The Capacity of Robust Private Information Retrieval with Colluding Databases

Abstract: Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of $K$ messages from $N$ non-communicating replicated databases (each holds all $K$ 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. $T$-private PIR is a generalization of PIR to include the requirement that even if any $T$ of the $N$ 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 $M \geq N$ databases, out of which any $M - N$ may fail to respond. For $K$ messages and $M\geq N$ databases out of which at least some $N$ must respond, we show that the capacity of $T$-private and Robust PIR is $\left(1+T/N+T^2/N^2+\cdots+T^{K-1}/N^{K-1}\right)^{-1}$. The result includes as special cases the capacity of PIR without robustness ($M=N$) or $T$-privacy constraints ($T=1$).

• The paper presents a novel capacity formula for robust T-private PIR that accounts for both collusion among databases and potential non-responses.
• The methodology uses symmetric querying and MDS-coded side information to ensure efficient, privacy-preserving retrieval across multiple databases.
• The results reveal how increases in colluding databases reduce capacity while more databases improve robustness, guiding practical PIR system design.

Essay on the Capacity of Robust Private Information Retrieval with Colluding Databases

In the paper titled "The Capacity of Robust Private Information Retrieval with Colluding Databases," Hua Sun and Syed A. Jafar undertake a detailed investigation into an extension of the Private Information Retrieval (PIR) problem. They explore the information-theoretic capacity of robust $T$-private PIR, where privacy against colluding databases is ensured even when multiple databases are non-responsive. This work is a significant progression in understanding the fundamental limits of PIR in more complex settings that involve collusion and robustness requirements.

Problem Statement and Theoretical Contributions

The authors generalize the conventional PIR model, introducing a scenario where $M \geq N$ databases are available, with any $M - N$ potentially failing to respond, necessitating a robustness aspect on top of privacy. They focus on maintaining the privacy of the desired message index even when up to $T$ of these $N$ databases collude to deduce which message a user retrives.

The main theoretical result of the paper is the derivation of the capacity formula for $T$-private and robust PIR:

$$C = \left(1 + \frac{T}{N} + \frac{T^2}{N^2} + \cdots + \frac{T^{K-1}}{N^{K-1}}\right)^{-1}.$$

This capacity expression reveals several important insights:

• Dependence on Parameters: The capacity decreases as the number of colluding databases $T$ increases, aligning with intuitive expectations that more stringent privacy requirements should naturally impose higher cost (in terms of reduced efficiency).
• Scalability with Messages and Databases: The capacity is a decreasing function of the number of messages $K$ and an increasing function of the number of databases $N$. The capacity approaches 1 (full efficiency) as the number of databases becomes large compared to the number of colluding databases.

Methodology and Proof Techniques

The authors employ a capacity-achieving scheme based on the principles of symmetric querying and the use of coded side information, leveraging properties of Maximum Distance Separable (MDS) codes. This innovative approach allows them to handle interference in downloaded data using algebraic properties, ensuring both privacy and robustness without loss of capacity.

The capacity-achieving scheme is built on three key principles:

1. Symmetry Across Databases: Symmetric queries across databases are used to prevent any database from gaining additional information.
2. Message Symmetry: Ensuring that the retrieved symbols for different messages are symmetrically handled to prevent leakage.
3. Exploiting Side Information: Leveraging side information already acquired about non-desired messages to download desired information more efficiently.

The authors employ a detailed proof strategy, using mathematical induction and the information-theoretic tool of Han’s inequality, to demonstrate that the derived capacity is indeed an upper bound across possible strategies. Their methodology solidifies the robustness of the proposed scheme against collusive information sharing among databases and unexpected database failures.

Implications and Future Directions

The findings have both theoretical and practical implications. Theoretically, the results connect PIR with broader areas such as coding theory and secure multiparty computation by demonstrating how robust coding strategies can maintain desired properties under practical constraints. Practically, the results provide a blueprint for designing PIR systems in distributed storage or cloud environments where user privacy against server collusion is a significant concern.

Looking forward, several interesting avenues open up. Considering computational constraints for robust $T$-private PIR, exploring optimal coding strategies within different field sizes to minimize operational overhead, and investigating the interplay between upload and download costs in practical settings could further refine the utility of these theoretical results. The challenge of balancing capacity with computational tractability remains an intriguing direction for future exploration in the domain of privacy-preserving information retrieval.