Secure Multi-User Linearly-Separable Distributed Computing

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 $\gamma$-sparse $\mathbf{E}$ defines the task allocation across $N$ servers, and a $\delta$-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 $\alpha_k$ server responses, the common randomness visible to that user must span a subspace of dimension greater than $\alpha_k-1$, 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.