  In this paper, we introduce the imperfect shuffle differential privacy model,
where messages sent from users are shuffled in an almost uniform manner before
being observed by a curator for private aggregation. We then consider the
private summation problem. We show that the standard split-and-mix protocol by
Ishai et. al. [FOCS 2006] can be adapted to achieve near-optimal utility bounds
in the imperfect shuffle model. Specifically, we show that surprisingly, there
is no additional error overhead necessary in the imperfect shuffle model.
