A private information retrieval scheme is a mechanism that allows a user to retrieve any one out of K messages from N non-communicating replicated databases, each of which stores all K messages, without revealing anything about the identity of the desired message index to any individual database. If the size of each message is L bits and the total download required by a PIR scheme from all N databases is D bits, then D is called the download cost and the ratio L/D is called an achievable rate.

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... or fixed K, N ∈ N, the capacity of PIR, denoted by C, is the supremum of achievable rates over all PIR schemes and over all message sizes, and was recently shown to be C = (1 + 1/N + 1/N 2 + · · · + 1/N K−1 ) −1 . In this work, for arbitrary K, N , we explore the minimum download cost D L across all PIR schemes (not restricted to linear schemes) for arbitrary message lengths L under arbitrary choices of alphabet (not restricted to finite fields) for the message and download symbols. If the same M -ary alphabet is used for the message and download symbols, then we show that the optimal download cost in M -ary symbols is D L = ⌈ L C ⌉. If the message symbols are in M -ary alphabet and the downloaded symbols are in M ′ -ary alphabet, then we show that the optimal download cost in M ′ -ary symbols,