Elliptic Curve Public-Key Cryptosystems — An Introduction

  • Erik De Win
  • Bart Preneel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1528)


We give a brief introduction to elliptic curve public-key cryptosystems. We explain how the discrete logarithm in an elliptic curve group can be used to construct cryptosystems. We also focus on practical aspects such as implementation, standardization and intellectual property.


Group Operation Elliptic Curve Signature Scheme Elliptic Curf Discrete Logarithm Problem 
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.


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  1. 1.
    ANSI X9.62-199x: Public-Key Cryptography for the Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA), November 11, 1997.Google Scholar
  2. 2.
    A. Atkin and F. Morain, “Elliptic curves and primality proving,” Mathematics of Computation, Vol. 61 (1993), pp. 29–68.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Bailey and C. Paar, “Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms,” Advances in Cryptology, Proc. Crypto’98, LNCS 1462, H. Krawczyk, Ed., Springer-Verlag, 1998, pp. 472–485.CrossRefGoogle Scholar
  4. 4.
    E. De Win, A. Bosselaers, S. Vandenberghe, P. De Gersem and J. Vandewalle, “A fast software implementation for arithmetic operations in GF(2n),” Advances in Cryptology, Proc. Asiacrypt’96, LNCS 1163, K. Kim and T. Matsumoto, Eds., Springer-Verlag, 1996, pp. 65–76.CrossRefGoogle Scholar
  5. 5.
    E. De Win, S. Mister, B. Preneel and M. Wiener, “On the performance of signature schemes based on elliptic curves,” Proceedings of the ANTS III conference, LNCS 1423, J. Buhler, Ed., Springer-Verlag, 1998, pp. 252–266.Google Scholar
  6. 6.
    D.M. Gordon, “A survey of fast exponentiation methods,” Journal of Algorithms, Vol. 27, pp. 129–146, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Guajardo and C. Paar, “Efficient algorithms for elliptic curve cryptosystems,” Advances in Cryptology, Proc. Crypto’97, LNCS 1294, B. Kaliski, Ed., Springer-Verlag, 1997, pp. 342–356.CrossRefGoogle Scholar
  8. 8.
    G. Harper, A. Menezes and S. Vanstone, “Public-key cryptosystems with very small key length, ”Advances in Cryptology, Proc. Eurocrypt’92, LNCS 658, R.A. Rueppel, Ed., Springer-Verlag, 1993, pp. 163–173.Google Scholar
  9. 9.
    IEEE P1363: Editorial Contribution to Standard for Public-Key Cryptography, July 27, 1998.Google Scholar
  10. 10.
    D. Knuth, The Art of Computer Programming, Vol. 2, Semi-Numerical Algorithms, 2nd Edition, Addison-Wesley, Reading, Mass., 1981.Google Scholar
  11. 11.
    ISO/IEC 9796-2, “Information technology — Security techniques — Digital signature schemes giving message recovery, Part 2: Mechanisms using a hash-function,” 1997.Google Scholar
  12. 12.
    N. Koblitz,“Elliptic curve cryptosystems,” Mathematics of Computation, Vol. 48, no. 177 (1987), pp. 203–209.Google Scholar
  13. 13.
    H. W. Lenstra Jr., “Factoring integers with elliptic curves,” Annals of Mathematics, Vol. 126 (1987), pp. 649–673.CrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Menezes, Elliptic Curve Public-Key Cryptosystems, Kluwer Academic Publishers, 1993.Google Scholar
  15. 15.
    A. Menezes, T. Okamoto and S. Vanstone, “Reducing elliptic curve logarithms to logarithms in a finite field,” IEEE Transactions on Information Theory, Vol. 39 (1993), pp. 1639–1646.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Menezes, P. van Oorschot and S. Vanstone, Handbook of Applied Cryptography, CRC Press, 1997.Google Scholar
  17. 17.
    V.S. Miller, “Use of elliptic curves in cryptography,” Advances in Cryptology Proc. Crypto’85, LNCS 218, H.C. Williams, Ed., Springer-Verlag, 1985, pp. 417–426.CrossRefGoogle Scholar
  18. 18.
    A. Miyaji, T. Ono and H. Cohen, “Efficient elliptic curve exponentiation,” Proceedings of ICICS’97, LNCS 1334, Y. Han, T. Okamoto and S. Qing, Eds., Springer-Verlag, 1997, pp. 282–290.Google Scholar
  19. 19.
    R. Mullin, I. Onyszchuk, S. Vanstone and R. Wilson, “Optimal normal bases in GF(pn),” Discrete Applied Mathematics, Vol. 22 (1988/1989), pp. 149–161.CrossRefMathSciNetGoogle Scholar
  20. 20.
  21. 21.
    T. Satoh and K. Araki, “Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves,” Commentarii Math. Univ. St. Pauli, Vol. 47 (1998), pp. 81–92.zbMATHMathSciNetGoogle Scholar
  22. 22.
    N. Smart, “The discrete logarithm problem on elliptic curves of trace one,” preprint, 1998.Google Scholar
  23. 23.
    R. Schroeppel, H. Orman, S. O’Malley and O. Spatscheck, “Fast key exchange with elliptic curve systems,” Advances in Cryptology, Proc. Crypto’95, LNCS 963, D. Coppersmith, Ed., Springer-Verlag, 1995, pp. 43–56.Google Scholar
  24. 24.
    M. Wiener, “Performance comparison of public-key cryptosystems,” CryptoBytes, Vol. 4., No. 1, Summer 1998, pp. 1–5.Google Scholar
  25. 25.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Erik De Win
    • 1
  • Bart Preneel
    • 1
  1. 1.Dept. Electrical Engineering-ESATKatholieke Universiteit LeuvenHeverleeBelgium

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