Private information retrieval from Byzantine and colluding databases

Abstract

We consider the problem of single-round private information retrieval (PIR) from N replicated databases. We consider the case when B databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of M messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the T-privacy constraint in this paper, where any T databases can collude, and exchange the queries submitted by the user. We determine the information-theoretic capacity of this problem, which is the maximum number of correct symbols that can be retrieved privately for every symbol of the downloaded data to be C = N−2B/N · 1−T/N−2B / 1−(T/N−2B)M, if 2B + T < N. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the errors introduced by the Byzantine databases. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes.