Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board Member: Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

2 Issues per year


CiteScore 2016: 0.35

SCImago Journal Rank (SJR) 2016: 0.372
Source Normalized Impact per Paper (SNIP) 2016: 0.517

Mathematical Citation Quotient (MCQ) 2016: 0.23

Online
ISSN
1869-6104
See all formats and pricing
More options …

A parallel evolutionary approach to solving systems of equations in polycyclic groups

Matthew J. Craven
  • Corresponding author
  • Centre for Mathematical Sciences, Plymouth University, Drake Circus, Plymouth, PL4 8AA, United Kingdom of Great Britain and Northern Ireland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Daniel Robertz
  • Centre for Mathematical Sciences, Plymouth University, Drake Circus, Plymouth, PL4 8AA, United Kingdom of Great Britain and Northern Ireland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-10-11 | DOI: https://doi.org/10.1515/gcc-2016-0012

Abstract

The Anshel–Anshel–Goldfeld (AAG) key exchange protocol is based upon the multiple conjugacy problem for a finitely-presented group. The hardness in breaking this protocol relies on the supposed difficulty in solving the corresponding equations for the conjugating element in the group. Two such protocols based on polycyclic groups as a platform were recently proposed and were shown to be resistant to length-based attack. In this article we propose a parallel evolutionary approach which runs on multicore high-performance architectures. The approach is shown to be more efficient than previous attempts to break these protocols, and also more successful. Comprehensive data of experiments run with a GAP implementation are provided and compared to the results of earlier length-based attacks. These demonstrate that the proposed platform is not as secure as first thought and also show that existing measures of cryptographic complexity are not optimal. A more accurate alternative measure is suggested. Finally, a linear algebra attack for one of the protocols is introduced.

Keywords: Evolutionary algorithms; polycyclic groups; cryptography; Anshel–Anshel–Goldfeld keyagreement protocol; high performance computing; parallel

MSC 2010: 20P05; 68W30; 90C27; 94A60

References

  • [1]

    Anshel I., Anshel M., Fisher B. and Goldfeld D., New key agreement protocols, Topics in Cryptology – CT-RSA 2001, Lecture Notes in Comput. Sci. 2020, Springer, Berlin (2001), 13–27. Google Scholar

  • [2]

    Anshel I., Anshel M. and Goldfeld D., An algebraic method for public-key cryptography, Math. Res. Lett. 6 (1999), 287–291. Google Scholar

  • [3]

    Cooperman G., ParGAP, Version 1.4.0, 2013, http://www.gap-system.org/Packages/pargap.html.

  • [4]

    Craven M. J. and Jimbo H. C., An evolutionary algorithm solution of the multiple conjugacy search problem in partially commutative groups with applications, Groups Complex. Cryptol. 4 (2012), 135–165. Google Scholar

  • [5]

    Eick B. and Kahrobaei D., Polycyclic groups: A new platform for cryptology?, preprint 2004, http://arxiv.org/abs/math/0411077.

  • [6]

    Eick B., Nickel W. and Horn M., Polycyclic, Version 2.1.1, 2013, http://www.gap-system.org/Packages/polycyclic.html.

  • [7]

    Franco N. and González-Meneses J., Conjugacy problem for braid groups and Garside groups, J. Algebra 266 (2003), no. 1, 112–132. Google Scholar

  • [8]

    Garber D., Kahrobaei D. and Lam H. T., Length-based attacks in polycyclic groups, J. Math. Cryptol. 9 (2015), no. 1, 33–43. Google Scholar

  • [9]

    Garber D., Kaplan S., Teicher M., Tsaban B. and Vishne U., Probabilistic solutions of equations in the braid group, Adv. Appl. Math. 35 (2005), 323–334. Google Scholar

  • [10]

    Garber D., Kaplan S., Teicher M., Tsaban B. and Vishne U., Length-based conjugacy search in the braid group, Algebraic Methods in Cryptography, Contemp. Math. 418, American Mathematical Society, Providence (2006), 75–88. Google Scholar

  • [11]

    Garside F. A., The braid group and other groups, Quart. J. Math. Oxford 20 (1969), 235–254. Google Scholar

  • [12]

    Gebhardt V., Efficient collection in infinite polycyclic groups, J. Symbolic Comput. 34 (2002), 213–228. Google Scholar

  • [13]

    Goldberg D. E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, 1989. Google Scholar

  • [14]

    Holt D. F., Eick B. and O’Brien E. A., Handbook of Computational Group Theory, Chapman & Hall/CRC, Chapman, 2005. Google Scholar

  • [15]

    Hughes J. and Tannenbaum A., Length-based attacks for certain group based encryption rewriting systems, preprint 2003, http://arxiv.org/abs/cs/0306032.

  • [16]

    Kahrobaei D. and Lam H. T., Heisenberg groups as platform for the AAG key-exchange protocol, Proceedings of the 22nd International Conference on Network Protocols (ICNP), IEEE Press, Piscataway (2014), 660–664. Google Scholar

  • [17]

    Ko K., Lee S., Cheon J., Han J., Kang J. and Park C., New public-key cryptosystem using braid groups, CRYPTO 2000, Lecture Notes in Comput. Sci. 1880, Springer Berlin (2000), 166–183. Google Scholar

  • [18]

    Kotov M. and Ushakov A., Analysis of a certain polycyclic group-based cryptosystem, J. Math. Cryptol. 9 (2015), 161–167. Google Scholar

  • [19]

    Myasnikov A. D. and Ushakov A., Length based attack and braid groups: Cryptanalysis of Anshel–Anshel–Goldfeld key exchange protocol, Public Key Cryptography, Lecture Notes in Comput. Sci. 4450, Springer, Berlin (2007), 76–88. Google Scholar

  • [20]

    Myasnikov A. G. and Ushakov A., Random subgroups and analysis of the length-based and quotient attacks, J. Math. Cryptol. 2 (2008), no. 1, 29–61. Google Scholar

  • [21]

    Nikolaev A. and Blaney K. R., A PTIME solution to the restricted conjugacy problem in generalized heisenberg groups, Groups Complex. Cryptol. 8 (2016), no. 1, 69–74. Google Scholar

  • [22]

    Ruinskiy D., Shamir A. and Tsaban B., Length-based cryptanalysis: The case of Thompson’s group, J. Math. Crypt. 1 (2007), 359–372. Google Scholar

  • [23]

    Sudholt D., Parallel evolutionary algorithms, Handbook of Computational Intelligence, Springer, Berlin (2015), 929–959. Google Scholar

  • [24]

    The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.7, 2015, http://www.gap-system.org.

About the article

Received: 2016-03-14

Published Online: 2016-10-11

Published in Print: 2016-11-01


Citation Information: Groups Complexity Cryptology, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2016-0012.

Export Citation

© 2016 by De Gruyter. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in