Private Set Intersection Using Multi -Message Symmetric Private Information Retrieval

We study the problem of private set intersection (PSI). In PSI, there are two entities, each storing a set P whose elements are picked from a finite set SK, on N replicated and non-colluding databases. It is required to determine the set intersection P1 11 P2 without leaking any information about the remaining elements to the other entity. 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 show that the exact capacity of MM-SPIR is CMM_SPIR = 1 k, when P < K 1, if the common randomness S satisfies H(S) > NP, per desired symbol. This result implies that there is no gain for MM-SPIR over successive single-message SPIR. We present a novel capacity -achieving scheme which builds seamlessly over the multi -message PIR (MM-PIR) scheme. Based on this capacity result for the MMSPIR problem, we show that the optimal download cost for the PSI problem is given by min {[ P1 N2 [ NCR1 where P is N2 -11 the cardinality of the set 'Pt.