On the Leaky Private Information Retrieval with Side Information

TL;DR

This paper characterizes the fundamental trade-offs in leaky private information retrieval with side information, proposing a unified framework that generalizes existing results and analyzes privacy-utility trade-offs at a scaling-law level.

Contribution

It introduces a probabilistic framework for L-PIR-SI schemes under differential privacy, deriving explicit bounds that unify and extend prior PIR results.

Findings

Derived upper bounds on download cost that generalize existing PIR capacities.

Showed the bounds recover perfect PIR-SI capacity as leakage vanishes.

Analyzed privacy-utility trade-offs, revealing how leakage ratios scale with message and side information sizes.

Abstract

This paper investigates the problem of Leaky Private Information Retrieval with Side Information (L-PIR-SI), providing a fundamental characterization of the trade-off among leaky privacy, side information, and download cost. We propose a unified probabilistic framework to design L-PIR-SI schemes under $\varepsilon$-differential privacy variants of both $W$-privacy and $(W, S)$-privacy. Explicit upper bounds on the download cost are derived, which strictly generalize existing results: our bounds recover the capacity of perfect PIR-SI as $\varepsilon \to 0$, and reduce to the known $\varepsilon$-leaky PIR rate in the absence of side information. Furthermore, we conduct a refined analysis of the privacy--utility trade-off at the scaling-law level, demonstrating that the leakage ratio exponent scales as $\mathcal{O}(\log \frac{K}{M + 1})$ under leaky $W$-privacy, and as $\mathcal{O}(\log K)$ under leaky $(W, S)$-privacy in the minimal non-trivial setting $M = 1$, where $K$ and $M$ denote the number of messages and the side information size, respectively.