On Families of Hash Functions via Geometric Codes and Concatenation

  • Jürgen Bierbrauer
  • Thomas Johansson
  • Gregory Kabatianskii
  • Ben Smeets
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 773)


In this paper we use coding theory to give simple explanations of some recent results on universal hashing. We first apply our approach to give a precise and elegant analysis of the Wegman-Carter construction for authentication codes. Using Reed-Solomon codes and the well known concept of concatenated codes we can then give some new constructions, which require much less key size than previously known constructions. The relation to coding theory allows the use of codes from algebraic curves for the construction of hash functions. Particularly, we show how codes derived from Artin-Schreier curves, Hermitian curves and Suzuki curves yield good classes of universal hash functions.


Hash Function Orthogonal Array Linear Code Algebraic Curf Authentication Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    J.L. Carter, M.N. Wegman, “Universal Classes of Hash Functions”, J. Computer and System Sci., Vol. 18, 1979, pp. 143–154.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    D.R. Stinson, Combinatorial techniques for universal hashing, University of Nebraska-Lincoln. Department of Computer Science and Engineering, 1990.Google Scholar
  3. 3.
    G.J. Simmons, “A survey of Information Authentication”, in Contemporary Cryptology, The science of information integrity, ed. G.J. Simmons, IEEE Press, New York, 1992.Google Scholar
  4. 4.
    M.N. Wegman, J.L. Carter, “New hash functions and their use in authentication and set equality”, J. Computer and System Sciences, Vol. 22, 1981, pp. 265–279.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    C.H. Bennett, G. Brassard, J-M. Roberts, “Privacy amplification by public discussion”, SIAM J. Comput., Vol. 17:2, 1988, pp. 210–229.CrossRefMathSciNetGoogle Scholar
  6. 6.
    D.R. Stinson, “Universal Hashing and Authentication Codes”, to appear in IEEE Transactions on Information Theory. This is a final version of [7].Google Scholar
  7. 7.
    D. R. Stinson, “Universal hashing and authentication codes” Proceedings of Crypto 91, Santa Barbara, USA, 1991, pp. 74–85.Google Scholar
  8. 8.
    T. Johansson, G. Kabatianskii, B. Smeets, “On the relation between A-codes and codes correcting independent errors” Proceedings Eurocrypt’93, to appear.Google Scholar
  9. 9.
    J. Bierbrauer, “Universal hashing and geometric codes”, manuscript.Google Scholar
  10. 10.
    G.D. Forney, Jr., Concatenated Codes, M.I.T. Press, Cambridge, MA, 1966.Google Scholar
  11. 11.
    J. Bierbrauer, “Construction of orthogonal arrays”, to appear in Journal of Statistical Planning and Inference.Google Scholar
  12. 12.
    T. Beth, D. Jungnickel, H. Lenz, Design Theory, Bibliographisches Institut, Zürich 1985.zbMATHGoogle Scholar
  13. 13.
    M.A. Tsfasman, S.G. VlĂduţ, Algebraic-Geometric codes, Kluwer Academic Publ., Dordrecht/Boston/London, 1991.zbMATHGoogle Scholar
  14. 14.
    R. Pellikaan, B.Z. Shen, and G.J.M. van Wee, “Which linear codes are algebraic-geometric?”, IEEE Trans. Information Theory, Vol. 37, 1991, pp. 583–602.CrossRefGoogle Scholar
  15. 15.
    B.H. Matzat, “Kanonische Codes auf einigen Überdeckungskurven”, Manuscripta Mathematica, Vol. 77, 1992, pp. 321–335.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    L. Gargano, J. Körner, U. Vaccaro, “Sperner capacities”, to appear in Graphs and Combinatorics.Google Scholar
  17. 17.
    L. Gargano, J. Körner, U. Vaccaro, “Capacities: from information theory to extremal set theory”, to appear in Journal of the AMS.Google Scholar
  18. 18.
    J.P. Hansen, H. Stichtenoth, “Group Codes on Certain algebraic curves with many rational points”, AAECC, Vol. 1, 1990, pp. 67–77.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    A. Garcia, S.J. Kim, R.F. Lax, “Consecutive Weierstrass gaps and minimum distance of Goppa codes”, Journal of Pure and Applied Algebra, Vol. 84, 1993, pp. 199–207.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    G. Kabatianskii, B. Smeets, T. Johansson, “Bounds on the size of a-codes and families of hash functions via coding theory”, manuscript.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Jürgen Bierbrauer
    • 1
  • Thomas Johansson
    • 2
  • Gregory Kabatianskii
    • 3
  • Ben Smeets
    • 2
  1. 1.Mathematisches Institut der UniversitätHeidelbergGermany
  2. 2.Dept. of Information TheoryUniversity of LundLundSweden
  3. 3.Inst. for Problems of Information TransmissionRussian Academy of SciencesMoscowRussia

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