On Sub-Packetization of Capacity-Achieving PIR Schemes for MDS Coded Databases

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. The capacity of coded-PIR with privacy against any individual server was determined by Banawan and Ulukus in 2016, i.e., CC-PIR = (1 + K N + · · · + K M−1 NM−1 ). They also presented a linear capacity-achieving scheme with sub-packetization KN . In this paper we focus on minimizing the sub-packetization for linear capacity-achieving coded-PIR schemes. We prove that the sub-packetization for all linear capacity-achieving coded-PIR schemes in the nontrivial cases (i.e. N > K ≥ 1 and M > 1) must be no less than Kn, where n = N/gcd(N,K). Moreover, we design a linear capacity-achieving coded-PIR scheme with sub-packetization Kn for all N > K ≥ 1 and M > 1. Therefore, Kn is the optimal sub-packetization for linear capacity-achieving coded-PIR schemes.