Optimal Download Cost of Private Information Retrieval for Arbitrary Message Length

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. For fixed $K,N\in\mathbb{N}$, 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 $C=(1+1/N+1/N^2+\cdots+1/N^{K-1})^{-1}$. 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 $D_L=\lceil\frac{L}{C}\rceil$. 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, $D_L\in\left\{\left\lceil \frac{L'}{C}\right\rceil,\left\lceil \frac{L'}{C}\right\rceil-1,\left\lceil \frac{L'}{C}\right\rceil-2\right\}$, where $L'= \lceil L \log_{M'} M\rceil$.