The question is very simple: Do you know an efficient algorithm to generate samples of the multivariate cauchy distribution

The pdf is given by

$f(x_i;\mu_i,\Sigma,k) = \frac{\Gamma\left(\frac{1+k}{2}\right)}{\pi^{(1+k)/2} \left[(x_i-\mu_i)\Sigma_{ij}(x_j-\mu_j) + 1\right]^{(1+k)/2}}$

where the indices $i,j=1,...,k$ and $\Sigma_{ij}$ is a $k\times k$ positive definite matrix.

Using that a multivariate cauchy distribution is really a multivariate Student's distribution with one degree of freedom, I would be happy also knowing how to generate samples of a Student's distribution.