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

Matthew J. Craven 1  and Daniel Robertz 2
  • 1 Centre for Mathematical Sciences, Plymouth University, Drake Circus, Plymouth, PL4 8AA, United Kingdom of Great Britain and Northern Ireland
  • 2 Centre for Mathematical Sciences, Plymouth University, Drake Circus, Plymouth, PL4 8AA, United Kingdom of Great Britain and Northern Ireland
Matthew J. Craven and Daniel Robertz

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.

  • [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.

  • [2]

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

    • Crossref
    • Export Citation
  • [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.

  • [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.

    • Crossref
    • Export Citation
  • [8]

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

  • [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.

    • Crossref
    • Export Citation
  • [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.

  • [11]

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

    • Crossref
    • Export Citation
  • [12]

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

    • Crossref
    • Export Citation
  • [13]

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

  • [14]

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

  • [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.

  • [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.

  • [18]

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

  • [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.

  • [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.

  • [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.

  • [22]

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

  • [23]

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

  • [24]

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

Purchase article
Get instant unlimited access to the article.
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Groups – Complexity – Cryptology is a journal for speedy publication of articles in the areas of combinatorial and computational group theory, computer algebra, complexity theory, and cryptology. GCC primarily publishes research papers, but comprehensive and timely survey articles on a topic inside the scope of the journal are also welcome.

Search