On PIR and Symmetric PIR From Colluding Databases With Adversaries and Eavesdroppers

We consider the problem of <italic>private information retrieval (PIR)</italic> and <italic>symmetric private information retrieval (SPIR)</italic> from replicated databases with colluding servers, in the presence of Byzantine adversaries and eavesdroppers. Specifically, there are <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> messages replicatively stored at <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> databases. A user wants to retrieve one message by communicating with the databases, without revealing the identity of the message retrieved. For <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-colluding databases, any <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> out of <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> databases may communicate their interactions with the user to guess the identity of the requested message. We consider the situation where the communication system can be vulnerable to attachers, namely, there is an adversary in the system that can tap in on or even try to corrupt the communication. The capacity is defined as the maximum number of information bits of the desired message retrieved per downloaded bit. For SPIR, it is further required that the user learns nothing about the other <inline-formula> <tex-math notation="LaTeX">$K-1$ </tex-math></inline-formula> messages in the database. Three types of adversaries are considered: a Byzantine adversary who can overwrite the transmission of any <inline-formula> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula> servers to the user; a passive eavesdropper who can tap in on the incoming and outgoing transmissions of any <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> servers; and a combination of both -- an adversary who can tap in on a set of any <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> nodes, and overwrite the transmission of a set of any <inline-formula> <tex-math notation="LaTeX">$B$ </tex-math></inline-formula> nodes. The problems of SPIR with colluding servers and the three types of adversaries are named T-BSPIR, T-ESPIR and T-BESPIR, respectively. We derive the capacities of the three secure SPIR problems. The results resemble those of secure network coding problems with adversaries and eavesdroppers. The capacity of <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-colluding PIR with Byzantine adversaries is characterized in <xref ref-type="bibr" rid="ref1">[1]</xref>. In this work, we consider <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-colluding PIR with an eavedropper (named T-EPIR). We derive the T-EPIR capacity when <inline-formula> <tex-math notation="LaTeX">$E \geq T$ </tex-math></inline-formula>; for the case where <inline-formula> <tex-math notation="LaTeX">$E \leq T$ </tex-math></inline-formula>, we find an outer bound (converse bound) and an inner bound (achievability) on the optimal achievable rate.

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