Revista Matemática Iberoamericana

Computation of Centralizers in Braid groups and Garside groups

Nuno Franco and Juan González-Meneses

Full-text: Open access

Abstract

We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conjugacy problem given by the authors in \cite{FGM}, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups \cite{AAG}. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.

Article information

Source
Rev. Mat. Iberoamericana, Volume 19, Number 2 (2003), 367-384.

Dates
First available in Project Euclid: 8 September 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1063050158

Mathematical Reviews number (MathSciNet)
MR2023190

Zentralblatt MATH identifier
1064.20040

Subjects
Primary: 20F36: Braid groups; Artin groups
Secondary: 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 94A60: Cryptography [See also 11T71, 14G50, 68P25, 81P94]

Keywords
Braid group Garside group centralizer cryptography

Citation

Franco, Nuno; González-Meneses, Juan. Computation of Centralizers in Braid groups and Garside groups. Rev. Mat. Iberoamericana 19 (2003), no. 2, 367--384. https://projecteuclid.org/euclid.rmi/1063050158


Export citation

References

  • Artin, E.: Theory of braids. Annals of Math. 48 (1946), 101-126.
  • Anshel, I., Anshel, M. and Goldfeld, D.: An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), no. 3-4, 287-291.
  • Birman, J., Ko, K. H. and Lee, S. J.: A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139 (1998), no. 2, 322-353.
  • Birman, J., Ko, K. H. and Lee, S. J.: The infimum, supremum and geodesic length of a braid conjugacy class. Adv. Math. 164 (2001), 41-56.
  • Brieskorn, E. and Saito, K.: Artin-Gruppen und Coxeter-Gruppen. Invent. Math. 17 (1972), 245-271.
  • Dehornoy, P.: Groupes de Garside. Ann. Sci. École Norm. Sup. (4) 35 (2002), 267-306.
  • Dehornoy, P. and Paris, L.: Gaussian groups and Garside groups, two generalizations of Artin groups. Proc. London Math. Soc. 79 (1999), no. 3, 569-604.
  • Elrifai, E. A. and Morton, H. R.: Algorithms for positive braids. Quart. J. Math. Oxford 45 (1994), 479-497.
  • Franco, N. and González-Meneses, J.: Conjugacy problem for braid groups and Garside groups, to appear in Journal of Algebra. Available at http://arxiv.org/math.GT/0112310
  • Garside, F. A.: The braid group and other groups. Quart. J. Math. Oxford 20 (1969), 235-154.
  • González-Meneses, J. and Wiest, B.: On the structure of the centralizer of a braid. In preparation.
  • Ko, K. H., Lee, S. J., Cheon, J. H., Han, J. W., Kang, J. and Park, C.: New public-key cryptosystem using braid groups. In Advances in cryptology-CRYPTO 2000 (Santa Barbara, CA), 166-183. Lecture Notes in Comput. Sci. 1880, Springer, Berlin, 2000.
  • Lyndon, R. C. and Schupp, P. E.: Combinatorial group theory. Reprint of the 1977 edition. Classics in Mathematics, Springer-Verlag, Berlin, 2001.
  • Makanin, G. S.: The normalizers in the braid group. Mat. Sb. (N. S.) 86 (128) (1971), 171-179.
  • Picantin, M.: Petits groupes gaussiens. Ph. D. Thesis, Université de Caen, 2000.
  • Picantin, M.: The conjugacy problem in small Gaussian groups. Comm. Algebra 29 (2001), no. 3, 1021-1039.
  • Thurston, W. P.: Braid Groups, Chapter 9 of Word processing in groups, D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson and W. P. Thurston. Jones and Bartlett Publishers, Boston, MA, 1992.