Vector Linear Secure Aggregation

TL;DR

This paper generalizes secure summation to vector linear functions, characterizing the minimal randomness needed for privacy when computing arbitrary linear combinations of user inputs.

Contribution

It introduces a vector linear secure aggregation framework and determines the optimal randomness cost based on the span of coefficient vectors.

Findings

Optimal randomness cost equals the dimension of the span of protected coefficient vectors.

Provides a characterization of minimal key variables for secure vector linear computation.

Extends secure summation to more general linear functions with privacy guarantees.

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.