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One-Way Functions and a Conditional Variant of MKTP

Authors Eric Allender , Mahdi Cheraghchi , Dimitrios Myrisiotis , Harsha Tirumala , Ilya Volkovich

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Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Cryptographic primitives
Keywords
  • Kolmogorov complexity
  • KT Complexity
  • Minimum KT-complexity Problem
  • MKTP
  • Conditional KT Complexity
  • Minimum Conditional KT-complexity Problem
  • McKTP
  • one-way functions
  • OWFs
  • average-case hardness
  • pseudorandom generators
  • PRGs
  • pseudorandom functions
  • PRFs
  • distinguishers
  • learning algorithms
  • NP-completeness
  • reductions

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Abstract

One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural NP-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows.  
1) First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs. 
2) Then, we observe that McKTP is NP-complete under polynomial-time randomized reductions. 
3) Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP.  Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem. In fact, building on recently-announced results of Ren and Santhanam [Rahul Ilango et al., 2021], we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs.

Cite As Get BibTex

Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-Way Functions and a Conditional Variant of MKTP. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.FSTTCS.2021.7

Author Details

Eric Allender
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Mahdi Cheraghchi
  • Department of EECS, University of Michigan, Ann Arbor, MI, USA
Dimitrios Myrisiotis
  • Department of Computing, Imperial College London, London, UK
Harsha Tirumala
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Ilya Volkovich
  • Computer Science Department, Boston College, Chestnut Hill, MA, USA

Funding

  • Allender, Eric: Partially supported by NSF Grants CCF-1909216 & CCF-1909683.
  • Cheraghchi, Mahdi: Mahdi Cheraghchi’s research was partially supported by the National Science Foundation under Grant No. CCF-2006455.
  • Myrisiotis, Dimitrios: This work was partly carried out during a visit of Dimitrios Myrisiotis to DIMACS, with support from the Special Focus on Lower Bounds in Computational Complexity funded under NSF Grant CCF-1836666.
  • Tirumala, Harsha: Harsha Tirumala was partially supported by NSF Grant CCF-1909216 and by the Simons Collaboration on Algorithms and Geometry.

Acknowledgements

We would like to thank Russell Impagliazzo for explaining his work [Russell Impagliazzo and Leonid A. Levin, 1990] to us, and Ján Pich and Ninad Rajgopal for illuminating discussions. We thank Ján Pich for bringing his work [Ján Pich, 2020] to our attention. We thank Mikito Nanashima and Hanlin Ren for helpful comments and for spotting bugs in the proofs of earlier versions of Lemma 20 and Lemma 21, respectively. In particular, we thank Hanlin Ren for asking the question of whether KT complexity would be an appropriate complexity measure to consider in the context of our work. We thank Yanyi Liu and Rafael Pass for the excellent correspondence regarding their work [Yanyi Liu and Rafael Pass, 2020; Yanyi Liu and Rafael Pass, 2021; Yanyi Liu and Rafael Pass, 2021], and Rahul Santhanam for bringing the work by Impagliazzo and Naor [Russell Impagliazzo and Moni Naor, 1996] to our attention. Finally, we would like to thank the anonymous reviewers for their helpful feedback.

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