  A protocol by Ishai et al.\ (FOCS 2006) showing how to implement distributed
$n$-party summation from secure shuffling has regained relevance in the context
of the recently proposed \emph{shuffle model} of differential privacy, as it
allows to attain the accuracy levels of the curator model at a moderate
communication cost. To achieve statistical security $2^{-\sigma}$, the protocol
by Ishai et al.\ requires the number of messages sent by each party to {\em
grow} logarithmically with $n$ as $O(\log n + \sigma)$. In this note we give an
improved analysis achieving a dependency of the form $O(1+\sigma/\log n)$.
Conceptually, this addresses the intuitive question left open by Ishai et al.\
of whether the shuffling step in their protocol provides a "hiding in the
crowd" amplification effect as $n$ increases. From a practical perspective, our
analysis provides explicit constants and shows, for example, that the method of
Ishai et al.\ applied to summation of $32$-bit numbers from $n=10^4$ parties
sending $12$ messages each provides statistical security $2^{-40}$.
