  Consider the setup where $n$ parties are each given a number $x_i \in
\mathbb{F}_q$ and the goal is to compute the sum $\sum_i x_i$ in a secure
fashion and with as little communication as possible. We study this problem in
the anonymized model of Ishai et al. (FOCS 2006) where each party may broadcast
anonymous messages on an insecure channel.
  We present a new analysis of the one-round "split and mix" protocol of Ishai
et al. In order to achieve the same security parameter, our analysis reduces
the required number of messages by a $\Theta(\log n)$ multiplicative factor. We
complement our positive result with lower bounds showing that the dependence of
the number of messages on the domain size, the number of parties, and the
security parameter is essentially tight.
  Using a reduction of Balle et al. (2019), our improved analysis of the
protocol of Ishai et al. yields, in the same model, an $\left(\varepsilon,
\delta\right)$-differentially private protocol for aggregation that, for any
constant $\varepsilon > 0$ and any $\delta = \frac{1}{\mathrm{poly}(n)}$,
incurs only a constant error and requires only a constant number of messages
per party. Previously, such a protocol was known only for $\Omega(\log n)$
messages per party.
