On sub-packetization and access number of capacity-achieving PIR schemes for MDS coded non-colluding servers

Consider the problem of private information retrieval (PIR) over a distributed storage system where M records are stored across N servers by using an [N,K] MDS code. For simplicity, this problem is usually referred as the coded PIR problem. In 2016, Banawan and Ulukus designed the first capacity-achieving coded PIR scheme with sub-packetization KNM and access number MKNM, where capacity characterizes the minimal download size for retrieving per unit of data, and sub-packetization and access number are two metrics closely related to implementation complexity. In this paper, we focus on minimizing the sub-packetization and the access number for linear capacity-achieving coded PIR schemes. We first determine the lower bounds on sub-packetization and access number, which are KnM−1 and MKnM−1, respectively, in the nontrivial cases (i.e., N >K ≥ 1 and M >1), where n= N/gcd(N,K). We then design a general linear capacity-achieving coded PIR scheme to simultaneously attain these two bounds, implying tightness of both bounds.

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