The Capacity of Private Information Retrieval From Coded Databases

We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> non-colluding databases, each storing an MDS-coded version of <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is <inline-formula> <tex-math notation="LaTeX">$C=(1+{K}/{N}+{K^{2}}/{N^{2}}+\cdots +{K^{M-1}}/{N^{M-1}})^{-1}=(1+R_{c}+R_{c}^{2}+\cdots +R_{c}^{M-1})^{-1}=({1-R_{c}})/({1-R_{c}^{M}})$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$R_{c}$ </tex-math></inline-formula> is the rate of the <inline-formula> <tex-math notation="LaTeX">$(N,K)$ </tex-math></inline-formula> MDS code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost when the storage code is restricted to the class of MDS codes. The result generalizes the achievability and converse results for the classical PIR with replicated databases to the case of MDS-coded databases.

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