We study the symmetric private information retrieval (SPIR) problem under arbitrary collusion and eavesdropping patterns for replicated databases. We find its capacity, which is the same as the capacity of the original SPIR problem with the number of serversNreplaced by a numberF*. The numberF* is the optimal solution to a linear programming problem, and it is a function of the joint pattern, which is the union of the collusion and eavesdropping pattern. This is the first result that shows how two arbitrary patterns collectively affect the capacity of the SPIR problem. We draw the conclusion that for SPIR problems, the collusion and eavesdropping constraints are interchangeable in terms of capacity, i.e., the two patterns play the same role in the SPIR problem and the capacity remains unchanged if we exchange the colluding and eavesdropping patterns. As corollaries of our result, the capacity of the SPIR problem under arbitrary collusion patterns, and the capacity of the PIR problem where each colluding set is included in some eavesdropping set, are also found. Some extensions with restrictions to finite message lengths are provided, and in this case, upper and lower bounds on the capacity are given. The lower bound is described with a solution to an integer linear programming problem.
Single-server private information retrieval (PIR) schemes are known to require linear query time. Recent works circumvent these classical lower bounds by leveraging preprocessing to answer queries in sublinear time. We prove computation lower bounds for PIR with preprocessing schemes making blackbox usage of any cryptography (such as random oracles or virtual blackbox obfuscation). If the client stores
s bits about an
n-bit database, then answering
k=Ω(s) queries requires either
Ω(n/s) amortized online communication or
Ω(n/s) amortized server cryptographic operations. This is tight, as known constructions match either bound while outperforming the other. Our lower bound is unconditional and allows arbitrary query protocols, weakened privacy, and server-side database encodings (including doubly efficient PIR) whenever the encoding is independent of the blackbox cryptography. Previous bounds were only known conditionally and for restricted classes of preprocessing, e.g., under non-encoding assumptions. Our framework also yields
Ω(n/s) communication lower bounds for schemes with
o(n/s) server cryptographic operations, communication-determined server cryptography, or perfect privacy in the idealized model. Finally, we prove lower bounds for symmetric PIR with client preprocessing in the random oracle model and give a matching construction using only one-way functions in the online phase.
Both the oblivious transfer (OT) problem and the symmetric private information retrieval (SPIR) problem studies the scenario where a client retrieves information privately and securely from databases, i.e., the privacy of the client is protected from the databases, and the undesired information is protected from the client. The OT problem studies the case of one database plus additional noisy resources between the database and the client. The SPIR problem studies the case of multiple replicated and non-colluding databases. In this paper, we combine the two models and propose a new problem of oblivious transfer (OT) with two replicated databases and a binary erasure multiple access channel connecting the databases and the client. We first provide an upper bound on the OT capacity. We then propose a protocol which achieves the upper bound. Therefore, we obtain the capacity of OT for this model. In our achievability and converse proofs, we utilized the techniques from both traditional OT and PIR. Compared to schemes that utilizes only techniques from OT, we see a 100% increase in the achieved OT rate.