Building Capacity-Achieving PIR Schemes with Optimal Sub-Packetization over Small Fields

Consider $N$ servers with replicated databases containing $M$ records. Suppose a user wants to privately retrieve one record by accessing the servers such that the identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a $T$ -private information retrieval ($T$ -PIR) scheme. Three indexes are concerned for PIR schemes: (1) rate, indicating the amount of retrieved information per unit of downloaded data. The highest achievable rate is characterized by the capacity; (2) sub-packetization, reflecting the implementation complexity for linear schemes; (3) field size. We consider linear schemes over a finite field. In this paper, a general $T$ - PIR scheme simultaneously attaining the optimality of almost all of the three indexes is presented. Specifically, we design a linear capacity-achieving T-PIR scheme with sub-packetization $dn^{M-1}$ over a finite field $\mathbb{F}_{q}, q\geq N$. The sub-packetization $dn^{M-1}$, where $d=\mathrm{gcd}(N,\ T)$ and $n=N/d$, has been proved to be optimal in our previous work. The field size is reduced by an exponential factor in our scheme comparing with existing capacity -achieving $T$ - PIR schemes.

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