Explicit Constructions of Capacity-Achieving T-PIR Schemes Over Small Fields via Generalized Minor Matrices

Abstract

Suppose a distributed storage system containing M files is replicated across N servers, and a user wants to privately retrieve one file by accessing the servers such that the identity of the retrieved file is kept secret from any subset of up to T servers, where each file can be viewed as a vector over the q-ary finite field $\mathbb {F}_{q}$ . A scheme designed for this purpose is called a T-private information retrieval (T-PIR) scheme. We consider the problem of explicitly constructing capacity-achieving T-PIR schemes over small finite fields. In this paper, we first provide a general framework for constructing explicit capacity-achieving T-PIR schemes for all parameters, which only relies on an MDS array matrix with a special information set. To construct such an MDS array matrix, we propose a new family of matrices over finite fields, called the generalized minor matrices of the Moore matrix, and establish a series of key identities. By combining favourable properties of generalized minor matrices with our framework, we construct an explicit capacity-achieving T-PIR scheme with optimal sub-packetization over the field $\mathbb {F}_{q}$ , as small as possible, for three classes of parameters $N,T,M\geq 3$ . Specifically, the first class of construction works for all $N=d(2t-1), T=dt$ , and the field size q is the least prime power satisfying $q^{t-1}\geq N$ . Moreover, this construction generalizes the scheme proposed by Xu and Wang in 2022, which only considers the case of $N=d(2t-1), T=dt$ with $2^{t-1}\geq N$ . For all $N=d(2t+1), T=dt$ , our second T-PIR scheme is the first explicit construction, and the field size q is the least prime power satisfying $q^{t} \geq N$ , which is the smallest field size among all known explicit capacity-achieving T-PIR schemes. Particularly, when $2^{t}\geq N$ , the field size of such constructions can be reduced to 2. In the case of $N=4s$ and $T=2s+1$ , our scheme is the first one to reduce the field size to $q=2$ . Compared with all known explicitly capacity-achieving T-PIR schemes, the required field of our schemes has the smallest size.