Abstract
The conjugacy search problem (CSP) is used as a primitive in several braid group-based public key encryption schemes. It has been pointed out that, in braid groups, it is unlikely to provide adequate security. Therefore, new structures need to be found. In this paper, we give a formulation of the CSP for left conjugacy closed loops. In order to construct a generalization of the Anshel–Anshel–Goldfeld key establishment method, we also define a partial conjugacy search problem PCSP and show it to be equivalent to the CSP, if the underlying structure is a group. We also study more closely the PCSP in a class of conjugacy closed loops of order p 2, where p is a prime.
Similar content being viewed by others
References
Anshel, I., Anshel, M., Fisher, B., Goldfeld, D.: New key agreement protocols in braid group cryptography. In: Topics in Cryptology—CT-RSA 2001 (San Francisco, CA). Lecture Notes in Computer Science, vol. 2020, pp. 13–27. Springer, Berlin (2001)
Anshel I., Anshel M. and Goldfeld D. (1999). An algebraic method for public-key cryptography. Math. Res. Lett. 6(3–4): 287–291
Bohli, J.M., Glas, B., Steinwandt, R.: Towards provably secure group key agreement building on group theory. In: Progress in Cryptology—VIETCRYPT 2006. Lecture Notes in Computer Science, vol. 4341, pp. 322–336. Springer, Heidelberg (2006)
Bohli J.M., Vasco M.I.G. and Steinwandt R. (2007). Secure group key establishment revisited. Int. J. Inf. Secur. 6(4): 243–254. doi:10.1007/s10207-007-0018-x
Cheon, J.H., Jun, B.: A polynomial time algorithm for the braid Diffie–Hellman conjugacy problem. In: Advances in Cryptology—CRYPTO 2003. Lecture Notes in Computer Science, vol. 2729, pp. 212–225. Springer, Berlin (2003)
Csörgő P. and Drápal A. (2005). Left conjugacy closed loops of nilpotency class two. Results Math. 47(3–4): 242–265
Csörgő P. and Drápal A. (2006). On left conjugacy closed loops in which the left multiplication group is normal. Abh. Math. Sem. Univ. Hamburg 76: 17–34
Drápal A. (2006). On extraspecial left conjugacy closed loops. J. Algebra 302(2): 771–792
Gebhardt V. (2005). A new approach to the conjugacy problem in Garside groups. J. Algebra 292(1): 282–302
Gebhardt V. (2006). Conjugacy search in braid groups: from a braid-based cryptography point of view. Appl. Algebra Eng. Comm. Comput. 17(3–4): 219–238
Hofheinz, D., Steinwandt, R.: A practical attack on some braid group based cryptographic primitives. In: Public key Cryptography—PKC 2003. Lecture Notes in Computer Science, vol. 2567, pp. 187–198. Springer, Berlin (2002)
Keedwell A.D. (2000). Construction, properties and applications of finite neofields. Comment. Math. Univ. Carolin. 41(2): 283–297. Loops’99 (Prague)
Ko, K.H., Lee, S.J., Cheon, J.H., Han, J.W., Kang, J.S., Park, C.: New public-key cryptosystem using braid groups. In: Advances in Cryptology—CRYPTO 2000 (Santa Barbara, CA). Lecture Notes in Computer Science, vol. 1880, pp. 166–183. Springer, Berlin (2000)
Kościelny, C.: NLPN sequences over GF(q). Quasigroups Relat. Syst. 4, 89–102 (1999) (1997)
Kościelny C. (2002). Generating quasigroups for cryptographic applications. Int. J. Appl. Math. Comput. Sci. 12(4): 559–569
Kunen K. (2000). The structure of conjugacy closed loops. Trans. Am. Math. Soc. 352(6): 2889–2911
Lee, E., Park, J.H.: Cryptanalysis of the public-key encryption based on braid groups. In: Advances in Cryptology—EUROCRYPT 2003. Lecture Notes in Computer Science, vol. 2656, pp. 477–490. Springer, Berlin (2003)
Lee, S.J., Lee, E.: Potential weaknesses of the commutator key agreement protocol based on braid groups. In: Advances in Cryptology—EUROCRYPT 2002 (Amsterdam). Lecture Notes in Computer Science, vol. 2332, pp. 14–28. Springer, Berlin (2002)
Patarin, J., Goubin, L.: Trapdoor one-way permutations and multivariate polynomials. In: International Conference on Information Security and Cryptology. Lecture Notes in Computer Science, vol. 1334, pp. 356–368. Springer, Berlin (1997)
Pflugfelder H.O. (1990). Quasigroups and Loops: Introduction, Sigma Series in Pure Mathematics, vol. 7. Heldermann, Berlin
Shpilrain, V.: Assessing security of some group based cryptosystems. In: Group Theory, Statistics, and Cryptography, Contemp. Math., vol. 360, pp. 167–177. American Mathematical Society, Providence (2004)
Shpilrain V. and Ushakov A. (2006). The conjugacy search problem in public key cryptography: unnecessary and insufficient. Appl. Algebra Eng. Comm. Comput. 17(3–4): 285–289
Shpilrain V. and Zapata G. (2006). Combinatorial group theory and public key cryptography. Appl. Algebra Eng. Comm. Comput. 17(3–4): 291–302
Author information
Authors and Affiliations
Corresponding author
Additional information
Authors wish to thank referees for valuable comments and suggestions.
Rights and permissions
About this article
Cite this article
Partala, J., Seppänen, T. On the conjugacy search problem and left conjugacy closed loops. AAECC 19, 311–322 (2008). https://doi.org/10.1007/s00200-008-0066-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-008-0066-0