Download Cost of Private Updating

TL;DR

This paper introduces a new private information retrieval scheme that efficiently updates a message from multiple databases by exploiting the correlation between outdated and current versions, significantly reducing download costs.

Contribution

The paper proposes a novel PIR scheme based on syndrome decoding that leverages message correlation to minimize download costs during private updates.

Findings

Significant reduction in download cost when the outdated message differs from the current one in less than half of its bits.

Derived bounds for the optimal download cost that match under certain conditions.

The scheme outperforms classical PIR approaches by utilizing message correlation.

Abstract

We consider the problem of privately updating a message out of $K$ messages from $N$ replicated and non-colluding databases. In this problem, a user has an outdated version of the message $\hat{W}_\theta$ of length $L$ bits that differ from the current version $W_\theta$ in at most $f$ bits. The user needs to retrieve $W_\theta$ correctly using a private information retrieval (PIR) scheme with the least number of downloads without leaking any information about the message index $\theta$ to any individual database. To that end, we propose a novel achievable scheme based on \emph{syndrome decoding}. Specifically, the user downloads the syndrome corresponding to $W_\theta$, according to a linear block code with carefully designed parameters, using the optimal PIR scheme for messages with a length constraint. We derive lower and upper bounds for the optimal download cost that match if the term $\log_2\left(\sum_{i=0}^f \binom{L}{i}\right)$ is an integer. Our results imply that there is a significant reduction in the download cost if $f < \frac{L}{2}$ compared with downloading $W_\theta$ directly using classical PIR approaches without taking the correlation between $W_\theta$ and $\hat{W}_\theta$ into consideration.