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On Search Complexity of Discrete Logarithm

Authors Pavel Hubáček , Jan Václavek

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Author Details

Pavel Hubáček
  • Charles University, Prague, Czech Republic
Jan Václavek
  • Charles University, Prague, Czech Republic

Funding

  • Hubáček, Pavel: Supported by the Grant Agency of the Czech Republic under the grant agreement no. 19-27871X and by the Charles University projects PRIMUS/17/SCI/9 and UNCE/SCI/004.

Acknowledgements

We wish to thank the anonymous reviewers for their helpful suggestions on the presentation of our results. The first author is grateful to Chethan Kamath for multiple enlightening discussions about TFNP and for suggesting Dove as a name for a total search problem contained in the complexity class PPP.

Cite AsGet BibTex

Pavel Hubáček and Jan Václavek. On Search Complexity of Discrete Logarithm. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 60:1-60:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.60

Abstract

In this work, we study the discrete logarithm problem in the context of TFNP - the complexity class of search problems with a syntactically guaranteed existence of solutions for all instances. Our main results establish that suitable variants of the discrete logarithm problem are complete for the complexity class PPP, respectively PWPP, i.e., the subclasses of TFNP capturing total search problems with a solution guaranteed by the pigeonhole principle, respectively the weak pigeonhole principle. Besides answering an open problem from the recent work of Sotiraki, Zampetakis, and Zirdelis (FOCS’18), our completeness results for PPP and PWPP have implications for the recent line of work proving conditional lower bounds for problems in TFNP under cryptographic assumptions. In particular, they highlight that any attempt at basing average-case hardness in subclasses of TFNP (other than PWPP and PPP) on the average-case hardness of the discrete logarithm problem must exploit its structural properties beyond what is necessary for constructions of collision-resistant hash functions. Additionally, our reductions provide new structural insights into the class PWPP by establishing two new PWPP-complete problems. First, the problem Dove, a relaxation of the PPP-complete problem Pigeon. Dove is the first PWPP-complete problem not defined in terms of an explicitly shrinking function. Second, the problem Claw, a total search problem capturing the computational complexity of breaking claw-free permutations. In the context of TFNP, the PWPP-completeness of Claw matches the known intrinsic relationship between collision-resistant hash functions and claw-free permutations established in the cryptographic literature.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • discrete logarithm
  • total search problems
  • completeness
  • TFNP
  • PPP
  • PWPP

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