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Asymmetry Hurts: Private Information Retrieval Under Asymmetric Traffic Constraints

9 January 2018
Karim A. Banawan
S. Ulukus
ArXiv (abs)PDFHTML
Abstract

We consider the classical setting of private information retrieval (PIR) of a single message (file) out of MMM messages from NNN distributed databases under the new constraint of \emph{asymmetric traffic} from databases. In this problem, the \emph{ratios between the traffic} from the databases are constrained, i.e., the ratio of the length of the answer string that the user (retriever) receives from the nnnth database to the total length of all answer strings from all databases is constrained to be τn\tau_nτn​. This may happen if the user's access to the databases is restricted due database availability, channel quality to the databases, and other factors. For this problem, for fixed MMM, NNN, we develop a general upper bound Cˉ(τ)\bar{C}(\boldsymbol{\tau})Cˉ(τ), which generalizes the converse proof of Sun-Jafar, where database symmetry was inherently used. Our converse bound is a piece-wise affine function in the traffic ratio vector τ=(τ1,⋯ ,τN)\boldsymbol{\tau}=(\tau_1, \cdots, \tau_N)τ=(τ1​,⋯,τN​). For the lower bound, we explicitly show the achievability of (M+N−1M)\binom{M+N-1}{M}(MM+N−1​) corner points. For the remaining traffic ratio vectors, we perform time-sharing between these corner points. The recursive structure of our achievability scheme is captured via a system of difference equations. The upper and lower bounds exactly match for M=2M=2M=2 and M=3M=3M=3 for any NNN and any τ\boldsymbol{\tau}τ. The results show strict loss of PIR capacity due to the asymmetric traffic constraints compared with the symmetric case of Sun-Jafar which implicitly uses τn=1N\tau_n=\frac{1}{N}τn​=N1​ for all nnn.

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