  Recent work in differential privacy has explored the prospect of combining
local randomization with a secure intermediary. Specifically, there are a
variety of protocols in the secure shuffle model (where an intermediary
randomly permutes messages) as well as the secure aggregation model (where an
intermediary adds messages). Most of these protocols are limited to approximate
differential privacy. An exception is the shuffle protocol by Ghazi, Golowich,
Kumar, Manurangsi, Pagh, and Velingker (arXiv:2002.01919): it computes bounded
sums under pure differential privacy. Its additive error is
$\tilde{O}(1/\varepsilon^{3/2})$, where $\varepsilon$ is the privacy parameter.
In this work, we give a new protocol that ensures $O(1/\varepsilon)$ error
under pure differential privacy. We also show how to use it to test uniformity
of distributions over $[d]$. The tester's sample complexity has an optimal
dependence on $d$. Our work relies on a novel class of secure intermediaries
which are of independent interest.
