Private Information Retrieval from Joint Systematic MDS-Coded with Non-Colluding Servers: Bounds and Constructions

Abstract

Consider a distributed storage system consisting of $N$ non-colluding servers that collectively store a database of $M$ files encoded using an $[N,K]$ maximum distance separable(MDS) code. A user wishes to retrieve one file privately by accessing the servers without revealing the identity of the requested file. A scheme designed for this purpose is called a joint MDS-coded private information retrieval(PIR) scheme, which was first introduced by Sun and Tian in 2019 to break the capacity $\frac{1-K/N}{1-(K/N)^M}$ of the separate MDS-coded PIR schemes established by Banawan and Ulukus. However, the capacity of joint MDS-coded PIR remains largely unexplored. In this paper, we study the capacity of joint MDS-coded PIR with systematic MDS array storage codes under prescribed storage patterns. Specifically, we first derive upper bounds on the capacity of joint MDS-coded PIR for $K=Mt$ and $K=Mt+1$, respectively. We then construct three joint MDS-coded PIR schemes for the cases $N\le K+t, K=Mt$, $N>K+t, K=Mt$ and $N\le K+t, K=Mt+1$. The proposed schemes require small file sizes and achieve higher retrieval rates: the first and third schemes exceed the capacity of separate MDS-coded PIR schemes, while the second scheme does so when the storage rate $\frac{K}{N}>r_M$ for some $0<r_M<\frac{M}{M+1}$. In particular, for $K=Mt$ and $N\leq K+t$, the proposed scheme achieves the derived upper bound, thereby establishing that the optimal joint MDS-coded PIR capacity under the considered storage pattern is $1-(1-\frac{1}{M})\frac{K}{N}$. Compared with capacity-achieving separate MDS-coded PIR schemes at the same storage-code rate, the proposed schemes may achieve a substantial relative retrieval-rate improvement: the maximum improvement can exceed $15\%$ when $M\geq 4$, exceed $20\%$ when $M\geq 9$, and asymptotically approach $1-2/e\approx 26.42\%$ as M increases.