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

Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Elder, Murray / 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 2017: 0.32

SCImago Journal Rank (SJR) 2017: 0.208
Source Normalized Impact per Paper (SNIP) 2017: 0.322

Mathematical Citation Quotient (MCQ) 2017: 0.32

See all formats and pricing
More options …

Public-key cryptosystem based on invariants of diagonalizable groups

František Marko / Alexandr N. Zubkov
  • Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Science (SORAN), Omsk, Pevtzova 13, 644043, Russia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Martin Juráš
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/gcc-2017-0003


We develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

Keywords: Cryptosystem; invariants; diagonalizable group; number field; supergroup

MSC 2010: 94A60; 11T71


  • [1]

    S. Agarwal and G. S. Frandsen, Binary GCD like algorithms for some complex quadratic rings, Algorithmic Number Theory, Lecture Notes in Comput. Sci. 3076, Springer, Berlin (2004), 57–71. Google Scholar

  • [2]

    L. Babai, Graph isomorphism in quasipolynomial time, preprint (2015), https://arxiv.org/abs/1512.03547.

  • [3]

    Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York, 1966. Google Scholar

  • [4]

    J. Buchsmann, M. J. Jacobson and E. Teske, On some computational problems in finite abelian groups, Math. Comp. 66 (1997), no. 220, 1663–1687. Google Scholar

  • [5]

    W. Burnside, On groups of linear substitutions of finite order which possess quadratic invariants, Proc. Lond. Math. Soc. (2) 12 (1913), no. 1, 89–93. Google Scholar

  • [6]

    H. Cohen, A course in Computational Algebraic Number Theory, Grad. Texts in Math. 138, Springer, Berlin, 1993. Google Scholar

  • [7]

    M. Conforti, G. Cornuéjols and G. Zambelli, Integer Programming, Grad. Texts in Math. 271, Springer, Cham, 2014. Google Scholar

  • [8]

    T. ElGamal, On computing logarithm over finite fields, Advances in Cryptology – CRYPTO 85 (Santa Barbara 1985), Lecture Notes in Comput. Sci. 218, Springer, Berlin (1985), 396–402. Google Scholar

  • [9]

    T. ElGamal, Public-key cryptosystem and a signature scheme based on discrete logarithms, IEEE Trans. Inform. Theory 31 (1985), no. 4, 469–472. CrossrefGoogle Scholar

  • [10]

    F. Gologlu, R. Granger, G. McGuire and J. Zumbragel, Discrete logarithms in GF(26120), NMBRTHRY List (2013). Google Scholar

  • [11]

    D. Grigoriev, Public-key cryptography and invariant theory, J. Math. Sci. (N. Y.) 126 (2005), no. 3, 1152–1157. Google Scholar

  • [12]

    D. Grigoriev, A. Kojevnikov and S. J. Nikolenko, Algebraic cryptography: New constructions and their security against provable break, St. Peterburg Math. J. 20 (2009), no. 6, 937–953. Google Scholar

  • [13]

    W. C. Huffman, Polynomial invariants of finite linear groups of degree two, Canad. J. Math. 32 (1980), no. 2, 317–330. Google Scholar

  • [14]

    J. E. Humphreys, Linear Algebraic Groups, Grad. Texts in Math. 21, Springer, New York, 1975. Google Scholar

  • [15]

    A. Joux, Discrete Logarithms in GF(24080), NMBRTHRY List (2013). Google Scholar

  • [16]

    E. Kaltofen and H. Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Math. Comp. 53 (1989), no. 188, 697–720. Google Scholar

  • [17]

    M. Laššák and Š. Porubský, Fermat–Euler theorem in algebraic number fields, J. Number Theory 60 (1996), 254–290. CrossrefGoogle Scholar

  • [18]

    F. Marko and A. N. Zubkov, Minimal degrees of invariants of (super)groups – A connection to cryptology, Linear Multilinear Algebra (2016), 10.1080/03081087.2016.1273876. Google Scholar

  • [19]

    B. McDonald, Finite Rings with Identity, Pure Appl. Math. 28, Marcel Dekker, New York, 1974. Google Scholar

  • [20]

    N. Nakagoshi, The structure of the multiplicative group of residue classes modulo 𝔭N+1, Nagoya Math. J. 73 (1979), 41–60. Google Scholar

  • [21]

    E. Noether, Der Endlichkeitssatz der invarianten endlicher Gruppen, Math. Ann. 77 (1916), 89–92. Google Scholar

  • [22]

    L. Smith, Polynomial invariants of finite groups – A survey of recent results, Bull. Amer. Math. Soc. 34 (1997), no. 3, 211–250. CrossrefGoogle Scholar

  • [23]

    A. V. Sutherland, Structure computation and discrete logarithms in finite abelian p-groups, Math. Comp. 80 (2011), no. 273, 477–500. Google Scholar

  • [24]

    P. Symonds, On the Castelnuovo–Mumford regularity of rings of polynomial invariants, Ann. of Math. (2) 174 (2011), no. 1, 499–517. Web of ScienceGoogle Scholar

  • [25]

    O. N. Vasilenko, Number-Theoretic Algorithms in Cryptography, Transl. Math. Monogr. 232, American Mathematical Society, Providence, 2007. Google Scholar

About the article

Received: 2016-09-08

Published Online: 2017-04-19

Published in Print: 2017-05-01

This publication was made possible by an NPRF award NPRP 6-1059-1-208 from the Qatar National Research Fund (a member of The Qatar Foundation). The statements made herein are solely the responsibility of the authors.

Citation Information: Groups Complexity Cryptology, Volume 9, Issue 1, Pages 31–54, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2017-0003.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in