPIcs.ITCS.2022.45},
  annote =	{Keywords: Average-case complexity, worst-case to average-case reduction}
}

Keywords: Average-case complexity, worst-case to average-case reduction
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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