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A private information retrieval scheme is a mechanism that allows a user toretrieve any one out of $K$ messages from $N$ non-communicating replicateddatabases, each of which stores all $K$ messages, without revealing anythingabout the identity of the desired message index to any individual database. Ifthe size of each message is $L$ bits and the total download required by a PIRscheme from all $N$ databases is $D$ bits, then $D$ is called the download costand 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 overall 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 $K, N$, weexplore the minimum download cost $D_L$ across all PIR schemes (not restrictedto linear schemes) for arbitrary message lengths $L$ under arbitrary choices ofalphabet (not restricted to finite fields) for the message and downloadsymbols. If the same $M$-ary alphabet is used for the message and downloadsymbols, 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 alphabetand the downloaded symbols are in $M'$-ary alphabet, then we show that theoptimal 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 $L'= \lceil L \log_{M'} M\rceil$.

Optimal Download Cost of Private Information Retrieval for Arbitrary Message Length