Skip to main content
Log in

Sharpening of the Explicit Lower Bounds for the Order of Elements in Finite Field Extensions Based on Cyclotomic Polynomials

  • Published:
Ukrainian Mathematical Journal Aims and scope

We explicitly construct elements of high multiplicative order in any extensions of finite fields based on cyclotomic polynomials.

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

Access this article

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. R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Press, Cambridge (1997).

    Google Scholar 

  2. G. L. Mullen and D. Panario, Handbook of Finite Fields, CRC Press (2013).

  3. S. Gao, “Elements of provable high orders in finite fields,” Proc. Amer. Math. Soc., 127, No. 6, 1615–1623 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  4. J. F. Voloch, “On the order of points on curves over finite fields,” Integers, 7, 9 (2007).

    MathSciNet  Google Scholar 

  5. J. F. Voloch, “Elements of high order on finite fields from elliptic curves,” Bull. Austral. Math. Soc., 81, No. 3, 425–429 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  6. Q. Cheng, “On the construction of finite field elements of large order,” Finite Fields Appl., 11, No. 3, 358–366 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Popovych, “Elements of high order in finite fields of the form F q[x]/(x m-a),” Finite Fields Appl., 19, No. 1, 86–92 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Popovych, “Elements of high order in Artin–Schreier extensions of finite fields,” Mat. Stud., 39, No. 2, 115–118 (2013).

    MathSciNet  Google Scholar 

  9. Q. Cheng, S. Gao, and D. Wan, “Constructing high order elements through subspace polynomials,” in: Proc. 23rd ACM-SIAM Symp. Discrete Algorithms (Kyoto, Japan, 17–19 January 2012), Philadelphia, USA (2011), pp. 1457–1463.

  10. J. Gathen and I. E. Shparlinski, “Orders of Gauss periods in finite fields,” Appl. Algebra Eng. Comm. Comput., 9, No. 1, 15–24 (1998).

    Article  MATH  Google Scholar 

  11. O. Ahmadi, I. E. Shparlinski, and J. F. Voloch, “Multiplicative order of Gauss periods,” Int. J. Number Theory, 6, No. 4, 877–882 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Popovych, “Elements of high order in finite fields of the form F q [x]/Φ r (x),” Finite Fields Appl., 18, No. 4, 700–710 (2012).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 6, pp. 815–825, June, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Popovych, R. Sharpening of the Explicit Lower Bounds for the Order of Elements in Finite Field Extensions Based on Cyclotomic Polynomials. Ukr Math J 66, 916–927 (2014). https://doi.org/10.1007/s11253-014-0981-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0981-0

Keywords

Navigation