Capacity-Achieving Private Information Retrieval Codes With Optimal Message Size and Upload Cost

Abstract

We propose a new capacity-achieving code for the private information retrieval (PIR) problem, and show that it has the minimum message size (being one less than the number of servers) and the minimum upload cost (being roughly linear in the number of messages) among a general class of capacity-achieving codes, and in particular, among all capacity-achieving linear codes. Different from existing code constructions, the proposed code is asymmetric, and this asymmetry appears to be the key factor leading to the optimal message size and the optimal upload cost. The converse results on the message size and the upload cost are obtained by an analysis of the information theoretic proof of the PIR capacity, from which a set of critical properties of any capacity-achieving code in the code class of interest is extracted. The symmetry structure of the PIR problem is then analyzed, which allows us to construct symmetric codes from asymmetric ones, yielding a meaningful bridge between the proposed code and existing ones in the literature.

On the Information Leakage in Private Information Retrieval Systems

“…Before presenting our main result, we provide the result of a simple analysis on the capacity-achieving PIR code given by Sun and Jafar [3], the capacity-achieving code proposed by Tian et al [4], and the SPIR code [5]; these three classes of codes will be referred to as the SJ code, the TSC code, and the SPIR code, respectively.…”

“…The message size L (in log |X | -ary unit) is also an important factor to consider in practice [4]. The minimum message size is highly dependent on the download cost.…”

“…The minimum message size is highly dependent on the download cost. It was shown in [4] that the message size of capacity-achieving PIR code is greater than or equal to N − 1 if the code is uniformly decomposable. Before stating our results on the message size, we first define the notion of uniformly decomposable code, similar to that in [4].…”

“…The capacity of a PIR system is defined as the maximum number of bits of desired message that can be retrieved per bit of downloaded information, which was shown in [3] to be C PIR = 1 + 1/N + 1/N 2 + · · · + 1/N K−1 −1 . There were also significant interests recently on various other variations of the PIR problem; see, e.g., [4]- [28].…”

“…The main result of this work is the characterizations of the optimal tradeoffs between the download cost and the amount of information leakage for both individual leakage and total leakage. By adapting the codes given in [4] and [5], we provide code constructions that can achieve these optimal tradeoffs; moreover, they are also shown to have the minimum message sizes. For the individual leakage case, the constructed codes do not require common randomness (unless there are only two messages).…”