  A fundamental problem in differential privacy is to release a privatized data
structure over a dataset that can be used to answer a class of linear queries
with small errors. This problem has been well studied in the static case. In
this paper, we consider the dynamic setting where items may be inserted into or
deleted from the dataset over time, and we need to continually release data
structures so that queries can be answered at any time. We present black-box
constructions of such dynamic differentially private mechanisms from static
ones with only a polylogarithmic degradation in the utility. For the
fully-dynamic case, this is the first such result. For the insertion-only case,
similar constructions are known, but we improve them over sparse update
streams.
