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Private Information Retrieval from Non-Replicated Databases

31 December 2018
Karim A. Banawan
S. Ulukus
ArXiv (abs)PDFHTML
Abstract

We consider the problem of private information retrieval (PIR) of a single message out of KKK messages from NNN 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 KKK messages, here, we consider the case of non-replicated databases under a special non-replication structure where each database stores MMM out of KKK messages and each message is stored across RRR different databases. This generates an RRR-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=2M=2M=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 \emph{fully-connected graphs}. We prove that the PIR capacity for the case of cyclic graphs is 2K+1\frac{2}{K+1}K+12​, and the PIR capacity for the case of fully-connected graphs is min⁡{2K,12}\min\{\frac{2}{K},\frac{1}{2}\}min{K2​,21​}. 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.

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