  Secure Aggregation protocols allow a collection of mutually distrust parties,
each holding a private value, to collaboratively compute the sum of those
values without revealing the values themselves. We consider training a deep
neural network in the Federated Learning model, using distributed stochastic
gradient descent across user-held training data on mobile devices, wherein
Secure Aggregation protects each user's model gradient. We design a novel,
communication-efficient Secure Aggregation protocol for high-dimensional data
that tolerates up to 1/3 users failing to complete the protocol. For 16-bit
input values, our protocol offers 1.73x communication expansion for $2^{10}$
users and $2^{20}$-dimensional vectors, and 1.98x expansion for $2^{14}$ users
and $2^{24}$ dimensional vectors.
