International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Permutation graphs, fast forward permutations, and

Boaz Tsaban
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Abstract: A permutation $P\in S_N$ is a \emph{fast forward permutation} if for each $m$ the computational complexity of evaluating $P^m(x)$ 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 $S_N$ is $\Theta(N)$ if one does not use queries of the form $P^m(x)$, but is only $\Theta(1)$ 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 $P^m(x)$ 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.
  title={Permutation graphs, fast forward permutations, and},
  booktitle={IACR Eprint archive},
  keywords={foundations /},
  note={Journal of Algorithms 47 (2), 104--121. 12250 received 16 Jul 2003},
  author={Boaz Tsaban},