Skip to main content

On Gaussian sums for finite fields and elliptic curves

  • Conference paper
  • First Online:
Algebraic Coding (Algebraic Coding 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 573))

Included in the following conference series:

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BACH E. Number-theoretic algorithms. Annual Review of Comp. Sci., 1990, v. 4, p. 119–172.

    Article  Google Scholar 

  2. BARG F.V., KATSMAN G.S., TSFASMAN M.A. Algebraic-geometric codes over curves of small genus. Problemy Peredachi Inform., 1987, v.23, no. 1, p.42–46 (in Russian).

    Google Scholar 

  3. BOMBIERI E. On exponential sums in finite fields. Amer. J. Math., 1966, v.88, p.71–105.

    Google Scholar 

  4. DRIENCOURT Y. Some properties of elliptic codes over a field of characteristic 2. Lecture Notes in Comp. Sci., 1985, v. 229.

    Google Scholar 

  5. DRIENCOURT Y., MICHON J. F. Elliptic codes over fields of characteristic 2. J. Pure and Appl. Algebra, 1987, v.45, p. 15–39.

    Article  Google Scholar 

  6. GARCIA A., VOLOCH J.F. Fermat curves over finite fields. J. Number Theory. 1988, v.30, p.345–356.

    Article  Google Scholar 

  7. GUPTA R. Division fields of Y2 = X3−aX. J. Number Theory, 1990, v.34, no.3, p.335–345.

    Article  Google Scholar 

  8. GUPTA R., MURTY M.R.P. Primitive points on elliptic curves. Compos. Math., 1986, v.58, no. 1, p.13–44.

    Google Scholar 

  9. GUPTA R., MURTY M.R.P. Cyclicity and generation of points mod p on elliptic curves. Invent. Math., 1990, v.111, no. 1, p.225–235.

    Article  Google Scholar 

  10. KALINSKI B.S. A pseudo-random bit generator based on elliptic logarithm. Lect. Notes in Comp. Sci., 1987, v.263, p.84–103.

    Google Scholar 

  11. KATSMAN G.L., TSFASMAN M.A. Spectrums of algebraicgeometric codes. Problemy Peredachi Inform. 1987, v.23, no. 4, p.19–34 (in Russian).

    Google Scholar 

  12. KOBLITZ N. Introduction to elliptic curves and modular forms. Springer-Verlag, 1984.

    Google Scholar 

  13. KOBLITZ N. Elliptic curve cryptosystem. Math. Comp., 1987, v.48, no.177, p.203–209.

    Google Scholar 

  14. KOBLITZ N. A course in number theory and cryptography. Springer-Verlag, 1987.

    Google Scholar 

  15. KOROBOV N.M. On the distribution of digits in periodic fractions. Matem. Sbornik, 1972, v.89, no. 4, p.654–670 (in Russian).

    Google Scholar 

  16. LANG S. Elliptic curves: Diophantine analysis. Springer-Verlag, 1978.

    Google Scholar 

  17. LANG S., TROTTER H. Primitive points on elliptic curves. Bull. Amer. Math. Soc., 1977, v.83, p.289–292.

    Google Scholar 

  18. LIDL R., NIEDERREITER H. Finite fields. Addison-Wesley, 1983.

    Google Scholar 

  19. MIYAMOTO I., MURTY M. R. P. Elliptic pseudoprimes. Math. Comp., 1989, v.53, no.187, p.415–430.

    Google Scholar 

  20. NIEDERREITER H. Quasy-Monte Carlo methods and pseudorandom numbers. Bull. Amer. Math. Soc., 1978, v.84, p.957–1041.

    Google Scholar 

  21. NIEDERREITER H. On a problem of Kodama concerning the Hasse-Witt matrix and distribution of residues. Proc. Japan Acad., Ser. A, 1987, v.63, no. 9, p.367–369.

    Google Scholar 

  22. SCHOOF R. J. Elliptic curves over finite fields and the computation of square roots mod p. Math. of Comp., 1985, v.44, no. 170, p.483–494.

    Google Scholar 

  23. SHANKS D. Five number theoretic algorithms. Proc. 2 Manitoba Conf. on Numerical Math., Univ. Manitoba, 1972, p. 51–70.

    Google Scholar 

  24. SHPARLINSKI I. E. On some properties of linear cyclic codes. Problemy Peredachi Inform., 1983, v.19, no. 3, p.106–110 (in Russian).

    Google Scholar 

  25. SHPARLINSKI I. E. On weigth enumerators of some codes. Problemy Peredachi Inform., 1986, v.22, no. 2, p.43–48 (in Russian).

    Google Scholar 

  26. SHPARLINSKI I. E. On primitive elements in finite fields and on elliptic curves. Matem. Sbornik, 1990, v.181, no. 9, p.1196–1206 (in Russian).

    Google Scholar 

  27. SHPARLINSKI I. E. On some problems of theory of finite fields. Uspechi Matem. Nauk, 1991, v.46, no. 1, p.165–200 (in Russian)

    Google Scholar 

  28. SHPARLINSKI I. E. On bounds of Gaussian sums. Matem. Zametki, 1991, (to appear).

    Google Scholar 

  29. SILVERMAN J. H. The arithmetic of elliptic curves. Springer-Verlag, 1984.

    Google Scholar 

  30. STECHKIN S. B. A bound of sums of Gauss. Matem. Zametki, 1975, v.17, no. 4. p.579–588 (in Russian).

    Google Scholar 

  31. TATE J. The arithmetic of elriptic curves. Invent. Math., 1974, v.23, p.179–206.

    Article  Google Scholar 

Download references

Authors

Editor information

Gérard Cohen Antoine Lobstein Gilles Zémor Simon Litsyn

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Shparlinski, I.E. (1992). On Gaussian sums for finite fields and elliptic curves. In: Cohen, G., Lobstein, A., Zémor, G., Litsyn, S. (eds) Algebraic Coding. Algebraic Coding 1991. Lecture Notes in Computer Science, vol 573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0034335

Download citation

  • DOI: https://doi.org/10.1007/BFb0034335

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55130-0

  • Online ISBN: 978-3-540-46739-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics