Hiding data in permutations

The latest issue of Paged Out! has an article by Stephen Hewitt “An off-line backup of your cryptographic key using playing cards.” The idea is to use a deck of 52 to store a 128-bit cryptographic key. To erase the key, shuffle the deck. In principle you could store a 225-bit key as a permutation of 52 cards because log2(52!) = 225.581. But then how would you number permutations so you could go from a number to a particular permutation and later decode the permutation to a number? Is this even practical? For a small number n, you could encode a number k < n by enumerating the first k permutations of a set of n items, and you could decode by enumerating permutations until you find the one you have. But this is completely impractical for large n, such as n = 52. The process of mapping permutation to an integer is called ranking, and the mapping from an integer to a permutation is called unranking. How efficiently can rankings and unrankings be calculated? Let n be the number of symbols…