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

Suppose a database containing M records is replicated across N servers, and a user wants to privately retrieve one record by accessing the servers such that 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, reflexing 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 over a finite field Fq , q ≥ N . The sub-packetization dn, where d=gcd(N,T ) and n=N/d, has been proved to be optimal in our previous work. The field size of all existing capacity-achieving T -PIR schemes must be larger than Nt where t = T/d, while our scheme reduces the field size by an exponential factor.