Optimal Download Cost of Private Information Retrieval for Arbitrary Message Length

A private information retrieval (PIR) scheme is a mechanism that allows a user to retrieve any one out of <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> messages from <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> non-communicating replicated databases, each of which stores all <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> messages, without revealing anything (in the information theoretic sense) about the identity of the desired message index to any individual database. If the size of each message is <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> bits and the total download required by a PIR scheme from all <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> databases is <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> bits, then <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> is called the download cost and the ratio <inline-formula> <tex-math notation="LaTeX">$L/D$ </tex-math></inline-formula> is called an achievable rate. For fixed <inline-formula> <tex-math notation="LaTeX">$K,N\in \mathbb {N}$ </tex-math></inline-formula>, the capacity of PIR, denoted by <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula>, is the supremum of achievable rates over all PIR schemes and over all message sizes, and was recently shown to be <inline-formula> <tex-math notation="LaTeX">$C=(1+1/N+1/N^{2}+\cdots +1/N^{K-1})^{-1}$ </tex-math></inline-formula>. In this paper, for arbitrary <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>, we explore the minimum download cost <inline-formula> <tex-math notation="LaTeX">$D_{L}$ </tex-math></inline-formula> across all PIR schemes (not restricted to linear schemes) for arbitrary message lengths <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> under arbitrary choices of alphabet (not restricted to finite fields) for the message and download symbols. If the same <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary alphabet is used for the message and download symbols, then we show that the optimal download cost in <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary symbols is <inline-formula> <tex-math notation="LaTeX">$D_{L}=\lceil \frac {L}{C}\rceil$ </tex-math></inline-formula>. If the message symbols are in <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary alphabet and the downloaded symbols are in <inline-formula> <tex-math notation="LaTeX">$M'$ </tex-math></inline-formula>-ary alphabet, then we show that the optimal download cost in <inline-formula> <tex-math notation="LaTeX">$M'$ </tex-math></inline-formula>-ary symbols, <inline-formula> <tex-math notation="LaTeX">$D_{L}\in \{\lceil ~({L'}/{C})\rceil, \lceil ~({L'}/{C}\rceil -1,\lceil ~({L'}/{C})\rceil -2\}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$L'= \lceil L \log _{M'} M\rceil$ </tex-math></inline-formula>, i.e., the optimal download cost is characterized to within two symbols.