Private Set Intersection: A Multi-Message Symmetric Private Information Retrieval Perspective

We study the problem of private set intersection (PSI). In this problem, there are two entities Ei, for i=1, 2, each storing a set Pi, whose elements are picked from a finite field mathbbFK, on Ni replicated and non-colluding databases. It is required to determine the set intersection P1 cap P2 without leaking any information about the remaining elements to the other entity with the least amount of downloaded bits. We first show that the PSI problem can be recast as a multi-message symmetric private information retrieval (MM-SPIR) problem. Next, as a stand-alone result, we derive the information-theoretic sum capacity of MM-SPIR, CMM-SPIR. We show that with K messages, N databases, and the size of the desired message set P, the exact capacity of MM-SPIR is CMM-SPIR = 1 - frac1N when P ≤ K-1, provided that the entropy of the common randomness S satisfies H(S) ≥ fracPN-1 per desired symbol. This result implies that there is no gain for MM-SPIR over successive single-message SPIR (SM-SPIR). For the MM-SPIR problem, we present a novel capacity-achieving scheme that builds on the near-optimal scheme of Banawan-Ulukus originally proposed for the multi-message PIR (MM-PIR) problem without database privacy constraints. Surprisingly, our scheme here is exactly optimal for the MM-SPIR problem for any P, in contrast to the scheme for the MM-PIR problem, which was proved only to be near-optimal. Our scheme is an alternative to the SM-SPIR scheme of Sun-Jafar. Based on this capacity result for MM-SPIR, and after addressing the added requirements in its conversion to the PSI problem, we show that the optimal download cost for the PSI problem is minleftleftlceilfracP1 N2N2-1rightrceil, leftlceilfracP2 N1N1-1rightrceilright, where Pi is the cardinality of set Pi