A New Design of Private Information Retrieval for Storage Constrained Databases

Abstract

Private information retrieval (PIR) allows a user to download one of K messages from N databases without revealing to any database which of the K messages is being downloaded. In general, the databases can be storage constrained where each database can only store up to µKL bits where 1 N ≤ µ ≤ 1 and L is the size of each message in bits. Let t = µN , a recent work showed that the capacity of Storage Constrained PIR (SC-PIR)−1 , which is achieved by a storage placement scheme inspired by the content placement scheme in the literature of coded caching and the original PIR scheme. Not surprisingly, this achievable scheme requires that each message is L = N t t K bits in length, which can be impractical. In this paper, without trying to make the connection between SC-PIR and coded caching problems, based on a general connection between the Full Storage PIR (FS-PIR) problem (µ = 1) and SC-PIR problem, we propose a new SC-PIR design idea using novel storage placement schemes. The proposed schemes significantly reduce the message size requirement while still meeting the capacity of SC-PIR. In particular, the proposed SC-PIR schemes require the size of each file to be only L = N t K−1 compared to the state-of-the-art L = N t t K . Hence, we conclude that PIR may not meet coded caching when the size of L is constrained.

Capacity-achieving private information retrieval scheme with a smaller sub-packetization

“…For arbitrary N , M and 1 ≤ t < N , we prove that t M −1 is a factor of sub-packetization L for any capacity-achieving PIR scheme from storage constrained servers, and design a PIR scheme with sub-packetization L = N t × t M −1 which achieves minimum possible download cost D(t) in (1). Comparing with the scheme in [16,17], our scheme has a reduced sub-packetization parameter L by a factor t. In particular, the improvement amounts to reduction by t times to the number of requests to servers. The difference between our scheme and the one in [16] lies in the requirement of a symmetric rule across servers.…”

“…Proof. It was shown in [17] that for arbitrary N and M , the maximum achievable rate R = L/D across all PIR schemes is the capacity C = (1 + 1/t + 1/t 2 + • • • + 1/t M −1 ) −1 . Therefore, for any capacity-achieving PIR scheme L/D = C, the download cost should be…”

Capacity-Achieving Private Information Retrieval Schemes from Uncoded Storage Constrained Servers with Low Sub-packetization

“…In this paper, we are interested in characterizing the optimal sub-packetization to achieve the capacity of SC-PIR systems. Note from the previous work [1], [31] that linear schemes are sufficient to achieve the capacity of SC-PIR. Additionally, it was proved in [1] that, for any µ with 1 N ≤ µ ≤ 1, the capacity of SC-PIR system can be achieved by memory-sharing technique between the discrete points such that M ∈ {1, 2, .…”

“…Though repetition coding can offer simplicity in designing PIR schemes and the high immunity against server failures, it suffers from extremely large storage cost. The storage cost in a PIR system has been widely investigated in terms of the coding structures in the storage design, such as specific Maximum Distance Separable (MDS) codes [3], [25], [37], an uncoded storage [26], [1], [31], and other more complicated coding techniques [20], [10], [5], [19], [36], [14], [2]. Moreover, the tradeoff between the storage cost and retrieval rate was considered without any explicit constraints on the storage codes [24], [29], [30].…”

“…Thus the problem of high sub-packetization shows up again in this SC-PIR model. Recently, Woolsey et al [31] proposed a general construction of SC-PIR schemes by establishing the connection between storage design and Storage Full PIR (SF-PIR, i.e., the case that each server can store all the files). Then, the sub-packetization to achieve the capacity of the SC-PIR system was reduced to N M K−1 in [31], which also increases exponentially with K.…”