  A canonical noise distribution (CND) is an additive mechanism designed to
satisfy $f$-differential privacy ($f$-DP), without any wasted privacy budget.
$f$-DP is a hypothesis testing-based formulation of privacy phrased in terms of
tradeoff functions, which captures the difficulty of a hypothesis test. In this
paper, we consider the existence and construction of both log-concave CNDs and
multivariate CNDs. Log-concave distributions are important to ensure that
higher outputs of the mechanism correspond to higher input values, whereas
multivariate noise distributions are important to ensure that a joint release
of multiple outputs has a tight privacy characterization. We show that the
existence and construction of CNDs for both types of problems is related to
whether the tradeoff function can be decomposed by functional composition
(related to group privacy) or mechanism composition. In particular, we show
that pure $\epsilon$-DP cannot be decomposed in either way and that there is
neither a log-concave CND nor any multivariate CND for $\epsilon$-DP. On the
other hand, we show that Gaussian-DP, $(0,\delta)$-DP, and Laplace-DP each have
both log-concave and multivariate CNDs.
