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Private Information Retrieval from Storage Constrained Databases -- Coded Caching meets PIR

14 November 2017
Maryam Abdul-Wahid
Firas Almoualem
Deepak Kumar
Ravi Tandon
ArXiv (abs)PDFHTML
Abstract

Private information retrieval (PIR) allows a user to retrieve a desired message out of KKK possible messages from NNN databases without revealing the identity of the desired message. Majority of existing works on PIR assume the presence of replicated databases, each storing all the KKK messages. In this work, we consider the problem of PIR from storage constrained databases. Each database has a storage capacity of μKL\mu KLμKL bits, where KKK is the number of messages, LLL is the size of each message in bits, and μ∈[1/N,1]\mu \in [1/N, 1]μ∈[1/N,1] is the normalized storage. In the storage constrained PIR problem, there are two key design questions: a) how to store content across each database under storage constraints; and b) construction of schemes that allow efficient PIR through storage constrained databases. The main contribution of this work is a general achievable scheme for PIR from storage constrained databases for any value of storage. In particular, for any (N,K)(N,K)(N,K), with normalized storage μ=t/N\mu= t/Nμ=t/N, where the parameter ttt can take integer values t∈{1,2,…,N}t \in \{1, 2, \ldots, N\}t∈{1,2,…,N}, we show that our proposed PIR scheme achieves a download cost of (1+1t+1t2+⋯+1tK−1)\left(1+ \frac{1}{t}+ \frac{1}{t^{2}}+ \cdots + \frac{1}{t^{K-1}}\right)(1+t1​+t21​+⋯+tK−11​). The extreme case when μ=1\mu=1μ=1 (i.e., t=Nt=Nt=N) corresponds to the setting of replicated databases with full storage. For this extremal setting, our scheme recovers the information-theoretically optimal download cost characterized by Sun and Jafar as (1+1N+⋯+1NK−1)\left(1+ \frac{1}{N}+ \cdots + \frac{1}{N^{K-1}}\right)(1+N1​+⋯+NK−11​). For the other extreme, when μ=1/N\mu= 1/Nμ=1/N (i.e., t=1t=1t=1), the proposed scheme achieves a download cost of KKK. The interesting aspect of the result is that for intermediate values of storage, i.e., 1/N<μ<11/N < \mu <11/N<μ<1, the proposed scheme can strictly outperform memory-sharing between extreme values of storage.

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