We introduce and study the iterated random permutation problem, which asks how hard it is to distinguish, in a black-box way, the r-th power of a random permutation from a uniformly random permutation of a set of size N. We show that this requires Omega(N) queries (even for a two-sided, adaptive adversary). As a direct application of this result, we show that cascading a block cipher with the same key cannot degrade its security (as a pseudorandom permutation) more than negligibly.
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