A General Coding Framework for Adaptive Private Information Retrieval

The problem of $T$-colluding private information retrieval (PIR) enables the user to retrieve one out of $M$ files from a distributed storage system with $N$ servers without revealing anything about the index of the desired file to any group of up to $T$ colluding servers. In the considered storage system, the $M$ files are stored across the $N$ distributed servers in an $X$-secure $K$-coded manner such that any group of up to $X$ colluding servers learns nothing about the files; the storage overhead at each server is reduced by a factor of $\frac{1}{K}$ compared to the total size of the files; and the files can be reconstructed from any $K+X$ servers. However, in practical scenarios, when the user retrieves the desired file from the distributed system, some servers may respond to the user very slowly or not respond at all. These servers are referred to as \emph{stragglers}, and particularly their identities and numbers are unknown in advance and may change over time. This paper considers the adaptive PIR problem that can be capable of tolerating the presence of a varying number of stragglers. We propose a general coding method for designing adaptive PIR schemes by introducing the concept of a \emph{feasible PIR coding framework}. We demonstrate that any \emph{feasible PIR coding framework} over a finite field $\mathbb{F}_q$ with size $q$ can be used to construct an adaptive PIR scheme that achieves a retrieval rate of $1-\frac{K+X+T-1}{N-S}$ simultaneously for all numbers of stragglers $0\leq S\leq N-(K+X+T)$ over the same finite field. Additionally, we provide an implementation of the \emph{feasible PIR coding framework}, ensuring that the adaptive PIR scheme operates over any finite field $\mathbb{F}_q$ with size $q\geq N+\max\{K, N-(K+X+T-1)\}$.