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The action of a few random permutations on r-tuples and an application to cryptography

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STACS 96 (STACS 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1046))

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Abstract

We prove that for every r and d≥2 there is a C such that for most choices of d permutations π 1, π2, ..., πd of S n , a product of less than C log n of these permutations is needed to map any r-tuple of distinct integers to another r-tuple. We came across this problem while studying a seemingly unrelated cryptographic problem, and use this result in order to show that certain cryptographic devices using permutation automata are highly insecure. The proof techniques we develop here give more general results, and constitute a first step towards the study of expansion properties of random Cayley graphs over the symmetric group, whose relevance to theoretical computer science is well-known (see [B&al90]).

Partially supported by Univ. of British Columbia.

Research supported in part by a CNET grant and a NSERC Postdoctoral Fellowship. This author enjoyed the hospitality of the University of British Colombia (Vancouver, Canada) while part of this research was carried out.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, Univ. of British Columbia, V6T 1Z2, Vancouver, Canada

    Joel Friedman

  2. CELAR, France

    Antoine Joux

  3. Dept. of Applied Mathematics, Massachussetts Instit. of Tech., 02138, Cambridge, MA, USA

    Yuval Roichman

  4. Ecole Normale Supérieure, 45 rue d'ULM, 75005, Paris, France

    Jacques Stern

  5. GREYC, Université de Caen, 14000, Caen, France

    Jean -Pierre Tillich

Authors
  1. Joel Friedman
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  2. Antoine Joux
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  3. Yuval Roichman
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  4. Jacques Stern
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  5. Jean -Pierre Tillich
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Claude Puech Rüdiger Reischuk

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© 1996 Springer-Verlag Berlin Heidelberg

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Friedman, J., Joux, A., Roichman, Y., Stern, J., Tillich, J.P. (1996). The action of a few random permutations on r-tuples and an application to cryptography. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_31

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  • DOI: https://doi.org/10.1007/3-540-60922-9_31

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60922-3

  • Online ISBN: 978-3-540-49723-3

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