Asymmetry Hurts: Private Information Retrieval Under Asymmetric Traffic Constraints

Abstract

We consider the classical setting of private information retrieval (PIR) of a single message (file) out of $M$ messages from $N$ distributed databases under the new constraint of asymmetric traffic from databases. In this problem, the 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 $n$ th database to the total length of all answer strings from all databases is constrained to be $\tau _{n}$ . This may happen if the user’s access to the databases is restricted due to database availability, channel quality to the databases, and other factors. For this problem, for fixed $M$ , $N$ , we develop a general upper bound $\bar {C}({\boldsymbol {\tau }})$ , 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 ${\boldsymbol {\tau }}=(\tau _{1}, \cdots , \tau _{N})$ . For the lower bound, we explicitly show the achievability of $\binom {M+N-1}{M}$ 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 $M=2$ and $M=3$ for any $N$ and any ${\boldsymbol {\tau }}$ . 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 $\tau _{n}=\frac {1}{N}$ for all $n$ .