Non Supersingular Elliptic Curves for Public Key Cryptosystems

  • T. Beth
  • F. Schaefer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 547)


For public key cryptosystems multiplication on elliptic curves can be used instead of exponentiation in finite fields. One attack to such a system is: embedding the elliptic curve group into the multiplicative group of a finite field via weilpairing; calculating the discrete logarithm on the curve by solving the discrete logarithm in the finite field. This attack can be avoided by constructing curves so that every embedding in a multiplicative group of a finite field requires a field of very large size.


Elliptic Curve Finite Field Elliptic Curf Multiplicative Group Discrete Logarithm 
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]
    T. Beth, D. Gollmann; Algorithm Engineering for Public Key Algorithms; IEEE Journal on Selected Areas in Comm., Vol. 7, No. 4, 1989, pp. 458–466.CrossRefGoogle Scholar
  2. [2]
    T. Beth, W. Geiselmann, F. Schaefer; Arithmetics on Elliptic Curves; Algebraic and Combinatorical Coding Theory, 2nd int. workshop, Leningrad, 1990, pp. 28–33.Google Scholar
  3. [3]
    D. Coppersmith; Fast evaluation of logarithms in fields of characteristic two; IEEE Trans. Inform. Theory, IT 30, 1984, pp. 587–594.CrossRefMathSciNetGoogle Scholar
  4. [4]
    M. Deuring; Die Typen der Multiplikatorenringe elliptischer Funktionenkoerper; Abh. Math. Sem. Hamburg, Bd. 14, 1941, pp. 197–272.CrossRefMathSciNetGoogle Scholar
  5. [5]
    W. Diffie, M. Hellman; New directions in cryptography; IEEE Trans. Inform. Theory, IT 22, 1976, pp. 644–654.CrossRefMathSciNetGoogle Scholar
  6. [6]
    T. ElGamal; A public key cryptosystem and a signature scheme based on discrete logarithms; IEEE Trans. Inform. Theory, IT 31, 1985, pp. 469–472.CrossRefMathSciNetGoogle Scholar
  7. [7]
    N. Koblitz; Elliptic Curve Cryptosystems; Mathematics of Computation, Vol. 48, No177, 1987, pp. 203–209.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Menezes, S. A. Vanstone; The Implementation fo Elliptic Curve Cryptosystems; Advances in Cryptology-Auscrypt 90, Springer LNCS 453, 1990, pp. 2–13.CrossRefGoogle Scholar
  9. [9]
    A. Menezes, T. Okamoto, S. A. Vanstone; Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field; University of Waterloo, preliminary version, sep. 1990.Google Scholar
  10. [10]
    V. S. Miller; Use of Elliptic Curves in Cryptography; Advances in Cryptology: Proceedings of Crypto 85, Springer LNCS 218, 1986, pp. 417–426.Google Scholar
  11. [11]
    P. Montgomery; Speeding the Pollard and elliptic curve methods of factorization; Math. Comp., Vol. 48, 1977, pp 243–264.CrossRefGoogle Scholar
  12. [12]
    J. H. Silverman; The Arithmetic of Elliptic Curves; Springer, New York, 1986.zbMATHGoogle Scholar
  13. [13]
    J. T. Tate; The Arithmetic of Elliptic Curves; Inventiones math. 23, Springer, 1974, pp. 179–206.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    W. C. Waterhouse, Abelian Varieties over finite fields; Ann. scient. Ec. Norm. Sup., 4th serie, 1969, pp. 521–560.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • T. Beth
    • 1
  • F. Schaefer
    • 1
  1. 1.Institut für Algorithmen und Kognitive SystemeUniversität KarlsruheKarlruhe 1

Personalised recommendations