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

Authors Lijie Chen, Shuichi Hirahara, Neekon Vafa

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  • Theory of computation → Complexity classes
Keywords
  • Average-case complexity
  • worst-case to average-case reduction

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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.

Cite As Get BibTex

Lijie Chen, Shuichi Hirahara, and Neekon Vafa. Average-Case Hardness of NP and PH from Worst-Case Fine-Grained Assumptions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.45

Author Details

Lijie Chen
  • MIT, Boston, MA, USA
Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Neekon Vafa
  • MIT, Boston, MA, USA

Funding

  • Chen, Lijie: supported by NSF CCF-2127597 and an IBM Fellowship.
  • Hirahara, Shuichi: supported by JST, PRESTO Grant Number JPMJPR2024, Japan.
  • Vafa, Neekon: supported by NSF fellowship DGE-1745302.

Acknowledgements

We are grateful to Rahul Ilango and Ryan Williams for helpful discussions. In particular, we want to thank Ryan Williams for the observation that an HSG with seed length O(log t) already suffices for our proof.

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