Skip to main content
SpringerLink
Log in

Primitive elements with prescribed trace

  • Original Paper
  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Let \(q\) be a power of a prime number \(p\). Let \(n\) be a positive integer. Let \(\mathbb {F}_{q^n}\) denote a finite field with \(q^n\) elements. In this paper, we consider the existence of the some specific elements in the finite field \(\mathbb {F}_{q^n}\). We get that when \(n\ge 29\), there are elements \(\xi \in \mathbb {F}_{q^n}\) such that \(\xi +\xi ^{-1}\) is a primitive element of \(\mathbb {F}_{q^n}\), and \(\mathrm{Tr}(\xi ) = a, \mathrm{Tr}(\xi ^{-1}) = b\) for any pair of prescribed \(a, b \in \mathbb {F}_q^*\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cochrane, T., Pinner, C.: Using Stepanov’s method for exponential sums involving rational functions. J. Number Theory 116, 270–292 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cohen, S.D.: In: Mullen, G.L., Shiue, P.J. (eds.) Primitive Elements and Polynomials: Existence Results, Lecture Notes in Pure and Appl. Math., vol. 141, pp. 43–45. Marcel Dekker, New York (1992)

  3. Cohen, S.D.: Primitive elements and polynomials with arbitrary trace. Discret. Math. 2, 1–7 (1990)

    Article  Google Scholar 

  4. Cohen, S.D.: Primitive polynomials with a prescribed coefficient. Finite Fields Their Appl. 12, 425–491 (2006)

    Article  MATH  Google Scholar 

  5. Fan, S., Han, W.: Primitive polynomials with three coefficients prescribed. Finite Fields Their Appl. 10, 506–521 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan, S., Han, W.: Primitive polynomials over finite fields of characteristic two. Appl. Alegbra Eng. Commun. Comput. 14, 381–395 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gao, S., von zur Gathen, J., Panario, D.: Gauss periods, primitive normal bases, and fast exponentiation in finite fields, preliminary version. In: Proc. Latin’95, Valparaiso, Chile, Springer, Lecture Notes on Computer Science, vol. 911, pp. 311–322 (1995); full version in Technical Report 296/95, Department of Computer Science, University of Toronto (1995)

  8. Gao, S., Vanstone, S.A.: On orders of optimal normal basis generators. Math. Comput. 64, 1227–1233 (1995). (MR 95j:11117)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gao, S., Von Zur Gathen, J., Panario, D.: Gauss periods: orders and cryptographical applications. Math. Comput. 67(221), 343–352 (1998)

    Article  MATH  Google Scholar 

  10. Jungnickel, D., Vanstone, S.A.: On primitive polynomials over finite fields. J. Algebra 124, 337–353 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lenstra, H.W. Jr., Schoof, R.J.: Primitive normal bases for finite fields. Math. Comput. 48, 217–231 (1987)

  12. Lidl, R., Niederreiter, H.: Finite Fields, Reading. Addison-Wesley, MA (1983)

    Google Scholar 

  13. Moreno, O.: On primitive elements of trace equal to 1 in \(GF(2^{m})^{*}\). Disctet. Math. 41, 53–56 (1982)

    Article  MATH  Google Scholar 

  14. Wassermann, A.: Zur Arithmetik in endlichen Korpern. Bayreuther Mathematische Schriften 44, 147–251 (1993). (MR 94g: 11114)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their grateful thankfulness to the referees for their valuable comments and suggestions which greatly improved the quality of the earlier version of this paper. The research of the work was supported by the NSF of China (No. 11371011).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Xiwang Cao & Peipei Wang

Authors
  1. Xiwang Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peipei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiwang Cao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, X., Wang, P. Primitive elements with prescribed trace. AAECC 25, 339–345 (2014). https://doi.org/10.1007/s00200-014-0228-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-014-0228-1

Keywords

Search

Navigation