  Using a similar random process to the one which yields the fractal
percolation sets, starting from the deterministic Menger sponge we get the
random Menger sponge. We examine its orthogonal projections from the point of
Hausdorff dimension, Lebesgue measure and existence of interior points. We
obtain these results as special cases of our theorems stated for random
self-similar IFSs.
  These are obatained by a random process similar to the fractal percolation,
applied for the cylinder sets of a deterministic self-similar IFS, as in
arXiv:1212.1345. In this paper the associated deterministic IFS on the line is
of the special form $\mathcal{S}=\left\{\frac{1}{L}x+t_i \right\} _{i=1}^{m}$,
where $L\in\mathbb{N}$, $L\geq 2$ and $t_i\in\mathbb{Q}$.
