A differentially private computation often begins with a bound on a $d$-dimensional statistic's $\ell_p$ sensitivity. The $K$-norm mechanism can yield more accurate additive noise by using a statistic-specific (and possibly non-$\ell_p$) norm. However, sampling such mechanisms requires sampling from the corresponding norm balls. These are $d$-dimensional convex polytopes, and the fastest known general algorithm for approximately sampling such polytopes takes time $\tilde O(d^{3+\omega})$, where $\omega \geq 2$ is the matrix multiplication exponent. For the simple problems of sum and ranked vote, this paper constructs samplers that run in time $\tilde O(d^2)$. More broadly, we suggest that problem-specific $K$-norm mechanisms may be an overlooked practical tool for private additive noise.

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