Double Blind T-Private Information Retrieval

Abstract

Double blind <inline-formula> <tex-math notation= LaTeX >$T$ </tex-math></inline-formula>-private information retrieval (DB-TPIR) enables two users, each of whom specifies an index (<inline-formula> <tex-math notation= LaTeX >$\\theta _{1}, \\theta _{2}$ </tex-math></inline-formula>, resp.), to efficiently retrieve a message <inline-formula> <tex-math notation= LaTeX >$W(\\theta _{1},\\theta _{2})$ </tex-math></inline-formula> labeled by the two indices, from a set of <inline-formula> <tex-math notation= LaTeX >$N$ </tex-math></inline-formula> servers that store all messages <inline-formula> <tex-math notation= LaTeX >$W(k_{1},k_{2}), k_{1}\\in \\{1,2,\\ldots, K_{1}\\}, k_{2}\\in \\{1,2,\\ldots, K_{2}\\}$ </tex-math></inline-formula>, such that the two users’ indices are kept private from any set of up to <inline-formula> <tex-math notation= LaTeX >$T_{1},T_{2}$ </tex-math></inline-formula> colluding servers, respectively, as well as from each other. A DB-TPIR scheme based on cross-subspace alignment is proposed in this paper, and shown to be capacity-achieving in the asymptotic setting of large number of messages and bounded latency. The scheme is then extended to <inline-formula> <tex-math notation= LaTeX >$M$ </tex-math></inline-formula>-way blind <inline-formula> <tex-math notation= LaTeX >$X$ </tex-math></inline-formula>-secure <inline-formula> <tex-math notation= LaTeX >$T$ </tex-math></inline-formula>-private information retrieval (MB-XS-TPIR) with multiple (<inline-formula> <tex-math notation= LaTeX >$M$ </tex-math></inline-formula>) indices, each belonging to a different user, arbitrary privacy levels for each index (<inline-formula> <tex-math notation= LaTeX >$T_{1}, T_{2},\\ldots, T_{M}$ </tex-math></inline-formula>), and arbitrary level of security (<inline-formula> <tex-math notation= LaTeX >$X$ </tex-math></inline-formula>) of data storage, so that the message <inline-formula> <tex-math notation= LaTeX >$W(\\theta _{1},\\theta _{2},\\ldots, \\theta _{M})$ </tex-math></inline-formula> can be efficiently retrieved while the stored data is held secure against collusion among up to <inline-formula> <tex-math notation= LaTeX >$X$ </tex-math></inline-formula> colluding servers, the <inline-formula> <tex-math notation= LaTeX >$m^{th}$ </tex-math></inline-formula> user’s index is private against collusion among up to <inline-formula> <tex-math notation= LaTeX >$T_{m}$ </tex-math></inline-formula> servers, and each user’s index <inline-formula> <tex-math notation= LaTeX >$\\theta _{m}$ </tex-math></inline-formula> is private from all other users. The general scheme relies on a tensor-product based extension of cross-subspace alignment and retrieves <inline-formula> <tex-math notation= LaTeX >$1-(X+T_{1}+\\cdots +T_{M})/N$ </tex-math></inline-formula> bits of desired message per bit of download.