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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.

References

  1. Miklós Ajtai. Generating hard instances of lattice problems (extended abstract). In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC), pages 99-108. ACM, 1996. URL: https://doi.org/10.1145/237814.237838.
  2. Adi Akavia, Oded Goldreich, Shafi Goldwasser, and Dana Moshkovitz. On basing one-way functions on NP-hardness. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 701-710. ACM, 2006. See also [Adi Akavia et al., 2010]. Google Scholar
  3. Adi Akavia, Oded Goldreich, Shafi Goldwasser, and Dana Moshkovitz. Erratum for: On basing one-way functions on NP-hardness. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pages 795-796. ACM, 2010. Google Scholar
  4. Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM J. Comput., 35(6):1467-1493, 2006. Google Scholar
  5. Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-way functions and a conditional variant of MKTP. Electron. Colloquium Comput. Complex., 28:9, 2021. Google Scholar
  6. Eric Allender, Mahdi Cheraghchi, Dimitrios Myrisiotis, Harsha Tirumala, and Ilya Volkovich. One-way functions and Partial MCSP. Electron. Colloquium Comput. Complex., 28:9, 2021. Google Scholar
  7. Andrej Bogdanov and Luca Trevisan. Average-case complexity. Found. Trends Theor. Comput. Sci., 2(1), 2006. Google Scholar
  8. Andrej Bogdanov and Luca Trevisan. On worst-case to average-case reductions for NP problems. SIAM J. Comput., 36(4):1119-1159, 2006. Google Scholar
  9. Chris Brzuska and Geoffroy Couteau. Towards fine-grained one-way functions from strong average-case hardness. IACR Cryptol. ePrint Arch., 2020:1326, 2020. Google Scholar
  10. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In David B. Shmoys, editor, Symposium on Theory of Computing (STOC), pages 624-633. ACM, 2014. Google Scholar
  11. Oded Goldreich, Shafi Goldwasser, and Silvio Micali. How to construct random functions. J. ACM, 33(4):792-807, 1986. Google Scholar
  12. Johan Håstad, Russell Impagliazzo, Leonid A. Levin, and Michael Luby. A pseudorandom generator from any one-way function. SIAM J. Comput., 28(4):1364-1396, 1999. Google Scholar
  13. Shuichi Hirahara. Non-black-box worst-case to average-case reductions within NP. In 59th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 247-258. IEEE Computer Society, 2018. Google Scholar
  14. Shuichi Hirahara and Rahul Santhanam. On the average-case complexity of MCSP and its variants. In Ryan O'Donnell, editor, 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, volume 79 of LIPIcs, pages 7:1-7:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  15. Rahul Ilango. Approaching MCSP from above and below: Hardness for a conditional variant and AC⁰[p]. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 34:1-34:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  16. Rahul Ilango, Hanlin Ren, and Rahul Santhanam. Hardness on any samplable distribution suffices: New characterizations of one-way functions by meta-complexity. Electron. Colloquium Comput. Complex., 28:82, 2021. Google Scholar
  17. Russell Impagliazzo. A personal view of average-case complexity. In Proceedings of the Tenth Annual Structure in Complexity Theory Conference, Minneapolis, Minnesota, USA, June 19-22, 1995, pages 134-147. IEEE Computer Society, 1995. Google Scholar
  18. Russell Impagliazzo and Leonid A. Levin. No better ways to generate hard NP instances than picking uniformly at random. In 31st Annual Symposium on Foundations of Computer Science, St. Louis, Missouri, USA, October 22-24, 1990, Volume II, pages 812-821. IEEE Computer Society, 1990. Google Scholar
  19. Russell Impagliazzo and Moni Naor. Efficient cryptographic schemes provably as secure as subset sum. J. Cryptol., 9(4):199-216, 1996. Google Scholar
  20. Yanyi Liu and Rafael Pass. On one-way functions and Kolmogorov complexity. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 1243-1254. IEEE, 2020. Google Scholar
  21. Yanyi Liu and Rafael Pass. Cryptography from sublinear-time average-case hardness of time-bounded Kolmogorov complexity. In Proceedings of the 53rd ACM Symposium on Theory of Computing (STOC). ACM, 2021. Google Scholar
  22. Yanyi Liu and Rafael Pass. A note on one-way functions and sparse languages. IACR Cryptol. ePrint Arch., 2021:890, 2021. Google Scholar
  23. Yanyi Liu and Rafael Pass. On one-way functions from NP-complete problems. Electron. Colloquium Comput. Complex., 28:59, 2021. Google Scholar
  24. Yanyi Liu and Rafael Pass. On the possibility of basing cryptography on EXP≠ BPP. Electron. Colloquium Comput. Complex., 28:56, 2021. Google Scholar
  25. Daniele Micciancio and Oded Regev. Worst-case to average-case reductions based on Gaussian measures. SIAM J. Comput., 37(1):267-302, 2007. URL: https://doi.org/10.1137/S0097539705447360.
  26. Mikito Nanashima. On basing auxiliary-input cryptography on NP-hardness via nonadaptive black-box reductions. In 12th Innovations in Theoretical Computer Science Conference (ITCS), volume 185 of LIPIcs, pages 29:1-29:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  27. Ján Pich. Learning algorithms from circuit lower bounds. CoRR, abs/2012.14095, 2020. URL: http://arxiv.org/abs/2012.14095.
  28. Hanlin Ren and Rahul Santhanam. Hardness of KT characterizes parallel cryptography. Electron. Colloquium Comput. Complex., 28:57, 2021. Google Scholar
  29. Rahul Santhanam. Pseudorandomness and the Minimum Circuit Size Problem. In 11th Innovations in Theoretical Computer Science Conference (ITCS), volume 151 of LIPIcs, pages 68:1-68:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  30. Andrew Chi-Chih Yao. Theory and applications of trapdoor functions (extended abstract). In 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, 3-5 November 1982, pages 80-91. IEEE Computer Society, 1982. Google Scholar
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