Private Information Retrieval from Non-Replicated Databases

Abstract

We consider the problem of private information retrieval (PIR) of a single message out of K messages from N non-colluding and non-replicated databases. Different from the majority of the existing literature, which considers the case of replicated databases where all databases store the same content in the form of all K messages, here, we consider the case of non-replicated databases under a special non-replication structure where each database stores M out of K messages and each message is stored across R different databases. This generates an R-regular graph structure for the storage system where the vertices of the graph are the messages and the edges are the databases. We derive a general upper bound for M = 2 that depends on the graph structure. We then specialize the problem to storage systems described by two special types of graph structures: cyclic graphs and fully-connected graphs. We prove that the PIR capacity for the case of cyclic graphs is 2 K+1 , and the PIR capacity for the case of fully-connected graphs is min{ 2 K , 1 2 }. To that end, we propose novel achievable schemes for both graph structures that are capacity-achieving. The central insight in both schemes is to introduce dependency in the queries submitted to databases that do not contain the desired message, such that the requests can be compressed. In both cases, the results show severe degradation in PIR capacity due to non-replication.

Bounds on the Capacity of PIR over Graphs

“…Proof: The proof goes along the same lines as the proof of Lemma 1 in [5]. The first inequality holds since conditioning reduces entropy, hence we proceed to prove the second inequality.…”

“…First, the upper bound ( 22) is tight for certain graphs. Indeed, it was shown in [5] that the PIR capacity of the cycle graph…”

“…Whereas for the lower bound the following is known. In [5] it was shown that C (K 3 ) = 1 2 and C (K 4 ) ≥ 3 10 . For general N we have C (K N ) ≥ 1/N .…”

“…In [17], Raviv et al who initially introduced this model, studied the resiliency of the system against colluding servers. In [5], Banawan et al explored the information-theoretic graph-based PIR capacity in the non-colluding model; however, their graph replication model is different from the model considered in this paper. Indeed, in [5] the files are the graph's vertices, and each server is an edge that connects between the files the server stores.…”

“…In [5], Banawan et al explored the information-theoretic graph-based PIR capacity in the non-colluding model; however, their graph replication model is different from the model considered in this paper. Indeed, in [5] the files are the graph's vertices, and each server is an edge that connects between the files the server stores. Hence, by assumption, each server stores only two files, whereas no such constraint is imposed in our model.…”