Skip to main content
SpringerLink
Log in

Security analysis of word problem-based cryptosystems

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We investigate two schemes based on the word problem on groups. From a complexity-theoretic point of view, we show that the problems underlying those schemes are equivalent. We then present a reaction attack on one of the schemes, thus easily transposed to the other. The attack, besides its efficiency, permits to recover an equivalent secret key.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abisha P.J., Thomas D.G., Subramanian K.G.: Public key cryptosystems based on free partially commutative monoids and groups. In: Proceedings of INDOCRYPT 2003, LNCS, vol. 2904, pp. 218–227. Springer, Berlin, Heidelberg (2003).

  2. Book R.V.: Confluent and other types of thue systems. J. ACM 29, 171–182 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Book R.V., Otto F.: String-Rewriting Systems. Texts and Monographs in Computer Science. Springer-Verlag, New York (1993).

  4. González-Vasco M.I., Steinwandt R.: Pitfalls in public key systems based on free partially commutative monoids and groups. Appl. Math. Lett. 19(10), 1037–1041 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. González-Vasco M.I., Steinwandt R.: A reaction attack on a public key cryptosystem based on the word problem. Appl. Algebra Eng. Commun. Comput. 14(5), 335–340 (2004)

    Article  MATH  Google Scholar 

  6. Levy-dit-Vehel F., Perret L.: On the Wagner–Magyarik cryptosystem. Revised Selected Papers of WCC 2005 Conference, LNCS, vol. 3969, pp. 316–329. Springer, Berlin (2006).

  7. Novikov P.S.: On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44, 1–143 (1955)

    Google Scholar 

  8. Wagner N.R., Magyarik M.R.: A public key cryptosystem based on the word problem. In: Proceedings of CRYPTO’84, LNCS, vol. 96, pp. 19–36. Springer-Verlag, Berlin.

  9. Wrathall C.: The word problem for free partially commutative groups. J. Symb. Comput. 6, 99–104 (1988)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ENSTA, 32 Boulevard Victor, 75739, Paris Cedex 15, France

    Françoise Levy-dit-Vehel

  2. LIP6/INRIA/SALSA, University Paris 6, Site Passy-Kennedy 104 avenue du Président Kennedy, 75016, Paris, France

    Ludovic Perret

Authors
  1. Françoise Levy-dit-Vehel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ludovic Perret
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Françoise Levy-dit-Vehel.

Additional information

Communicated by P. Wild.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Levy-dit-Vehel, F., Perret, L. Security analysis of word problem-based cryptosystems. Des. Codes Cryptogr. 54, 29–41 (2010). https://doi.org/10.1007/s10623-009-9307-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-009-9307-x

Keywords

Mathematics Subject Classification (2000)

Search

Navigation