  We consider a foundational unsupervised learning task of $k$-means data
clustering, in a federated learning (FL) setting consisting of a central server
and many distributed clients. We develop SecFC, which is a secure federated
clustering algorithm that simultaneously achieves 1) universal performance: no
performance loss compared with clustering over centralized data, regardless of
data distribution across clients; 2) data privacy: each client's private data
and the cluster centers are not leaked to other clients and the server. In
SecFC, the clients perform Lagrange encoding on their local data and share the
coded data in an information-theoretically private manner; then leveraging the
algebraic structure of the coding, the FL network exactly executes the Lloyd's
$k$-means heuristic over the coded data to obtain the final clustering.
Experiment results on synthetic and real datasets demonstrate the universally
superior performance of SecFC for different data distributions across clients,
and its computational practicality for various combinations of system
parameters. Finally, we propose an extension of SecFC to further provide
membership privacy for all data points.
