Star Product PIR Schemes with Colluding Servers over Small Fields

Private Information Retrieval (PIR) was first proposed by B. Chor, O. Goldreich, E. Kushilevitz and M. Sudan in their 1995 FOCS paper. For MDS coded distributed storage system private information retrieval was proposed and the capacity of PIR schemes for MDS coded distributed storage was studied. Star product PIR schemes from general coded distributed storage system with colluding servers were constructed over general finite fields. These star product schemes has no restriction on the sizes of fields and can be constructed for coded distributed storage across large number of servers. In this paper we first propose and prove the Singleton type upper bound on the storage rate, ratio of colluding servers and the retrieval rate of the star product PIR schemes. Secondly star product PIR schemes for coded distributed storage from algebraic geometry (AG) codes are analysed. We prove that when the number of the servers goes to the infinity, star product PIR schemes with colluding servers for AG-coded distributed storage have parameters closing to the Singleton type upper bound if the field is large. Comparing with the star product PIR schemes for Reed-Solomon coded and Reed-Muller coded distributed storage we show that PIR schemes with colluding servers for AG coded distributed storage have their performance advantages. AG-code based star product PIR schemes with colluding, Byzantine and unresponsive servers are discussed. $q$-ary cyclic code based star product PIR schemes for replicated data storage are also studied. When the storage code is the Reed-Muller code, the best choice of the retrieval code is not always the Reed-Muller code.