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This paper investigates reducing sub-packetization of capacity-achieving schemes for uncoded Storage Constrained Private Information Retrieval (SC-PIR) systems. In the SC-PIR system, a user aims to download one out of K files from N servers while revealing nothing about the identity of the requested file to any individual server, in which the K files are stored at the N servers in an uncoded form and each server can store up to &#x03BC;K equivalent files, where &#x03BC; is the normalized storage capacity of each server. We first prove that there exists a capacity-achieving SC-PIR scheme for a given storage design if and only if all the packets are stored exactly at M\triangleq &#x03BC;N servers for &#x03BC; such that M=&#x03BC;N &#x2208; {2,3,...,N}. Then, the optimal sub-packetization for capacity-achieving linear SC-PIR schemes is characterized as the solution to an optimization problem, which is typically hard to solve since it involves non-continuous indicator functions. Moreover, a new notion of array called Storage Design Array (SDA) is introduced for the SC-PIR system. With any given SDA, an associated capacity-achieving SC-PIR scheme is constructed. Next, the SC-PIR schemes that have equal-size packets are investigated. Furthermore, the optimal equal-size sub-packetization among all capacity-achieving linear SC-PIR schemes characterized by Woolsey et al. is proved to be \frac N(M-1)gcd(N,M), which is achieved by a construction of SDA. Finally, by allowing unequal size of packets, a greedy SDA construction is proposed, where the sub-packetization of the associated SC-PIR scheme is upper bounded by \frac N(M-1)gcd(N,M). Among all capacity-achieving linear SC-PIR schemes, the sub-packetization is optimal when min{M,N-M}|N or M=N, and within a multiplicative gap \frac min{M,N-M}gcd(N,M) of the optimal one in general. In particular, for the special case N=d&#x00B7;M&#x00B1;1 where the positive integer d &#x2265; 2, we propose another SDA construction to obtain lower sub-packetization.

Capacity-Achieving Private Information Retrieval Schemes From Uncoded Storage Constrained Servers With Low Sub-Packetization