  Differentially private noise mechanisms commonly use symmetric noise
distributions. This is attractive both for achieving the differential privacy
definition, and for unbiased expectations in the noised answers. However, there
are contexts in which a noisy answer only has utility if it is conservative,
that is, has known-signed error, which we call a padded answer. Seemingly, it
is paradoxical to satisfy the DP definition with one-sided error, but we show
how it is possible to bury the paradox into approximate DP's delta parameter.
We develop a few mechanisms for one-sided padding mechanisms that always give
conservative answers, but still achieve approximate differential privacy. We
show how these mechanisms can be applied in a few select areas including making
the cardinalities of set intersections and unions revealed in Private Set
Intersection protocols differential private and enabling multiparty computation
protocols to compute on sparse data which has its exact sizes made differential
private rather than performing a fully oblivious more expensive computation.
