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 $E_{i}$ , for notation="LaTeX">$i=1, 2$ each storing a notation="LaTeX">$\mathcal {P}_{i}$ whose elements picked from finite notation="LaTeX">$\mathbb {S}_{K}$ on notation="LaTeX">$N_{i}$ replicated and non-colluding databases. It is required to determine notation="LaTeX">${\mathcal {P}}_{1} \cap {\mathcal {P}} _{2}$ without leaking any information about remaining other entity, do with least amount downloaded bits. first show that PSI can be recast as multi-message symmetric retrieval (MM-SPIR) certain added restrictions. Next, stand-alone result, we derive information-theoretic sum capacity MM-SPIR, notation="LaTeX">$C_{MM-SPIR}$ . notation="LaTeX">$K$ messages, notation="LaTeX">$N$ databases, given size desired message notation="LaTeX">$P$ exact MM-SPIR notation="LaTeX">$C_{MM-SPIR} = 1 - \frac {1}{N}$ when notation="LaTeX">$P \leq K-1$ provided entropy common randomness notation="LaTeX">$S$ satisfies notation="LaTeX">$H(S) \geq {P}{N-1}$ per symbol. When K$ trivially need This result implies no gain over successive single-message SPIR (SM-SPIR). For present novel capacity-achieving scheme which builds seamlessly near-optimal Banawan-Ulukus originally proposed PIR (MM-PIR) database privacy constraints. Surprisingly, our here exactly optimal in contrast MM-PIR was proved only near-optimal. Our an alternative usage SM-SPIR Sun-Jafar. Based after addressing requirements its conversion download cost by notation="LaTeX">$\min \left \{{\left \lceil{ {P_{1} N_{2}}{N_{2}-1}}\right \rceil, {P_{2} N_{1}}{N_{1}-1}}\right \rceil }\right \}$ where notation="LaTeX">$P_{i}$ cardinality {P}}_{i}$