Journal of Cryptology

, Volume 1, Issue 3, pp 139–150 | Cite as

Hyperelliptic cryptosystems

  • Neal Koblitz


In this paper we discuss a source of finite abelian groups suitable for cryptosystems based on the presumed intractability of the discrete logarithm problem for these groups. They are the jacobians of hyperelliptic curves defined over finite fields. Special attention is given to curves defined over the field of two elements. Explicit formulas and examples are given, and the problem of finding groups of almost prime order is discussed.

Key words

Cryptosystem Public key Discrete logarithm Hyperelliptic curve Jacobian 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr.,Factorization of b n±1, b=2, 3, 5, 6, 7, 10, 11, 12 up to High Powers, American Mathematical Society, Providence, RI, 1983.Google Scholar
  2. [2]
    D. Cantor, Computing in the jacobian of a hyperelliptic curve,Math. Comp.,48 (1987), 95–101.Google Scholar
  3. [3]
    W. Diffie and M. Hellman, New directions in cryptography,IEEE Trans. Inform. Theory,22 (1976), 644–654.Google Scholar
  4. [4]
    T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms,IEEE Trans. Inform. Theory,31 (1985), 469–472.Google Scholar
  5. [5]
    W. Fulton,Algebraic Curves, Benjamin, New York, 1969.Google Scholar
  6. [6]
    N. Koblitz,Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1984.Google Scholar
  7. [7]
    N. Koblitz,A Course in Number Theory and Cryptography, Springer-Verlag, New York, 1987.Google Scholar
  8. [8]
    N. Koblitz, Elliptic curve cryptosystems,Math. Comp. 48 (1987), 203–209.Google Scholar
  9. [9]
    N. Koblitz, Primality of the number of points on an elliptic curve over a finite field,Pacific J. Math.,131 (1988), 157–165.Google Scholar
  10. [10]
    S. Lang,Introduction to Algebraic Geometry, Interscience, New York, 1958.Google Scholar
  11. [11]
    R. Lidl and H. Niederreiter,Finite Fields, Addison-Wesley, Reading, MA, 1983.Google Scholar
  12. [12]
    V. Miller, Use of elliptic curves in cryptography,Advances in Cryptology-Crypto '85, Springer-Verlag, New York, 1986, pp. 417–426.Google Scholar
  13. [13]
    A. M. Odlyzko, Discrete logarithms and their cryptographic significance,Advances in Cryptography: Proceedings of Eurocrypt 84, Springer-Verlag, New York, 1985, pp. 224–314.Google Scholar
  14. [14]
    E. Seah and H. C. Williams, Some primes of the form (a n − 1)/(a − 1),Math. Comp.,33 (1979), 1337–1342.Google Scholar
  15. [15]
    D. Shanks,Solved and Unsolved Problems in Number Theory, 3rd edn., Chelsea, New York, 1985.Google Scholar
  16. [16]
    W. C. Waterhouse, Abelian varieties over finite fields,Ann. Sci. École Norm. Sup. (4),2 (1969), 521–560.Google Scholar

Copyright information

© International Association for Cryptologic Research 1988

Authors and Affiliations

  • Neal Koblitz
    • 1
  1. 1.Department of Mathematics GN-50University of WashingtonSeattleU.S.A.

Personalised recommendations