This paper introduces the problem of Private Linear Transformation (PLT) which generalizes the problems of private information retrieval and private linear computation. The PLT problem includes one or more remote server(s) storing (identical copies of) $K$ messages and a user who wants to compute $L$ independent linear combinations of a $D$-subset of messages. The objective of the user is to perform the computation by downloading minimum possible amount of information from the server(s), while protecting the identities of the $D$ messages required for the computation. In this work, we focus on the single-server setting of the PLT problem when the identities of the $D$ messages required for the computation must be protected jointly. We consider two different models, depending on whether the coefficient matrix of the required $L$ linear combinations generates a Maximum Distance Separable (MDS) code. We prove that the capacity for both models is given by $L/(K-D+L)$, where the capacity is defined as the supremum of all achievable download rates. Our converse proofs are based on linear-algebraic and information-theoretic arguments that establish connections between PLT schemes and linear codes. We also present an achievability scheme for each of the models being considered.

中文翻译：

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单服务器私有线性转换：联合隐私案例

本文介绍了私有线性变换 (PLT) 的问题，该问题概括了私有信息检索和私有线性计算的问题。PLT 问题包括一个或多个远程服务器存储（相同副本）$K$ 消息和想要计算 $D$-消息子集的 $L$ 独立线性组合的用户。用户的目标是通过从服务器下载尽可能少的信息来执行计算，同时保护计算所需的 $D$ 消息的身份。在这项工作中，当必须共同保护计算所需的 $D$ 消息的身份时，我们专注于 PLT 问题的单服务器设置。我们考虑两种不同的模型，取决于所需的 $L$ 线性组合的系数矩阵是否生成最大距离可分离 (MDS) 代码。我们证明两种模型的容量均由 $L/(K-D+L)$ 给出，其中容量定义为所有可实现下载速率的最高值。我们的逆证明基于线性代数和信息论论证，这些论证在 PLT 方案和线性代码之间建立了联系。我们还为正在考虑的每个模型提供了一个可实现性方案。我们的逆证明基于线性代数和信息论论证，这些论证在 PLT 方案和线性代码之间建立了联系。我们还为正在考虑的每个模型提供了一个可实现性方案。我们的逆证明基于线性代数和信息论论证，这些论证在 PLT 方案和线性代码之间建立了联系。我们还为正在考虑的每个模型提供了一个可实现性方案。