[owiwfzux] shuffling one bit

persons A, B, and C in separate rooms with point-to-point communication.

1. person C describes to person A how they intend to shuffle cards, e.g., 3 riffle shuffles.
2. person A defines two decks, specifying the order of cards in each deck, and tells the information to person B.
3. person B secretly randomly chooses one of person A's card orderings, constructs a deck with the chosen ordering, then gives the deck to person C.  we go through this person B intermediary so that person A cannot secretly mark decks to tell them apart afterward.
4. person C shuffles the received deck as declared in step 1, and gives the shuffled deck to person A.
5. person A tries to determine which of the two decks person B gave to person C.

I strongly suspect there are ways that person A can construct decks so that it is very difficult to shuffle them enough destroy the one bit of information that is encoded in them.  you will need the full 7 riffle shuffles as proved by Diaconis.  (practically perhaps more, if your riffle shuffles are not very good.)

optionally omit step 1 to give person A less of an advantage.

a "shuffling" step that person C can do to possibly make things difficult is the following: flip a coin.  if and only if tails, reverse the order of the deck.  (I am not aware of any way to do this quickly.  perhaps cardistry magicians know of a way.)  then, do other shuffles.

variation: split the deck into 2.  flip a coin.  reverse one half or the other corresponding to the coin flip.  then riffle shuffle.

also consider person C hand scrambling a speedcube instead of shuffling a deck of cards.  person B needs to be able to construct arbitrary (legal) Rubik's cube states given by person A.