  Advances in communications, storage and computational technology allow
significant quantities of data to be collected and processed by distributed
devices. Combining the information from these endpoints can realize significant
societal benefit but presents challenges in protecting the privacy of
individuals, especially important in an increasingly regulated world.
Differential privacy (DP) is a technique that provides a rigorous and provable
privacy guarantee for aggregation and release. The Shuffle Model for DP has
been introduced to overcome challenges regarding the accuracy of local-DP
algorithms and the privacy risks of central-DP. In this work we introduce a new
protocol for vector aggregation in the context of the Shuffle Model. The aim of
this paper is twofold; first, we provide a single message protocol for the
summation of real vectors in the Shuffle Model, using advanced composition
results. Secondly, we provide an improvement on the bound on the error achieved
through using this protocol through the implementation of a Discrete Fourier
Transform, thereby minimizing the initial error at the expense of the loss in
accuracy through the transformation itself. This work will further the
exploration of more sophisticated structures such as matrices and
higher-dimensional tensors in this context, both of which are reliant on the
functionality of the vector case.
