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An Algebraic Approach to Maliciously Secure Private Set Intersection

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Book cover Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11478))

Abstract

Private set intersection (PSI) is an important area of research and has been the focus of many works over the past decades. It describes the problem of finding an intersection between the input sets of at least two parties without revealing anything about the input sets apart from their intersection.

In this paper, we present a new approach to compute the intersection between sets based on a primitive called Oblivious Linear Function Evaluation (OLE). On an abstract level, we use this primitive to efficiently add two polynomials in a randomized way while preserving the roots of the added polynomials. Setting the roots of the input polynomials to be the elements of the input sets, this directly yields an intersection protocol with optimal asymptotic communication complexity \(O(m\kappa )\). We highlight that the protocol is information-theoretically secure against a malicious adversary assuming OLE.

We also present a natural generalization of the 2-party protocol for the fully malicious multi-party case. Our protocol does away with expensive (homomorphic) threshold encryption and zero-knowledge proofs. Instead, we use simple combinatorial techniques to ensure the security. As a result we get a UC-secure protocol with asymptotically optimal communication complexity \(O((n^2+nm)\kappa )\), where n is the number of parties, m is the set size and \(\kappa \) is the security parameter. Apart from yielding an asymptotic improvement over previous works, our protocols are also conceptually simple and require only simple field arithmetic. Along the way we develop techniques that might be of independent interest.

S. Ghosh—Supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement #669255 (MPCPRO), the European Union’s Horizon 2020 research and innovation programme under grant agreement #731583 (SODA) and the Independent Research Fund Denmark project BETHE.

T. Nilges—Part of the research leading to these results was done while the author was at Aarhus University. Supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement #669255 (MPCPRO).

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Notes

  1. 1.

    Please see the full version of the paper [15].

  2. 2.

    Here we mean an OPE is used for each element of the receiver’s input set. This can be circumvented by clever hashing strategies, e.g.  [13, 19].

  3. 3.

    Since this check leaks some information about the inputs, we have to perform the check in a secure manner.

  4. 4.

    Our multi-party PSI functionality uses similar idea as augmented semi-honest multi-party PSI as in previous works [27].

  5. 5.

    The commitment we implicitly use has been used previously in [11], as has the check sub-protocol.

  6. 6.

    In order to achieve our claimed efficiency we actually use UC commitments, but non-malleable commitments are sufficient for the security of the protocol.

  7. 7.

    For ease of notation, here we assume that the commitments are completely sent before \(\mathcal {A} \) commits himself. The very same argument also holds if \(\mathcal {A} \) only received synchronized messages of \(\mathsf {com}_j\) and has to start committing concurrently.

  8. 8.

    For reference see Figure 8 of [36].

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Authors and Affiliations

  1. Department of Computer Science, Aarhus University, Aarhus, Denmark

    Satrajit Ghosh

  2. ITK Engineering GmbH, Rülzheim, Germany

    Tobias Nilges

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  1. Satrajit Ghosh
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Correspondence to Satrajit Ghosh .

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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Ghosh, S., Nilges, T. (2019). An Algebraic Approach to Maliciously Secure Private Set Intersection. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11478. Springer, Cham. https://doi.org/10.1007/978-3-030-17659-4_6

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