Capacity-achieving private information retrieval scheme with a smaller sub-packetization

Abstract

<p style='text-indent:20px;'>Private information retrieval (PIR) allows a user to retrieve one out of <inline-formula><tex-math id="M1">\begin{document}$ M $\end{document}</tex-math></inline-formula> messages from <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> servers without revealing the identity of the desired message. Every message consists of <inline-formula><tex-math id="M3">\begin{document}$ L $\end{document}</tex-math></inline-formula> symbols (packets) from an additive group and the length <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula> is called the sub-packetization. A PIR scheme with download cost (DC) <inline-formula><tex-math id="M5">\begin{document}$ D/L $\end{document}</tex-math></inline-formula> is implemented by querying <inline-formula><tex-math id="M6">\begin{document}$ D $\end{document}</tex-math></inline-formula> sums of the symbols to servers. We assume that each uncoded server can store up to <inline-formula><tex-math id="M7">\begin{document}$ tLM/N $\end{document}</tex-math></inline-formula> symbols, <inline-formula><tex-math id="M8">\begin{document}$ t\in\{1,2,\cdots,N\} $\end{document}</tex-math></inline-formula>. The minimum DC of storage constrained PIR was determined by Attia <i>et al.</i> in 2018 to be <inline-formula><tex-math id="M9">\begin{document}$ DC_{min} = 1+1/t+1/t^{2}+\cdots+1/t^{M-1} $\end{document}</tex-math></inline-formula>. The capacity of storage constrained PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired symbols that can be privately retrieved per bit of downloaded symbols. Tandon <i>et al.</i> designed a capacity-achieving PIR scheme with sub-packetization <inline-formula><tex-math id="M10">\begin{document}$ L' = {N\choose t}t^{M} $\end{document}</tex-math></inline-formula> of each message. In this paper, we design a PIR scheme with <inline-formula><tex-math id="M11">\begin{document}$ t $\end{document}</tex-math></inline-formula> times smaller sub-packetization <inline-formula><tex-math id="M12">\begin{document}$ L'/t $\end{document}</tex-math></inline-formula> and with the minimum DC for any parameters <inline-formula><tex-math id="M13">\begin{document}$ N, M, t $\end{document}</tex-math></inline-formula>. We also prove that <inline-formula><tex-math id="M14">\begin{document}$ t^{M-1} $\end{document}</tex-math></inline-formula> is a factor of sub-packetization <inline-formula><tex-math id="M15">\begin{document}$ L $\end{document}</tex-math></inline-formula> for any capacity-achieving PIR scheme from storage constrained servers.