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

31 December 2019
Zhusheng Wang
Karim A. Banawan
S. Ulukus
ArXiv (abs)PDFHTML
Abstract

We study the problem of private set intersection (PSI). In this problem, there are two entities EiE_iEi​, for i=1,2i=1, 2i=1,2, each storing a set Pi\mathcal{P}_iPi​, whose elements are picked from a finite field FK\mathbb{F}_KFK​, on NiN_iNi​ replicated and non-colluding databases. It is required to determine the set intersection P1∩P2\mathcal{P}_1 \cap \mathcal{P}_2P1​∩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−SPIRC_{MM-SPIR}CMM−SPIR​. We show that with KKK messages, NNN databases, and the size of the desired message set PPP, the exact capacity of MM-SPIR is CMM−SPIR=1−1NC_{MM-SPIR} = 1 - \frac{1}{N}CMM−SPIR​=1−N1​ when P≤K−1P \leq K-1P≤K−1, provided that the entropy of the common randomness SSS satisfies H(S)≥PN−1H(S) \geq \frac{P}{N-1}H(S)≥N−1P​ 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 PPP, 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 min⁡{⌈P1N2N2−1⌉,⌈P2N1N1−1⌉}\min\left\{\left\lceil\frac{P_1 N_2}{N_2-1}\right\rceil, \left\lceil\frac{P_2 N_1}{N_1-1}\right\rceil\right\}min{⌈N2​−1P1​N2​​⌉,⌈N1​−1P2​N1​​⌉}, where PiP_iPi​ is the cardinality of set Pi\mathcal{P}_iPi​

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