The introduction of the new multi-user linearly-separable distributed computing framework, has recently revealed how a parallel treatment of users can yield large parallelization gains with relatively low computation and communication costs. These gains stem from a new approach that converts the computing problem into a sparse matrix factorization problem; a matrix \(\mathbf{F}\) that describes the users' requests, is decomposed as \(\mathbf{F} = \mathbf{DE}\), where a \(γ\)-sparse \(\mathbf{E}\) defines the task allocation across \(N\) servers, and a \(δ\)-sparse \(\mathbf{D}\) defines the connectivity between \(N\) servers and \(K\) users as well as the decoding process. While this approach provides near-optimal performance, its linear nature has raised data secrecy concerns. We adopt an information-theoretic secrecy framework requiring that each user learns nothing more than its own requested function. Our main results provide (i) a necessary condition stating that for each user $k$ observing \(α_k\) server responses, the common randomness visible to that user must span a subspace of dimension greater than \(α_k-1\), and (ii) a necessary and sufficient condition requiring that removing from \(\mathbf{D}\) the columns corresponding to the servers observed by a user leaves a matrix of rank at least \(K-1\). Based on these conditions, we design a general, cost-preserving secrecy-enforcing transformation valid over both finite and real fields, obtained by appending to \(\mathbf{E}\) a basis of \(\mathrm{Null}(\mathbf{D})\) and carefully injecting shared randomness. This scheme preserves communication and computation costs, guarantees perfect information-theoretic secrecy over finite fields, and in the real case yields an explicit mutual-information bound that can be made arbitrarily small by increasing the variance of Gaussian common randomness.

翻译：新引入的多用户线性可分分布式计算框架近期揭示了如何通过对用户进行并行处理，以相对较低的计算和通信成本实现显著的并行化增益。这些增益源于一种新方法，该方法将计算问题转化为稀疏矩阵分解问题：描述用户请求的矩阵\(\mathbf{F}\)被分解为\(\mathbf{F} = \mathbf{DE}\)，其中γ-稀疏矩阵\(\mathbf{E}\)定义了跨\(N\)个服务器的任务分配，而δ-稀疏矩阵\(\mathbf{D}\)定义了\(N\)个服务器与\(K\)个用户之间的连接以及解码过程。尽管该方法提供了接近最优的性能，但其线性特性引发了数据保密性的担忧。我们采用信息论保密框架，要求每个用户只能获知其自身请求的函数，而不得了解任何其他信息。我们的主要结果包括：（i）一个必要条件，表明对于每个观察到\(\alpha_k\)个服务器响应的用户\(k\)，该用户可见的公共随机性必须张成一个维度大于\(\alpha_k-1\)的子空间；（ii）一个充分必要条件，要求从\(\mathbf{D}\)中移除用户所观察服务器的对应列后，剩余矩阵的秩至少为\(K-1\)。基于这些条件，我们设计了一种通用的、保持成本不变的保密增强变换，该变换在有限域和实数域上均有效，通过将\(\mathrm{Null}(\mathbf{D})\)的一组基附加到\(\mathbf{E}\)并精心注入共享随机性来实现。该方案在保持通信和计算成本的同时，在有限域上保证了完美信息论保密性，而在实数情况下，通过增加高斯公共随机性的方差，可得到显式的互信息上界，且该上界可被任意缩小。