Limits of Preprocessing for Single-Server PIR

We present lower bounds for the static cryptographic data structure problem of single-server private information retrieval (PIR). PIR considers the setting where a server holds a database of n entries and a client wishes to privately retrieve the i-th entry without revealing the index i to the server. In our work, we focus on PIR with preprocessing where an r-bit hint may be computed in a preprocessing stage and stored by the server to be used to perform private queries in expected time t. As our main result, we prove that for any single-server, computationally secure PIR with preprocessing, it must be that tr = Ω(n log n) when r = Ω(log n). If r = O(log n), then we show that t = Ω(n). Our lower bound holds even when the scheme errs with probability 1/n2 and the adversary's distinguishing advantage is 1/n. Our work improves upon the tr = Ω(n) lower bound of Beimel, Ishai and Malkin [JoC'04]. For information-theoretic security, we present a stronger lower bound of t + r = Ω(n) and show a matching construction. Both our lower bounds apply for public-key doubly-efficient PIRs of Boyle, Ishai, Pass and Wootters [TCC'17]. Additionally, our lower bound for information-theoretic security also applies for offline-online PIRs as defined by Corrigan-Gibbs and Kogan [Eurocrypt'20], where the hint is private and only viewed by the client. We prove our lower bounds in a variant of the cell probe model where only accesses to the database are charged cost and computation and accesses to the hint are free. Our main technical contribution is a novel use of the cell sampling technique (also known as the incompressibility technique) used to obtain lower bounds on data structures. In previous works, this technique only leveraged the correctness guarantees to prove lower bounds even when used for cryptographic primitives. Our work combines the cell sampling technique with the privacy guarantees of PIR to construct a powerful, polynomial-time adversary that is critical to proving our higher lower bounds.