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

- 117 Downloads

## Abstract

We prove that for every *r* and *d*≥2 there is a *C* such that for most choices of *d* permutations *π*_{1}, π_{2}, ..., 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]).

## Keywords

Directed Graph Undirected Graph Regular Graph Cayley Graph Finite Automaton## Preview

Unable to display preview. Download preview PDF.

## References

- [AKS83]M. Ajtai, J. Komlòs, E. Szemerédi, “Sorting in
*c*log*n*parallel steps”,*Combinatorica***3**(1983), 1–19.Google Scholar - [Ang78]D. Angluin. “On the complexity of minimum inference of regular sets”,
*Information and Control***39**(1978), 302–320.Google Scholar - [AS83]D. Angluin and C.H. Smith. “Inductive inference, theory and methods”,
*Computing Surveys***15**(**3**) (1983), 237–269.CrossRefGoogle Scholar - [AM85]N. Alon and V.D. Milman. “gl
_{1}, isoperimetric inequalities for graphs and superconcentrators”,*J. Comb. Theory*, Ser. B,**38**, (1985), 73–88.Google Scholar - [Bab94]L. Babai. “Transparent proofs and limits to approximation”,
*preprint*, (1994).Google Scholar - [B&al90]L. Babai, G. Hetyei, W.M. Kantor, A. Lubotzky, A. Seres. “On the diameter of finite groups”,
*31st annual Symposium on Foundations of Computer Science*, (1990), 857–865.Google Scholar - [BGG90]M. Bellare, O. Goldreich, S. Goldwasser. “Randomness in interactive proofs”,
*31st Annual Symposium on Foundations of Computer Science*, IEEE Computer Society Press, (1990), 563–572.Google Scholar - [Bol85]B. Bollobas.
*Random Graphs*, Academic Press, London (1985).Google Scholar - [Bol88]B. Bollobas. “The isoperimetric number of random regular graphs”,
*Europ. J. Combinatorics***9**(1988), 241–244.Google Scholar - [BV82]B. Bollobas and W. F. de la Vega. “The diameter of random-regular graphs”, Combinatorica,
**2**, (1982), 125–134.Google Scholar - [BS87]A. Broder, E. Shamir. “On the second eigenvalue of random regular graphs”,
*28th annual Symposium on Foundations of Computer Science*, (1987), 286–284.Google Scholar - [Del89]C. Delorme. “Counting closed paths in trees”,
*Technical Report*n.516, University of Paris-Sud, Laboratoire de recherche en informatique Orsay, September 1989 (in French).Google Scholar - [Fil91]J. Fill. “Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains with an application to the exclusion processes” Ann. Appl. Prob. 1, (1991), 62–87.Google Scholar
- [F&al93]Y. Freund, M. Kearns, D. Ron, R. Rubinfeld, R.E. Schapire and L. Sellie. “Efficient learning of typical finite automata from random walks”,
*25th ACM Symposium on the Theory of Computing*(1993), 315–324.Google Scholar - [FJRST95]J. Friedman,A. Joux,Y. Roichman,J. Stern,J.P. Tillich. “The action of a few permutations on
*r*-tuples is quickly transitive”,*submitted*.Google Scholar - [Fri91]J. Friedman. “On the second eigenvalue and random walks in random
*d*-regular graphs”,*Combinatorica***11**(4) (1991), 331–362.CrossRefGoogle Scholar - [FKS89]J. Friedman, J. Kahn, E. Szemeredi. “On the second eigenvalue in random regular graphs”,
*21st annual Symposium on Theory of Computing*, ACM press, (1989), 587–598.Google Scholar - [Gol78]E.M. Gold. “Complexity of automaton identification from given data”,
*Information and Control***37**(1978), 302–320.CrossRefGoogle Scholar - [G&al90]O. Goldreich, R. Impagliazzo, L. Levin, R. Venkatesen, D. Zuckerman. “Security preserving amplification of randomness”,
*31st Annual Symposium on Foundations of Computer Science*, IEEE Computer Society Press, (1990), 318–326.Google Scholar - [IZ89]R. Impagliazzo, D. Zuckerman. “How to recycle random bits”,
*30th Annual Symposium on Foundations of Computer Science*, IEEE Computer Society Press, (1989), 248–253.Google Scholar - [JST93]A. Joux, J. Stern, J.P. Tillich. “Inferring finite automata by queries of fixed length”,
*Preprint*.Google Scholar - [Kah91]N. Kahale. “Better expansions for Ramanujan graphs”,
*32nd Annual Symposium on Foundations of Computer Science*(1991), 398–404.Google Scholar - [Kah92]N. Kahale. “On the second eigenvalue and linear expansion of regular graphs”,
*33rd Annual Symposium on Foundations of Computer Science*(1992), 296–303.Google Scholar - [LR92]J. Lafferty, D. Rockmore. “Fast Fourier analysis for
*SL*_{2}over a finite field, and related numerical experiments”,*Experimental Mathematics***1**, (1992), 115–139.Google Scholar - [Lubl]A. Lubotzky.
*Discrete groups, expanding graphs and invariant measures*, Progress in Mathematics, Vol. 125, Birkhäuser 1994.Google Scholar - [Lub2]A. Lubotzky. “Cayley graphs: eigenvalues, expanders and random walks”, to appear in Survey in Combinatorics, 1995.Google Scholar
- [McK81]B. McKay. “The expected eigenvalue distribution of a large regular graph”,
*Linear Algebra and its Applications*,**40**, (1981), 203–216.CrossRefGoogle Scholar - [Mih89]M. Mihail. “Conductance and convergence of Markov chains—a combinatorial treatment of expanders”,
*Proceedings of the 30th Annual Symposium on Foundations of Computer Science*, 1989.Google Scholar - [Moh89]B. Mohar. “Isoperimetric number of graphs”,
*Journal of Comb. Theory*(**B**) (1989), 274–291.CrossRefGoogle Scholar - [Pip77]
- [RS87]R.L. Rivest and R.E. Schapire. “Diversity based inference of finite automata”
*Proceedings of the 28th Annual Symposium on the Foundations of Computer Science*(1987), 78–87.Google Scholar - [RS89]R.L. Rivest and R.E. Schapire. “Inference of finite automata using homing sequences”
*Proceedings of the 21st ACM Symposium on the Theory of Computing*(1989), 411–420.Google Scholar - [Tan84]R.M. Tanner. “Explicit constructions of concentrators from generalized
*N*-gons”,*SIAM J. Alg. Disc. Meth.*,**5**, (1984), 287–293.Google Scholar - [TZ93]J.P. Tillich, G. Zémor. “Group-theoretic hash functions”,
*Proceedings of the 1st French-Israeli Workshop in algebraic coding 1993*, Springer Verlag, Lecture Notes**781**, 90–110.Google Scholar - [TZ94]J.P. Tillich, G. Zémor. “Hashing with SL2”,
*Advances in Cryptology, Proceedings of CRYPTO94*, Springer Verlag, Lecture Notes**839**, 40–49.Google Scholar - [Vaz91]U. Vazirani. “Rapidly mixing markov chains”,
*Proceedings of Symposia in Applied Mathematics*, Volume**44**, (1991), 99–121.Google Scholar - [Zem94]G. Zémor. “Hash Functions and Cayley graphs”, to appear in
*Design, Codes and Cryptography*, of October 1994.Google Scholar