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 In this problem, a user has an outdated version of the message $\hat{W}_\theta$ of length $L$ bits that differ from the current version $W_\theta$ in at most $f$ bits.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.

中文翻译：

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私人更新的下载费用

我们考虑从$ N $复制和非冲突数据库的$ K $消息中私下更新消息的问题。在此问题中，用户具有消息$ \ hat {W} _ \ theta $的长度为$ L $位的过时版本，在最多$ f $位中与当前版本$ W_ \ theta $不同。用户需要使用下载次数最少的私有信息检索（PIR）方案正确检索$ W_ \ theta $，而不会将有关消息索引$ \ theta $的任何信息泄漏到任何单独的数据库。为此，我们提出了一种基于\ emph {syndrome解码}的新颖可实现方案。具体地，用户使用具有长度约束的消息的最优PIR方案，根据具有精心设计的参数的线性分组代码，下载与$ W_ \ theta $相对应的校正子。如果术语$ \ log_2 \ left（\ sum_ {i = 0} ^ f \ binom {L} {i} \ right）$是整数，我们将得出匹配的最佳下载成本的上限和下限。我们的结果表明，与直接使用经典PIR方法下载$ W_ \ theta $相比，如果$ f <\ frac {L} {2} $，而无需考虑$ W_ \ theta $与$ \ hat {W} _ \ theta $在内。