# Is Triple-DES a group?

I know for a fact that DES is not a group, but are any of the Triple-Des versions a group? Why, or why not?

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Keith Campbell & Michael Wiener's paper proving that DES is not a group is (currently) available for free from the publisher, in a much better format than I once paid for. –  fgrieu Jun 28 '13 at 11:14

Justification: We know that the subgroup generated by DES is very large. If (any of the variants of) 3DES formed a closed group, then the subgroup generated by 3DES would be no larger than $2^{168}$. We know the latter is not the case. Therefore, the former is not the case, either.
I initialy wondered if "the subgroup generated by DES" is the subgroup of the group of (even) permutations generated by the set of (likely, $2^{56}$) permutations defined by DES encryption; or by the set of (likely, $2^{57}-4$) permutations defined by DES encryption and decryption. Now I realize these definitions are equivalent. –  fgrieu Jun 28 '13 at 9:11