Secure Multi-User Linearly-Separable Distributed Computing

TL;DR

This paper introduces a secure multi-user distributed computing framework that enhances parallelization while ensuring information-theoretic secrecy through a novel matrix-based approach and shared randomness.

Contribution

It develops necessary and sufficient secrecy conditions and proposes a cost-preserving transformation to enforce perfect secrecy in the framework.

Findings

Secrecy requires the shared randomness to span a subspace larger than _k-1 for each user.

Removing certain columns from the task matrix affects its rank, impacting secrecy.

The proposed scheme guarantees perfect secrecy over finite fields and arbitrarily small mutual information over real fields.

Abstract

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.