Multi-Party Threshold Private Set Intersection with Sublinear Communication

In multi-party threshold private set intersection (PSI), n parties each with a private set wish to compute the intersection of their sets if the intersection is sufficiently large. Previously, Ghosh and Simkin (CRYPTO 2019) studied this problem for the two-party case and demonstrated interesting lower and upper bounds on the communication complexity. In this work, we investigate the communication complexity of the multi-party setting (n ≥ 2). We consider two functionalities for multi-party threshold PSI. In the first, parties learn the intersection if each of their sets and the intersection differ by at most T . In the second functionality, parties learn the intersection if the union of all their sets and the intersection differ by at most T . For both functionalities, we show that any protocol must have communication complexity Ω(nT ). We build protocols with a matching upper bound of O(nT ) communication complexity for both functionalities assuming threshold FHE. We also construct a computationally more efficient protocol for the second functionality with communication complexity Õ(nT ) under a weaker assumption of threshold additive homomorphic encryption. As a consequence, we achieve the first “regular” multi-party PSI protocol where the communication complexity only grows with the size of the set difference and does not depend on the size of the input sets.

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