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

Joel Friedman; Antoine Joux; Yuval Roichman; Jacques Stern; Jean Pierre Tillich

doi:10.1007/3-540-60922-9_31

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

Joel Friedman
, Antoine Joux
, Yuval Roichman
, Jacques Stern
, Jean Pierre Tillich

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

3 Scopus citations

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, ..., π_d 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]).

Original language	English
Title of host publication	STACS 1996 - 13th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings
Editors	Claude Puech, Rudiger Reischuk
Publisher	Springer Verlag
Pages	375-386
Number of pages	12
ISBN (Print)	9783540609223
DOIs	https://doi.org/10.1007/3-540-60922-9_31
State	Published - 1996
Externally published	Yes
Event	13th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1996 - Grenoble, France Duration: 22 Feb 1996 → 24 Feb 1996

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	1046
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	13th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1996
Country/Territory	France
City	Grenoble
Period	22/02/96 → 24/02/96

Bibliographical note

Publisher Copyright:
© 1996, Springer Verlag. All rights reserved.

Access to Document

10.1007/3-540-60922-9_31

Cite this

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 C. Puech, & R. Reischuk (Eds.), STACS 1996 - 13th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings (pp. 375-386). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1046). Springer Verlag. https://doi.org/10.1007/3-540-60922-9_31

Friedman, Joel ; Joux, Antoine ; Roichman, Yuval et al. / The action of a few random permutations on r-tuples and an application to cryptography. STACS 1996 - 13th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. editor / Claude Puech ; Rudiger Reischuk. Springer Verlag, 1996. pp. 375-386 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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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 Sn , 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]).",

author = "Joel Friedman and Antoine Joux and Yuval Roichman and Jacques Stern and Tillich, \{Jean Pierre\}",

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Friedman, J, Joux, A, Roichman, Y, Stern, J & Tillich, JP 1996, The action of a few random permutations on r-tuples and an application to cryptography. in C Puech & R Reischuk (eds), STACS 1996 - 13th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1046, Springer Verlag, pp. 375-386, 13th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1996, Grenoble, France, 22/02/96. https://doi.org/10.1007/3-540-60922-9_31

The action of a few random permutations on r-tuples and an application to cryptography. / Friedman, Joel; Joux, Antoine; Roichman, Yuval et al.
STACS 1996 - 13th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. ed. / Claude Puech; Rudiger Reischuk. Springer Verlag, 1996. p. 375-386 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1046).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - 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 Sn , 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]).

AB - 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 Sn , 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]).

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Friedman J, Joux A, Roichman Y, Stern J, Tillich JP. The action of a few random permutations on r-tuples and an application to cryptography. In Puech C, Reischuk R, editors, STACS 1996 - 13th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings. Springer Verlag. 1996. p. 375-386. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-60922-9_31

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

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