Permutation graphs, fast forward permutations, and sampling the cycle structure of a permutation

P ∈ SN is a fast forward permutation if for each m the computational complexity of evaluating Pm(x) is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions. By studying the evolution of permutation graphs, we prove that the number of queries needed to distinguish a random cyclus from a random permutation in SN is (N) if one does not use queries of the form P m(x), but is only (1) if one is allowed to make such queries. We construct fast forward permutations which are indistinguishable from random permutations even when queries of the form Pm(x) are allowed. This is done by introducing an efficient method to sample the cycle structure of a random permutation, which in turn solves an open problem of Naor and Reingold.  2003 Elsevier Science (USA). All rights reserved.