Say I have a logic circuit of $n$ bits. And I have a reversible gate $G_n$ that can perform arbitrary (even) permutations of the values of these $n$ bits. Is it possible to replace this by some number of $3$-bit (arbitrary permutation) logic gates $G_3$? How?
(An example of a 3-bit permutation gate would be say the Toffoli. The $G_3$ and $G_n$ gates can perform arbitrary even permutations.)
P.S. For some more context see my previous question.