Advertisement

Thompson’s Group and Public Key Cryptography

  • Vladimir Shpilrain
  • Alexander Ushakov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3531)

Abstract

Recently, several public key exchange protocols based on symbolic computation in non-commutative (semi)groups were proposed as a more efficient alternative to well established protocols based on numeric computation. Notably, the protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al. exploited the conjugacy search problem in groups, which is a ramification of the discrete logarithm problem. However, it is a prevalent opinion now that the conjugacy search problem alone is unlikely to provide sufficient level of security no matter what particular group is chosen as a platform.

In this paper we employ another problem (we call it the decomposition problem), which is more general than the conjugacy search problem, and we suggest to use R. Thompson’s group as a platform. This group is well known in many areas of mathematics, including algebra, geometry, and analysis. It also has several properties that make it fit for cryptographic purposes. In particular, we show here that the word problem in Thompson’s group is solvable in almost linear time.

Keywords

Normal Form Word Problem Braid Group Negative Word Empty Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anshel, I., Anshel, M., Goldfeld, D.: An algebraic method for public-key cryptography. Math. Res. Lett. 6, 287–291 (1999)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompson’s groups. L’Enseignement Mathematique 42(2), 215–256 (1996)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cha, J.C., Ko, K.H., Lee, S.J., Han, J.W., Cheon, J.H.: An efficient implementation of braid groups. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 144–156. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Hofheinz, D., Steinwandt, R.: A practical attack on some braid group based cryptographic primitives. In: Desmedt, Y.G. (ed.) PKC 2003. LNCS, vol. 2567, pp. 187–198. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Hughes, J., Tannenbaum, A.: Length-based attacks for certain group based encryption rewriting systems, Workshop SECI02 Securitè de la Communication sur Intenet, Tunis, Tunisia (September 2002), http://www.storagetek.com/hughes/
  6. 6.
    Ko, K.H., Lee, S.J., Cheon, J.H., Han, J.W., Kang, J., Park, C.: New public-key cryptosystem using braid groups. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 166–183. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Shpilrain, V.: Assessing security of some group based cryptosystems. Contemp. Math., Amer. Math. Soc. 360, 167–177 (2004)MathSciNetGoogle Scholar
  8. 8.
    Shpilrain, V., Ushakov, A.: The conjugacy search problem in public key cryptography: unnecessary and insufficient, Applicable Algebra in Engineering, Communication and Computing (to appear), http://eprint.iacr.org/2004/321/
  9. 9.
    Shpilrain, V., Zapata, G.: Combinatorial group theory and public key cryptography. Applicable Algebra in Engineering, Communication and Computing (to appear)Google Scholar
  10. 10.
    Sims, C.: Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Vladimir Shpilrain
    • 1
  • Alexander Ushakov
    • 2
  1. 1.Department of MathematicsThe City College of New YorkNew YorkUSA
  2. 2.Department of MathematicsCUNY Graduate CenterNew YorkUSA

Personalised recommendations