**Fast Private Set Intersection from Homomorphic Encryption**

*Hao Chen and Kim Laine and Peter Rindal*

**Abstract: **Private Set Intersection (PSI) is a cryptographic technique that allows two parties to compute the intersection of their sets without revealing anything except the intersection. We use fully homomorphic encryption to construct a fast PSI protocol with a small communication overhead that works particularly well when one of the two sets is much smaller than the other, and is secure against semi-honest adversaries.

The most computationally efficient PSI protocols have been constructed using tools such as hash functions and oblivious transfer, but a potential limitation with these approaches is the communication complexity, which scales linearly with the size of the larger set. This is of particular concern when performing PSI between a constrained device (cellphone) holding a small set, and a large service provider (e.g. \emph{WhatsApp}), such as in the Private Contact Discovery application.

Our protocol has communication complexity linear in the size of the smaller set, and logarithmic in the larger set. More precisely, if the set sizes are $N_Y < N_X$, we achieve a communication overhead of $O(N_Y \log N_X)$. Our running-time-optimized benchmarks show that it takes $36$ seconds of online-computation, $71$ seconds of non-interactive (receiver-independent) pre-processing, and only $12.5$MB of round trip communication to intersect five thousand $32$-bit strings with $16$ million $32$-bit strings. Compared to prior works, this is roughly a $38$--$115 \times$ reduction in communication with minimal difference in computational overhead.

**Category / Keywords: **private set intersection; fully homomorphic encryption

**Date: **received 31 Mar 2017, last revised 6 Sep 2017

**Contact author: **kim laine at microsoft com

**Available format(s): **PDF | BibTeX Citation

**Note: **Added a reference to [Kiss et al., 2017]

**Version: **20170906:224834 (All versions of this report)

**Short URL: **ia.cr/2017/299

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