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The Communication Complexity of Threshold Private Set Intersection

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Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11693))

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Abstract

Threshold private set intersection enables Alice and Bob who hold sets \(S_{\mathsf {A}}\) and \(S_{\mathsf {B}}\) of size n to compute the intersection \(S_{\mathsf {A}} \cap S_{\mathsf {B}} \) if the sets do not differ by more than some threshold parameter \(t\). In this work, we investigate the communication complexity of this problem and we establish the first upper and lower bounds. We show that any protocol has to have a communication complexity of \(\varOmega (t)\). We show that an almost matching upper bound of \(\tilde{\mathcal {O}}(t)\) can be obtained via fully homomorphic encryption. We present a computationally more efficient protocol based on weaker assumptions, namely additively homomorphic encryption, with a communication complexity of \(\tilde{\mathcal {O}}(t ^2)\). For applications like biometric authentication, where a given fingerprint has to have a large intersection with a fingerprint from a database, our protocols may result in significant communication savings.

Prior to this work, all previous protocols had a communication complexity of \(\varOmega (n)\). Our protocols are the first ones with communication complexities that mainly depend on the threshold parameter \(t\) and only logarithmically on the set size n.

S. Ghosh and M. Simkin—Supported by the European Unions’s Horizon 2020 research and innovation program under grant agreement No 669255 (MPCPRO) and under grant agreement No 731583 (SODA) and the Independent Research Fund Denmark project BETHE.

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Notes

  1. 1.

    A rational function is the fraction of two polynomials. See Sect. 2.1 for details.

  2. 2.

    See [MTZ03] for details on rational function interpolation over a field.

  3. 3.

    The lemma we are referring to here is Lemma 2 in the paper of Kissner and Song.

  4. 4.

    For the case of the Paillier cryptosystem this is strictly speaking not the case, since not every element from the message space has an inverse. However, finding an element that does not have an inverse is as hard as breaking the security of the cryptosystem. Therefore, we can treat the message space as if it was an actual field.

  5. 5.

    See Theorem 3 in their work.

  6. 6.

    separating the numerator and denominator from a given rational function is easy here, because we obtain the coefficients of both separately during the interpolation step.

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Acknowledgments

We would like to thank Ivan Damgård, Claudio Orlandi, and Tobias Nilges for the fruitful discussions that have helped shape the current work.

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  1. Aarhus University, Aarhus, Denmark

    Satrajit Ghosh & Mark Simkin

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Correspondence to Satrajit Ghosh or Mark Simkin .

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

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Ghosh, S., Simkin, M. (2019). The Communication Complexity of Threshold Private Set Intersection. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_1

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