The Capacity of Private Information Retrieval from Coded Databases (1609.08138v1)

Abstract: We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of $N$ non-colluding databases, each storing a coded version of $M$ messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is $C=\left(1+\frac{K}{N}+\frac{K^2}{N^2}+\cdots+\frac{K^{M-1}}{N^{M-1}}\right)^{-1}=(1+R_c+R_c^2+\cdots+R_c^{M-1})^{-1}=\frac{1-R_c}{1-R_c^M}$, where $R_c$ is the rate of the $(N,K)$ code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost. The result generalizes the achievability and converse results for the classical PIR with replicating databases to the case of coded databases.

• The paper derives the PIR capacity expression for coded databases using novel interference decoding techniques.
• It combines an achievability scheme with an induction-based converse proof, ensuring mathematical rigor and practical relevance.
• Findings reveal a trade-off between storage cost and retrieval efficiency, guiding optimal design in privacy-preserving systems.

Overview of "The Capacity of Private Information Retrieval from Coded Databases"

This paper addresses a significant problem in the field of distributed storage systems: Private Information Retrieval (PIR) from coded databases. Specifically, the authors seek to determine the information-theoretic capacity for retrieving data from multiple non-colluding databases wherein each stores encoded versions of multiple messages. The primary challenge is to acquire the desired data without divulging the identity of that data to any database.

Key Contributions

The major contribution of the paper is the derivation of the PIR capacity when considering databases that employ coding schemes rather than simple replication. The authors prove that this capacity is given by the expression:

$C = \frac{1 - R_{c}}{1 - R_{c}^{M}}$

where $R_{c}$ is the coding rate of the (N, K) storage code, and $M$ is the number of messages. This is a generalized extension of the classical PIR capacity results which typically consider replication-coded databases.

The authors illustrate that the capacity is dependent solely on the code rate and the number of messages, ignoring details about the specific structure of the storage code or the number of databases involved. This universal property suggests an optimal separation between the storage code design and the retrieval scheme for a fixed code rate.

Analytical Methodology

The paper presents a detailed analysis combining mathematical rigor and practical coding theories. The authors utilize both achievable schemes and converse proofs to establish the stated capacity rigorously. In the achievability proof, they propose a PIR scheme that adapts the techniques from earlier works on PIR with replication but incorporates additional steps for coded databases, such as handling interference decoding. The converse proof is achieved through an induction-based argument that generalizes known results for simple and colluding adversary models to the coded setting.

Implications and Future Work

The results imply a trade-off between the storage cost and retrieval efficiency, influencing how future systems might design storage architectures with privacy as a core requirement. For example, systems could balance between higher redundancy (simpler retrieval) or more complex coding structures (lower storage costs).

Although the paper does not claim groundbreaking new coding schemes, it paves the way for more nuanced private data retrieval techniques that incorporate varying levels of redundancy. Future work might explore extending these results to more complex scenarios where databases may collude or are subject to varying reliability conditions (e.g., node failures). Additionally, open-door avenues could be investigated around the optimization of storage codes themselves to streamline retrieval processes further, which might involve novel erasure codes or other modern coding mechanisms.

Conclusion

This work is a noteworthy addition to the landscape of data privacy in distributed systems, providing both a theoretical benchmark for PIR capacity and a structured method for achieving it in practice. As data privacy becomes an ever more critical issue, studies like these not only help establish foundational limits but also guide practical application in data-intensive industries. The separation of storage code design from retrieval scheme design demonstrated here might well become a blueprint for both academic research and industrial applications moving forward. The authors’ contribution lies in offering a mathematically sound, logically coherent approach to a complex problem that continues to gain importance in our data-driven age.