On locally decodable codes, self-correctable codes, and t-private PIR

A k-query locally decodable code (LDC) allows to probabilistically decode any bit of an encoded message by probing only k bits of its corrupted encoding. A stronger and desirable property is that of self-correction, allowing to efficiently recover not only bits of the message but also arbitrary bits of its encoding. In contrast to the initial constructions of LDCs, the recent and most efficient constructions are not known to be self-correctable. The existence of self-correctable codes of comparable efficiency remains open. A closely related problem with a very different motivation is that of private information retrieval (PIR). A k-server PIR protocol allows a user to retrieve the i-th bit of a database, which is replicated among k servers, without revealing information about i to any individual server. A natural generalization is t-private PIR, which keeps i hidden from any t colluding servers. In contrast to the initial PIR protocols, it is not known how to generalize the recent and most efficient protocols to yield t-private protocols of comparable efficiency. In this work we study both of the above questions, showing that they are in fact related. We start by presenting a general transformation of any t-private PIR protocol (equivalently, LDC) into a t-private protocol with a similar amount of communication per server. Combined with the recent result of Yekhanin (STOC 2007), this yields a significant improvement over previous t-private PIR protocols. A major weakness of our transformation is that the number of servers in the resulting t-private protocols grows exponentially with t. We show that if the underlying LDC satisfies the stronger self-correction property, then there is a similar transformation in which the number of servers grows only linearly with t, which is the best one can hope for. Finally, we study the question of closing the current gap between the complexity of the best known LDC and that of self-correctable codes, and relate this question to a conjecture of Hamada concerning the algebraic rank of combinatorial designs.