Cryptology ePrint Archive: Report 2003/138

Permutation graphs, fast forward permutations, and

Boaz Tsaban

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.

Category / Keywords: foundations /

Publication Info: Journal of Algorithms 47 (2), 104--121.

Date: received 16 Jul 2003

Contact author: tsaban at math huji ac il

Available format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

Note: It seems that a recent result of Goldwasser, Goldreich, and Nussbaum can be extended to prove the conjecture at the end of this paper.

Version: 20030717:170702 (All versions of this report)

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