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Hunting and Gathering – Verifiable Random Functions from Standard Assumptions with Short Proofs

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Public-Key Cryptography – PKC 2019 (PKC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11443))

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Abstract

A verifiable random function (VRF) is a pseudorandom function, where outputs can be publicly verified. That is, given an output value together with a proof, one can check that the function was indeed correctly evaluated on the corresponding input. At the same time, the output of the function is computationally indistinguishable from random for all non-queried inputs.

We present the first construction of a VRF which meets the following properties at once: It supports an exponential-sized input space, it achieves full adaptive security based on a non-interactive constant-size assumption and its proofs consist of only a logarithmic number of group elements for inputs of arbitrary polynomial length.

Our construction can be instantiated in symmetric bilinear groups with security based on the decision linear assumption. We build on the work of Hofheinz and Jager (TCC 2016), who were the first to construct a verifiable random function with security based on a non-interactive constant-size assumption. Basically, their VRF is a matrix product in the exponent, where each matrix is chosen according to one bit of the input. In order to allow verification given a symmetric bilinear map, a proof consists of all intermediary results. This entails a proof size of \(\varOmega (L)\) group elements, where L is the bit-length of the input.

Our key technique, which we call hunting and gathering, allows us to break this barrier by rearranging the function, which – combined with the partitioning techniques of Bitansky (TCC 2017) – results in a proof size of \(\ell \) group elements for arbitrary \(\ell \in \omega (1)\).

Supported by ERC Project PREP-CRYPTO (724307), by DFG grant HO 4534/2-2 and by a DAAD scholarship. This work was done in part while visiting the FACT Center at IDC Herzliya, Israel.

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Notes

  1. 1.

    For matrix \(\mathbf {M}\in \mathbb {Z}_p^3\) and subspaces \(\mathcal {U},\mathcal {V}\subseteq \mathbb {Z}_p^3\), by \(\mathbf {M}^\top \cdot \mathcal {U}=\mathcal {V}\) we denote the property that for all \(\mathbf {u}\in \mathcal {U}\) we have \(\mathbf {M}^\top \mathbf {u}\in \mathcal {V}\) and for each \(\mathbf {v}\in \mathcal {V}\) there exists a \(\mathbf {u}\) with \(\mathbf {M}^\top \mathbf {u}=\mathbf {v}\).

  2. 2.

    More precisely, with probability at least \(1-(d-1)/p-1/p=1-d/p\).

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Acknowledgments

I would like to thank the anonymous reviewers of TCC 2018 and PKC 2019 for their helpful comments. Further, I would like to thank my advisor Dennis Hofheinz for his support and helpful feedback.

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Correspondence to Lisa Kohl .

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Kohl, L. (2019). Hunting and Gathering – Verifiable Random Functions from Standard Assumptions with Short Proofs. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11443. Springer, Cham. https://doi.org/10.1007/978-3-030-17259-6_14

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