Building Capacity-Achieving PIR Schemes with Optimal Sub-Packetization over Small Fields
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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^M-1 over a finite field F_q, q≥ N. The sub-packetization dn^M-1, 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^M-2 where t=T/d, while our scheme reduces the field size by an exponential factor.
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