Stream Ciphers by Andreas Klein

Author:Andreas Klein
Language: eng
Format: epub
Publisher: Springer London, London

for any permutation π of {0,…,n−1}.

Theorem 9.10

(Diaconis [79])

Let Q be a shuffling process and let τ be a strongly uniform stopping time. Then for all times t, the variation distance between Q t and the uniform shuffle U satisfies

Proof

See [79]. □

So we have to find a uniform stopping time τ and compute the probabilities P(τ>t).

We define τ by the following procedure. 1.At the beginning the elements S[0],…,S[n−2] are unmarked and S[n−1] is marked.

2.Every time we swap S[i] with S[j], S[i] becomes marked if either j=i or S[j] is already marked. Note this is not symmetric in i and j, we will not mark S[j] if S[i] is already marked.

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