Multi-Message Private Information Retrieval

We consider the problem of multi-message private information retrieval (MPIR) from N non-communicating replicated databases. In MPIR, the user is interested in retrieving P messages out of M stored messages without leaking the identity of the retrieved messages. The information-theoretic sum capacity of MPIR C-s(P) is the maximum number of desired message symbols that can be retrieved privately per downloaded symbol. For the case P >= M/2, we determine the exact sum capacity of MPIR as C-s(P) = 1/1+M-P/PN. For P <= M/2, we develop lower and upper bounds, for all M, P, N. These bounds match if the number of messages M is an integer multiple of number of desired messages P, in which case, C-s(P) = 1-1/N/1 - (1/N)(M/P). Our results indicate that joint retrieval of desired messages is more efficient than successive use of single-message retrieval schemes.