A New Capacity-Achieving Private Information Retrieval Scheme With (Almost) Optimal File Length for Coded Servers

Abstract

In a distributed storage system, private information retrieval (PIR) guarantees that a user retrieves one file from the system without revealing any information about the identity of its interested file to any individual server. In this paper, we investigate an $(N,K,M)$ coded server model of PIR, where each of $M$ files is distributed to $N$ servers in the form of $(N,K)$ maximum distance separable (MDS) code for some $N>K$ and $M>1$ . As a result, we propose a new capacity-achieving $(N,K,M)$ coded linear PIR scheme such that it can be implemented with file length $\frac {K(N-K)}{\gcd (N,K)}$ , which is much smaller than the previous best result $K\left({\frac {N}{\gcd (N,K)}}\right)^{M-1}$ . Notably, among all the capacity-achieving coded linear PIR schemes, we show that the file length is optimal if $M>\big \lfloor \frac {K}{\gcd (N,K)}-\frac {K}{N-K}\big \rfloor +1$ or $\min (K,N-K)|N$ , and within a multiplicative gap $\frac {\min (K,N-K)}{\gcd (N,K)}$ of a lower bound on the minimum file length otherwise.