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.

翻译：本文研究了带侧信息的泄露式隐私信息检索问题，为泄露隐私、侧信息与下载成本之间的权衡关系提供了基础性刻画。我们提出了一个统一的概率框架，用于在$W$-隐私和$(W, S)$-隐私的$\varepsilon$-差分隐私变体下设计L-PIR-SI方案。推导了下载成本的显式上界，该上界严格推广了现有结果：当$\varepsilon \to 0$时，我们的界可退化为完美PIR-SI的容量；在无侧信息时，则简化为已知的$\varepsilon$-泄露PIR速率。此外，我们对缩放律层面的隐私-效用权衡进行了精细化分析，证明在最小非平凡设定$M = 1$（其中$K$和$M$分别表示消息数量和侧信息规模）下，泄露率指数在泄露式$W$-隐私下按$\mathcal{O}(\log \frac{K}{M + 1})$缩放，在泄露式$(W, S)$-隐私下按$\mathcal{O}(\log K)$缩放。