The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in \({{\mathsf {T}}}{{\mathsf {C}}}^0\)

  • Alexei Miasnikov
  • Svetla Vassileva
  • Armin Weiß
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10304)


We show that the conjugacy problem in a wreath product \(A \wr B\) is uniform-\({{\mathsf {T}}}{{\mathsf {C}}}^0\)-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. Moreover, if B is torsion free, the power problem for B can be replaced by the slightly weaker cyclic submonoid membership problem for B, which itself turns out to be uniform-\({{\mathsf {T}}}{{\mathsf {C}}}^0\)-Turing-reducible to the conjugacy problem in \(A \wr B\) if A is non-abelian.

Furthermore, under certain natural conditions, we give a uniform \({{\mathsf {T}}}{{\mathsf {C}}}^0\) Turing reduction from the power problem in \(A \wr B\) to the power problems of A and B. Together with our first result, this yields a uniform \({{\mathsf {T}}}{{\mathsf {C}}}^0\) solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also for free solvable groups.


Wreath products Conjugacy problem Word problem \({{\mathsf {T}}}{{\mathsf {C}}}^0\) Free solvable group 


  1. 1.
    Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity within NC\({^1}\). J. Comput. Syst. Sci. 41(3), 274–306 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dehn, M.: Über unendliche diskontinuierliche Gruppen. Math. Ann. 71(1), 116–144 (1911)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Diekert, V., Myasnikov, A.G., Weiß, A.: Conjugacy in Baumslag’s Group, generic case complexity, and division in power circuits. In: LATIN Symposium, pp. 1–12 (2014)Google Scholar
  4. 4.
    Grigoriev, D., Shpilrain, V.: Authentication from matrix conjugation. Groups Complex. Cryptology 1, 199–205 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gul, F., Sohrabi, M., Ushakov, A.: Magnus embedding and algorithmic properties of groups \(F/N^{(d)}\). ArXiv e-prints, abs/1501.01001, January 2015Google Scholar
  6. 6.
    Hesse, W.: Division is in uniform TC\({^0}\). In: Orejas, F., Spirakis, P.G., Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 104–114. Springer, Heidelberg (2001). doi: 10.1007/3-540-48224-5_9 CrossRefGoogle Scholar
  7. 7.
    Hesse, W., Allender, E., Barrington, D.A.M.: Uniform constant-depth threshold circuits for division and iterated multiplication. JCSS 65, 695–716 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kargapolov, M.I., Remeslennikov, V.N.: The conjugacy problem for free solvable groups. Algebra i Logika Sem. 5(6), 15–25 (1966)MathSciNetzbMATHGoogle Scholar
  9. 9.
    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). doi: 10.1007/3-540-44598-6_10 CrossRefGoogle Scholar
  10. 10.
    König, D., Lohrey, M.: Evaluating matrix circuits. CoRR, abs/1502.03540 (2015)Google Scholar
  11. 11.
    Krebs, A., Lange, K., Reifferscheid, S.: Characterizing TC\(^{0}\) in terms of infinite groups. Theory Comput. Syst. 40(4), 303–325 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lange, K.-J., McKenzie, P.: On the complexity of free monoid morphisms. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, pp. 247–256. Springer, Heidelberg (1998). doi: 10.1007/3-540-49381-6_27 CrossRefGoogle Scholar
  13. 13.
    Maciel, A., Thérien, D.: Threshold circuits of small majority-depth. Inf. Comput. 146(1), 55–83 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Magnus, W.: On a theorem of Marshall Hall. Ann. Math. 40, 764–768 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matthews, J.: The conjugacy problem in wreath products and free metabelian groups. Trans. Am. Math Soc. 121, 329–339 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Miller III, C.F.: On group-theoretic decision problems and their classification, vol. 68. Annals of Mathematics Studies. Princeton University Press (1971)Google Scholar
  17. 17.
    Myasnikov, A., Roman’kov, V., Ushakov, A., Vershik, A.: The word, geodesic problems in free solvable groups. Trans. Amer. Math. Soc. 362(9), 4655–4682 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Myasnikov, A.G., Vassileva, S., Weiß, A.: Log-space complexity of the conjugacy problem in wreath products. Groups Complex. Cryptol. (2017, to appear)Google Scholar
  19. 19.
    Miasnikov, A., Vassileva, S., Weiß, A.: The conjugacy problem in free solvable groups and wreath product of abelian groups is in TC\(^{0}\). ArXiv e-prints, abs/1612.05954 (2016)Google Scholar
  20. 20.
    Remeslennikov, V., Sokolov, V.G.: Certain properties of the Magnus embedding. Algebra i logika 9(5), 566–578 (1970)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Robinson,D.: Parallel Algorithms for Group Word Problems. PhD thesis, University of California, San Diego (1993)Google Scholar
  22. 22.
    Shpilrain, V., Zapata, G.: Combinatorial group theory and public key cryptography. Appl. Algebra Engrg. Comm. Comput. 17, 291–302 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vassileva, S.: Polynomial time conjugacy in wreath products and free solvable groups. Groups Complex. Cryptol. 3(1), 105–120 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Vollmer, H.: Introduction to Circuit Complexity. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  25. 25.
    Waack, S.: The parallel complexity of some constructions in combinatorial group theory (abstract). In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 492–498. Springer, Heidelberg (1990). doi: 10.1007/BFb0029647 CrossRefGoogle Scholar
  26. 26.
    Wang, L., Wang, L., Cao, Z., Okamoto, E., Shao, J.: New constructions of public-key encryption schemes from conjugacy search problems. In: Lai, X., Yung, M., Lin, D. (eds.) Inscrypt 2010. LNCS, vol. 6584, pp. 1–17. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-21518-6_1 CrossRefGoogle Scholar
  27. 27.
    Weiß, A.: A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups. In: Algebra and computer science, vol. 677. Contemporary Mathematics, pp. 185–212. American Mathematical Society, Providence, RI (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alexei Miasnikov
    • 1
  • Svetla Vassileva
    • 2
  • Armin Weiß
    • 1
  1. 1.Stevens Institute of TechnologyHobokenUSA
  2. 2.Champlain CollegeSt-lambertCanada

Personalised recommendations