On Sub-Packetization of Capacity-Achieving PIR Schemes for MDS Coded Databases

Consider the problem of private information retrieval (PIR) over a distributed storage system where $M$ records are stored across $N$ servers by using an $[N,K]$ MDS code. For simplicity, this problem is usually referred as the coded-PIR problem. The capacity of coded-PIR with privacy against any individual server was determined by Banawan and Ulukus in 2016, i.e., $\mathcal{C}_{\tiny C-PIR}=(1+\frac{K}{N}+\dots+\frac{K^{M-1}}{N^{M-1}})^{-1}$. They also presented a linear capacity-achieving scheme with sub-packetization $KN^{M}$. In this paper we focus on minimizing the sub-packetization for linear capacity-achieving coded-PIR schemes. We prove that the sub-packetization for all linear capacity-achieving coded-PIR schemes in the nontrivial cases (i.e. $N>K\geq 1$ and $M>1$) must be no less than $Kn^{M-1}$, where $n=N/{\rm gcd}(N,K)$. Moreover, we design a linear capacity-achieving coded-PIR scheme with sub-packetization $Kn^{M-1}$ for all $N>K\geq 1$ and $M>1$. Therefore, $Kn^{M-1}$ is the optimal sub-packetization for linear capacity-achieving coded-PIR schemes.