Vector Linear Secure Aggregation
Xihang Yuan, Hua Sun
TL;DR
This work introduces vector linear secure aggregation, where $K$ users compute a linear function $F W$ of their inputs while safeguarding information about another linear function $G W$. It provides an exact information-theoretic characterization of the minimum randomness cost: $R_{ZSigma}^* = rank([F;G]) - rank(F) = rank(G|F)$, and gives explicit achievability via a noise-precoding scheme using matrices $V$ and $V^\bot$ to protect exactly the subspace needed for $G$ without obstructing the recovery of $F W$. A matching converse bound confirms optimality, and the construction specializes to the classical secure summation case. The results illuminate how linear leakage constraints shape randomness requirements, with potential impact on federated learning and distributed computation where linear aggregations and privacy of certain components are essential.
Abstract
The secure summation problem, where $K$ users wish to compute the sum of their inputs at a server while revealing nothing about all $K$ inputs beyond the desired sum, is generalized in two aspects - first, the desired function is an arbitrary linear function (multiple linear combinations) of the $K$ inputs instead of just the sum; second, rather than protecting all $K$ inputs, we wish to guarantee that no information is leaked about an arbitrary linear function of the $K$ inputs. For this vector linear generalization of the secure summation problem, we characterize the optimal randomness cost, i.e., to compute one instance of the desired vector linear function, the minimum number of the random key variables held by the users is equal to the dimension of the vector space that is in the span of the vectors formed by the coefficients of the linear function to protect but not in the span of the vectors formed by the coefficients of the linear function to compute.