Building Capacity-Achieving T-PIR Schemes for Some Parameters Over Binary Field via Subfield Sub-Codes
Xu Jingke, Libo Wang · IEEE Transactions on Communications · 2021
The$T$-Private Information Retrieval ($T$-PIR) problem is that a user wishes to retrieve a single record from$N$servers, without revealing the identity of the desired record even if the number of colluding servers up to$T$. Here every server stores all$M$records and each record can be viewed as a vector over the$q$-ary finite field$\mathbb {F}_{q}$. To reduce the field size, a capacity-achieving$T$-PIR scheme based on some MDS array codes for all possible$N>T,M\geq 2$was constructed by Xu and Zhang in 2019. A key idea in that scheme is to construct some MDS array codes in query phase, whose generator matrices satisfy the recovery property (see (C1) inExample 1, page 6),i.e.,there exist some specific columns in each generator matrix such that the number of them is equal to the rank of such matrix. We call the set of each column index of those columns in each generator matrix as the column index set. However, that scheme didn’t give an explicit selection of the column index set satisfying the recovery property. In this paper, let$N=d(2t-1)$,$N>T=dt>1, M\geq 3$and$d\geq 1$, under the constraint of$2^{t-1}\geq N$, we present a novel method to construct capacity-achieving$T$-PIR schemes over the binary field with an explicit selection of the column index set. Moreover, our scheme has sub-packetization$N(2t-1)^{M-2}$, which is optimal for all linear capacity-achieving$T$-PIR schemes determined by Zhang and Xu in 2019. The key to our method is that all locators and column multipliers are precisely selected such that a Generalized Reed-Solomon (GRS) code defined by them has a zero-dimensional subfield sub-code. Compared with all the known capacity-achieving$T$-PIR schemes for the same non-trivial parameters, ours is the first explicit capacity-achieving$T$-PIR scheme over the binary field.