  Differential privacy has emerged as a standard requirement in a variety of
applications ranging from the U.S. Census to data collected in commercial
devices, initiating an extensive line of research in accurately and privately
releasing statistics of a database. An increasing number of such databases
consist of data from multiple sources, not all of which can be trusted. This
leaves existing private analyses vulnerable to attacks by an adversary who
injects corrupted data. Despite the significance of designing algorithms that
guarantee privacy and robustness (to a fraction of data being corrupted)
simultaneously, even the simplest questions remain open. For the canonical
problem of estimating the mean from i.i.d. samples, we introduce the first
efficient algorithm that achieves both privacy and robustness for a wide range
of distributions. This achieves optimal accuracy matching the known lower
bounds for robustness, but the sample complexity has a factor of $d^{1/2}$ gap
from known lower bounds. We further show that this gap is due to the
computational efficiency; we introduce the first family of algorithms that
close this gap but takes exponential time. The innovation is in exploiting
resilience (a key property in robust estimation) to adaptively bound the
sensitivity and improve privacy.
