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We show that any permutation of ${1,2,...,N}$ can be written as the productof two involutions. As a consequence, any permutation of the elements of anarray can be performed in-place in parallel in time O(1). In the case where thepermutation is the $k$-way perfect shuffle we develop two methods forefficiently computing such a pair of involutions.  The first method works whenever $N$ is a power of $k$; in this case the timeis O(N) and space $O(\log^2 N)$. The second method applies to the general casewhere $N$ is a multiple of $k$; here the time is $O(N \log N)$ and the space is$O(\log^2 N)$. If $k=2$ the space usage of the first method can be reduced to$O(\log N)$ on a machine that has a SADD (population count) instruction.

Parallel and sequential in-place permuting and perfect shuffling using involutions