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A dynamic two-stage Delta network ($N$ inputs and outputs) is introduced and analyzed for permutation routing. The notion of evil twins is introduced and a deterministic procedure is given to route any permutation in no more than $2 \root 4 \of N$ network cycles. Two limited randomized routing schemes are then analyzed. The first called Single Randomization yields on average at most $N {\bar !}+1+{\frac{1}{N}}$ ($N {\bar !} =O({\frac{\log{N}}{\log\log{N}}})$1 and is the greatest integer such that $(N {\bar !} )! \leq N$) network cycles and the second called Multiple Randomization yields on average at most $\lfloor \log(\log{N}+1)\rfloor+2+{\frac{1}{N}}$ network cycles for any input permutation. The probability of any permutation requiring at least $c$ network cycles more than the above average bounds is then shown to be at most ${\frac{1}{(c+1)!}}$ for Single Randomization and ${\frac{1}{N^{2^{c}}}}$ for Multiple Randomization, respectively. It is then shown how the dynamic two-stage network can be physically realized as a three-stage network. Both the evil twin and Multiple Randomization algorithms have been integrated into an off-the-shelf ASIC from PMC-Sierra, Inc. (PM-73488) which has been designed as a building block for such a three-stage implementation. These routing schemes are also adapted to run on a recirculating network. Recirculation is used to effect a reshuffling of data as in the dynamic network, but with a considerable reduction in network cost.