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On the Black-Box Complexity of Correlation Intractability

Authors Nico Döttling, Tamer Mour

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Author Details

Nico Döttling
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Tamer Mour
  • Bocconi University, Milan, Italy

Funding

  • Döttling, Nico: Funded by the European Union (ERC, LACONIC, 101041207). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Mour, Tamer: Supported by the Binational Science Foundation (Grant No. 2016726), by the European Union Horizon 2020 Research and Innovation Program (ERC Projects REACT - Grant No. 756482, PROMETHEUS - 780701, and FGC - 101019547), and by the Azrieli Fellows Program.

Acknowledgements

We thank Zvika Brakerski for comments and discussion, and Venkata Koppula for his helpful feedback on a prior version of this work.

Cite AsGet BibTex

Nico Döttling and Tamer Mour. On the Black-Box Complexity of Correlation Intractability. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 40:1-40:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.40

Abstract

Correlation intractability is an emerging cryptographic paradigm that enabled several recent breakthroughs in establishing soundness of the Fiat-Shamir transform and, consequently, basing non-interactive zero-knowledge proofs and succinct arguments on standard cryptographic assumptions. In a nutshell, a hash family is said to be correlation intractable for a class of relations ℛ if, for any relation R ∈ ℛ, it is hard given a random hash function h ← H to find an input z s.t. (z,h(z)) ∈ R, namely a correlation. Despite substantial progress in constructing correlation intractable hash functions, all constructions known to date are based on highly-structured hardness assumptions and, further, are of complexity scaling with the circuit complexity of the target relation class. In this work, we initiate the study of the barriers for building correlation intractability. Our main result is a lower bound on the complexity of any black-box construction of CIH from collision resistant hash (CRH), or one-way permutations (OWP), for any sufficiently expressive relation class. In particular, any such construction for a class of relations with circuit complexity t must make at least Ω(t) invocations of the underlying building block. We see this as a first step in developing a methodology towards broader lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Correlation Intractability
  • Fiat-Shamir
  • Black-box Complexity
  • Black-box Separations

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