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Private Polynomial Commitments and Applications to MPC

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

Polynomial commitment schemes allow a prover to commit to a polynomial and later reveal the evaluation of the polynomial on an arbitrary point along with proof of validity. This object is central in the design of many cryptographic schemes such as zero-knowledge proofs and verifiable secret sharing. In the standard definition, the polynomial is known to the prover whereas the evaluation points are not private. In this paper, we put forward the notion of private polynomial commitments that capture additional privacy guarantees, where the evaluation points are hidden from the verifier while the polynomial is hidden from both.

We provide concretely efficient constructions that allow simultaneously batch the verification of many evaluations with a small additive overhead. As an application, we design a new concretely efficient multi-party private set-intersection with malicious security and improved asymptotic communication and space complexities.

We demonstrate the concrete efficiency of our construction via an implementation. Our scheme can prove \(2^{10}\) evaluations of a private polynomial of degree \(2^{10}\) in 157 s. The proof size is only 169 KB and the verification time is 11.8 s. Moreover, we also implemented the multi-party private set intersection protocol and scale it to 1000 parties (which has not been shown before). The total running time for \(2^{14}\) elements per party is 2,410 s. While existing protocols offer better computational complexity, our scheme offers significantly smaller communication and better scalability (in the number of parties) owing to better memory usage.

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Acknowledgements

We thank the anonymous PKC‘23 reviewers for their helpful comments. The first and second authors are supported by ISF grant No. 1316/18. The second, third and fifth authors are supported by DARPA under Contract No. HR001120C0087. The third author was supported by Technology and Humanity Fund from Georgetown University’s McCourt School of Public Policy. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA.

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Authors and Affiliations

  1. Bar Ilan University, Ramat Gan, Israel

    Rishabh Bhadauria & Carmit Hazay

  2. Georgetown University, Washington, DC, USA

    Muthuramakrishnan Venkitasubramaniam

  3. Ligero Inc., Rochester, USA

    Carmit Hazay & Muthuramakrishnan Venkitasubramaniam

  4. Texas A&M University, College Station, USA

    Wenxuan Wu & Yupeng Zhang

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  1. Rishabh Bhadauria
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  2. Carmit Hazay
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  3. Muthuramakrishnan Venkitasubramaniam
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  4. Wenxuan Wu
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Correspondence to Rishabh Bhadauria .

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

    Alexandra Boldyreva

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

    Vladimir Kolesnikov

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Bhadauria, R., Hazay, C., Venkitasubramaniam, M., Wu, W., Zhang, Y. (2023). Private Polynomial Commitments and Applications to MPC. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_5

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