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Multi-Message Private Information Retrieval: Capacity Results and Near-Optimal Schemes

6 February 2017
Karim A. Banawan
S. Ulukus
ArXiv (abs)PDFHTML
Abstract

We consider the problem of multi-message private information retrieval (MPIR) from NNN non-communicating replicated databases. In MPIR, the user is interested in retrieving PPP messages out of MMM stored messages without leaking the identity of the retrieved messages. The information-theoretic sum capacity of MPIR CsPC_s^PCsP​ is the maximum number of desired message symbols that can be retrieved privately per downloaded symbol. For the case P≥M2P \geq \frac{M}{2}P≥2M​, we determine the exact sum capacity of MPIR as CsP=11+M−PPNC_s^P=\frac{1}{1+\frac{M-P}{PN}}CsP​=1+PNM−P​1​. The achievable scheme in this case is based on downloading MDS-coded mixtures of all messages. For P≤M2P \leq \frac{M}{2}P≤2M​, we develop lower and upper bounds for all M,P,NM,P,NM,P,N. These bounds match if the total number of messages MMM is an integer multiple of the number of desired messages PPP, i.e., MP∈N\frac{M}{P} \in \mathbb{N}PM​∈N. In this case, CsP=1−1N1−(1N)M/PC_s^P=\frac{1-\frac{1}{N}}{1-(\frac{1}{N})^{M/P}}CsP​=1−(N1​)M/P1−N1​​. The achievable scheme in this case generalizes the single-message capacity achieving scheme to have unbalanced number of stages per round of download. For all the remaining cases, the difference between the lower and upper bound is at most 0.00820.00820.0082, which occurs for M=5M=5M=5, P=2P=2P=2, N=2N=2N=2. Our results indicate that joint retrieval of desired messages is more efficient than successive use of single-message retrieval schemes.

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