The Asymptotic Capacity of Byzantine Symmetric Private Information Retrieval and Its Consequences

TL;DR

This paper determines the asymptotic capacity of symmetric private information retrieval with Byzantine servers, clarifies the definition of Byzantine servers, and proposes a scheme that achieves the capacity in the case of infinitely many messages.

Contribution

It defines Byzantine servers in the context of symmetric PIR, derives an upper bound on capacity, and constructs a scheme that achieves this bound for infinite messages.

Findings

Established the capacity of symmetric PIR with Byzantine servers.

Provided a new definition of Byzantine servers suitable for symmetric privacy.

Developed a scheme that attains the capacity bound.

Abstract

We consider the problem of finding the asymptotic capacity of symmetric private information retrieval (SPIR) with $B$ Byzantine servers. Prior to finding the capacity, a definition for the Byzantine servers is needed since in the literature there are two different definitions. In \cite{byzantine_tpir}, where it was first defined, the Byzantine servers can send any symbol from the storage, their received queries and some independent random symbols. In \cite{unresponsive_byzantine_1}, Byzantine servers send any random symbol independently of their storage and queries. It is clear that these definitions are not identical, especially when \emph{symmetric} privacy is required. To that end, we define Byzantine servers, inspired by \cite{byzantine_tpir}, as the servers that can share everything, before and after the scheme initiation. In this setting, we find an upper bound, for an infinite number of messages case, that should be satisfied for all schemes that protect against this setting and develop a scheme that achieves this upper bound. Hence, we identify the capacity of the problem.