Capacity-Achieving Private Information Retrieval Schemes from Uncoded Storage Constrained Servers with Low Sub-packetization
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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 retrieve one out of K files from N servers while revealing nothing about its identity 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 μ K equivalent files, where μ 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≜μ N servers for μ such that M=μ N∈{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 because of involving 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 N(M-1)/(N,M). 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 N(M-1)/(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 min{M,N-M}/(N,M) of the optimal one otherwise. In particular, for the case N=d· M±1 where d≥ 2, another SDA is constructed to obtain lower sub-packetization.
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