The Optimal Sub-Packetization of Linear Capacity-Achieving PIR Schemes With Colluding Servers

Abstract

Suppose $M$ records are replicated in $N$ servers (each storing all $M$ records), a user wants to privately retrieve one record by accessing the servers such that the identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a $T$ -private information retrieval (PIR) scheme. In practice, capacity-achieving and small sub-packetization are both desired for PIR schemes, because the former implies the highest download rate and the latter means simple realization. Meanwhile, sub-packetization is the key technique for achieving capacity. In this paper, we characterize the optimal sub-packetization for linear capacity-achieving $T$ -PIR schemes. First, a lower bound on the sub-packetization $L$ for linear capacity-achieving $T$ -PIR schemes is proved, i.e., $L\geq dn^{M-1}$ , where $d={\mathrm{ gcd}}(N,T)$ and $n=N/d$ . Then, for general values of $M$ and $N>T\geq 1$ , a linear capacity-achieving $T$ -PIR scheme with sub-packetization $dn^{M-1}$ is designed. Comparing with the first capacity-achieving $T$ -PIR scheme given by Sun and Jafar in 2016, our scheme reduces the sub-packetization from $N^{M}$ to the optimal and further reduces the field size by a factor of $Nd^{M-2}$ .