The Capacity of Robust Private Information Retrieval With Colluding Databases

• Published in: IEEE transactions on information theory Vol. 64; no. 4; pp. 2361 - 2370
• Main Authors: Sun, Hua, Jafar, Syed Ali
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-04-2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Capacity; colluding databases; Communication; Data privacy; Distributed databases; Downloading; Encoding; Indexes; Information retrieval; Information theory; Messages; private information retrieval; Robustness; unresponsive databases
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2017.2777490

Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> messages from <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> non-communicating replicated databases (each holds all <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> messages) while keeping the identity of the desired message index a secret from each individual database. The information theoretic capacity of PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula>-private PIR is a generalization of PIR to include the requirement that even if any <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> of the <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> databases collude, the identity of the retrieved message remains completely unknown to them. Robust PIR is another generalization that refers to the scenario where we have <inline-formula> <tex-math notation="LaTeX">M \geq N </tex-math></inline-formula> databases, out of which any <inline-formula> <tex-math notation="LaTeX">M - N </tex-math></inline-formula> may fail to respond. For <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> messages and <inline-formula> <tex-math notation="LaTeX">M\geq N </tex-math></inline-formula> databases out of which at least some <inline-formula> <tex-math notation="LaTeX">N </tex-math></inline-formula> must respond, we show that the capacity of <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula>-private and Robust PIR is <inline-formula> <tex-math notation="LaTeX">(1+T/N+T^{2}/N^{2}+\cdots +T^{K-1}/N^{K-1})^{-1} </tex-math></inline-formula>. The result includes as special cases the capacity of PIR without robustness (<inline-formula> <tex-math notation="LaTeX">M=N </tex-math></inline-formula>) or <inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula>-privacy constraints (<inline-formula> <tex-math notation="LaTeX">T=1 </tex-math></inline-formula>).