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A Toolbox for Barriers on Interactive Oracle Proofs

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Theory of Cryptography (TCC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13747))

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Abstract

Interactive oracle proofs (IOPs) are a proof system model that combines features of interactive proofs (IPs) and probabilistically checkable proofs (PCPs). IOPs have prominent applications in complexity theory and cryptography, most notably to constructing succinct arguments.

In this work, we study the limitations of IOPs, as well as their relation to those of PCPs. We present a versatile toolbox of IOP-to-IOP transformations containing tools for: (i) length and round reduction; (ii) improving completeness; and (iii) derandomization.

We use this toolbox to establish several barriers for IOPs:

  • Low-error IOPs can be transformed into low-error PCPs. In other words, interaction can be used to construct low-error PCPs; alternatively, low-error IOPs are as hard to construct as low-error PCPs. This relates IOPs to PCPs in the regime of the sliding scale conjecture for inverse-polynomial soundness error.

  • Limitations of quasilinear-size IOPs for 3SAT with small soundness error.

  • Limitations of IOPs where query complexity is much smaller than round complexity.

  • Limitations of binary-alphabet constant-query IOPs.

We believe that our toolbox will prove useful to establish additional barriers beyond our work.

G. Arnon—Supported in part by a grant from the Israel Science Foundation (no. 2686/20) and by the Simons Foundation Collaboration on the Theory of Algorithmic Fairness.

A. Chiesa—Supported in part by the Ethereum Foundation.

E. Yogev—Supported in part by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office, and by the Alter Family Foundation.

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Notes

  1. 1.

    There exists a function in \({\textsf{E}}\) with circuit complexity \(2^{\varOmega (n)}\) for circuits with \(\textrm{PSPACE}\) gates.

  2. 2.

    \(\texttt{RETH}\) states that there exists a constant \(c>0\) such that \(\textrm{3SAT}\notin \textrm{BPTIME}[2^{c \cdot n}]\).

  3. 3.

    Assuming \(\texttt{ETH}\), the proof length of the PCP can be \(2^{o(n)}\).

  4. 4.

    It is sufficient to assume that \(\beta = \frac{1}{2} \cdot \left( \frac{2\cdot e\cdot \textsf{l}\cdot \log \lambda }{c \cdot n}\right) ^{-\textsf{q}}\) to find contradiction in \(\beta \le \frac{1}{2} \cdot \left( \frac{2\cdot e \cdot \textsf{l}\cdot \log \lambda }{c\cdot n}\right) ^{-\textsf{q}}\) since we can always increase the soundness error without loss of generality.

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Author information

Authors and Affiliations

  1. Weizmann Institute, Rehovot, Israel

    Gal Arnon

  2. UC Riverside, Riverside, USA

    Amey Bhangale

  3. EPFL, Lausanne, Switzerland

    Alessandro Chiesa

  4. Bar-Ilan University, Ramat Gan, Israel

    Eylon Yogev

Authors
  1. Gal Arnon
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  2. Amey Bhangale
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  3. Alessandro Chiesa
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  4. Eylon Yogev
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Correspondence to Gal Arnon .

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Editors and Affiliations

  1. Ruhr University Bochum, Bochum, Germany

    Eike Kiltz

  2. Massachusetts Institute of Technology, Cambridge, MA, USA

    Vinod Vaikuntanathan

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Arnon, G., Bhangale, A., Chiesa, A., Yogev, E. (2022). A Toolbox for Barriers on Interactive Oracle Proofs. In: Kiltz, E., Vaikuntanathan, V. (eds) Theory of Cryptography. TCC 2022. Lecture Notes in Computer Science, vol 13747. Springer, Cham. https://doi.org/10.1007/978-3-031-22318-1_16

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  • DOI: https://doi.org/10.1007/978-3-031-22318-1_16

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