General constructions for information-theoretic private information retrieval

A Private Information Retrieval (PIR) protocol enables a user to retrieve a data item from a database while hiding the identity of the item being retrieved; specifically, in a t-private k-server PIR protocol the database is replicated among k servers, and the user's privacy is protected from any collusion of up to t servers. The main cost-measure of such protocols is the communication complexity of retrieving a single bit of data. This work addresses the information-theoretic setting for PIR, where the user's privacy should be unconditionally protected against computationally unbounded servers. We present a general construction, whose abstract components can be instantiated to yield both old and new families of PIR protocols. A main ingredient in the new protocols is a generalization of a solution by Babai, Gál, Kimmel, and Lokam for a communication complexity problem in the multiparty simultaneous messages model. Our protocols simplify and improve upon previous ones, and resolve some previous anomalies. In particular, we get (1) 1-private k-server PIR protocols with O(k3n1/(2k-1)) communication bits, where n is the database size; (2) t-private k-server protocols with O(n1/⌊(2k-1)/t⌋) communication bits, for any constant integers k>t≥1; and (3) t-private k-server protocols in which the user sends O(logn) bits to each server and receives O(nt/k+ε) bits in return, for any constant integers k>t≥1 and constant ε>0. The latter protocols have applications to the construction of efficient families of locally decodable codes over large alphabets and to PIR protocols with reduced work by the servers.