The Capacity of Private Information Retrieval With Private Side Information Under Storage Constraints

We consider the problem of private information retrieval (PIR) of a single message out of <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> messages from <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> replicated and non-colluding databases where a cache-enabled user (retriever) of cache-size <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> possesses side information in the form of uncoded portions of the messages where the message identities are unknown to the databases. The identities of these side information messages need to be kept private from the databases, i.e., we consider PIR with private side information (PSI). We characterize the optimal normalized download cost for this PIR-PSI problem under the storage constraint <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> as <inline-formula> <tex-math notation="LaTeX">$D^{*}=1+\frac {1}{N}+\frac {1}{N^{2}}+ {\dots }+\frac {1}{N^{K-1-M}}+ \frac {1-r_{M}}{N^{K-M}}+\frac {1-r_{M-1}}{N^{K-M+1}}+ {\dots }+\frac {1-r_{1}}{N^{K-1}}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> is the number of side information messages and <inline-formula> <tex-math notation="LaTeX">$r_{i}$ </tex-math></inline-formula> is the portion of the <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>th side information message that is cached with <inline-formula> <tex-math notation="LaTeX">$\sum _{i=1}^{M} r_{i}=S$ </tex-math></inline-formula>. Based on this capacity result, we prove two facts: First, for a fixed memory size <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> and a fixed number of accessible messages <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>, uniform caching achieves the lowest normalized download cost, i.e., <inline-formula> <tex-math notation="LaTeX">$r_{i}=\frac {S}{M}$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$i=1, {\dots }, M$ </tex-math></inline-formula>, is optimum. Second, for a fixed memory size <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>, among all possible <inline-formula> <tex-math notation="LaTeX">$K-\left \lceil{ {S} }\right \rceil +1$ </tex-math></inline-formula> uniform caching schemes, the uniform caching scheme which caches <inline-formula> <tex-math notation="LaTeX">$M=K$ </tex-math></inline-formula> messages achieves the lowest normalized download cost.

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