Double blind T -private information retrieval (DB-TPIR) enables two users, each of whom specifies an index (θ 1 , θ 2 , resp.), to efficiently retrieve a message W (θ 1 , θ 2 ) labeled by the two indices, from a set of N servers that store all messagesthat the two users' indices are kept private from any set of up to T 1 , T 2 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 M -way blind X-secure T -private information retrieval (MB-XS-TPIR) with multiple (M ) indices, each belonging to a different user, arbitrary privacy levels for each index (T 1 , T 2 , • • • , T M ), and arbitrary level of security (X) of data storage, so that the message W (θ 1 , θ 2 , • • • , θ M ) can be efficiently retrieved while the stored data is held secure against collusion among up to X colluding servers, the m th user's index is private against collusion among up to T m servers, and each user's index θ m is private from all other users. The general scheme relies on a tensor-product based extension of cross-subspace alignment and retrieves 1 − (X + T 1 + • • • + T M )/N bits of desired message per bit of download.
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