Advertisement

Primitive Roots in Cubic Extensions of Finite Fields

  • Donald Mills
  • Gavin McNay

Abstract

N. Katz has shown that the absolute value of sums of the form \(\sum\limits_{b \in {F_q}} X \left( {\theta + b} \right)\), F q the finite field of q elements, χ a nontrivial multiplicative character of \({F_{{q^n}}}\), and θ a F q -generator of \({F_{{q^n}}}\), is bounded from above by \(\left( {n - 1} \right)\sqrt q\). We use this result in conjunction with a sieve due to S. Cohen to show the following for n = 3: For any prime power q and any F q -generator θ of \({F_{{q^n}}}\), there exists a primitive element of the form + b\({F_{{q^n}}}\) for some a, bF q , a ≠ 0. We discuss an application of these primitive sums in their use as pseudorandom vector generators, and conclude by discussing the harder problem of guaranteeing the existence of such roots when a is forced to be 1.

Keywords

Finite Field Prime Power Primitive Element Primitive Root Primitive Polynomial 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bombieri, E. Counting points on curves over finite fields (d’après S. A. Stepanov),Sèm. Bourbaki (1972/3), Exp. 430.Google Scholar
  2. 2.
    Carlitz, L. Distribution of primitive roots in a finite field, Quart. J. Math. (2), 4 (1953), 4–10.zbMATHGoogle Scholar
  3. 3.
    Chou, W.-S.; Cohen, S.D. Primitive elements with zero traces, Finite Fields Appl. 7 (2001), 125–141.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chung, F.R.K. Diameters and eigenvalues, J. Amer. Math. Soc. 2 (1989), 187–196.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cohen, S.D. Primitive roots in the quadratic extension of a finite field, J. London Math. Soc. (2), 27 (1983), 221–228.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cohen, S.D. Consecutive primitive roots in a finite field, Proc. Amer. Math. Soc. (2), 93 (1985), 189–197.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cohen, S.D. Consecutive primitive roots in a finite field II, Proc. Amer. Math. Soc. (2), 94 (1985), 605–611.zbMATHGoogle Scholar
  8. 8.
    Cohen, S.D.: Mullen, G.L. Primitive elements in finite fields and Costas arrays, App. Alg. Engr. Comm. Comp. 2 (1992), 297–299.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Davenport, H. On primitive roots in finite fields, Quart. J. Math. Oxford 8 (1937), 308–312.CrossRefGoogle Scholar
  10. 10.
    Deligne, P. La conjecture de Weil (I), Publ. Math. IHES 43 (1974), 273–307.MathSciNetGoogle Scholar
  11. 11.
    Deligne, P. La conjecture de Weil (II), Publ. Math. IHES 52 (1981), 313–428.Google Scholar
  12. 12.
    Ireland, K.; Rosen, M. A classical introduction to modern number theory, Springer-Verlag, New York, 1982.zbMATHGoogle Scholar
  13. 13.
    Jungnickel, D. Finite fields: structure and arithmetic, Wissenschaftsverlag, Mannheim, 1993.zbMATHGoogle Scholar
  14. 14.
    Katz, N.M. An estimate for character sums, J. Amer. Math. Soc. 2 (1989), 197–200.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lidl, R.; Niederreiter, H. Finite fields, Encyclopedia of Mathematics and Its Applications 20, Cambridge Univ. Press, Cambridge, 1983.Google Scholar
  16. 16.
    Mullen, G.L. A note on a finite field pseudorandom vector generator of Niki, Math. Japonica 38 (1993), 59–60.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Niederreiter, H. A pseudorandom vector generator based on finite field arithmetic, Math. Japonica 31 (1986), 759–774.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Niki, N. Generation of n-space uniform pseudorandom numbers based on computations in a finite field (Japanese), Proc. Symp. Applications of Number Theory on Numerical Analysis, Kyoto, 1984.Google Scholar
  19. Niki, N. Generation of n-space uniform pseudorandom numbers based on computations in a finite field (Japanese), Lecture Notes No. 537, 100–111, Research Inst. Math. Sciences, Kyoto, 1984.Google Scholar
  20. 19.
    Niki, N. Finite field arithmetics and multidimensional uniform pseudorandom numbers (Japanese), Proc. Inst. Statist. Math. 32 (1984), 231–239.MathSciNetzbMATHGoogle Scholar
  21. 20.
    Perel’muter, C.I.; Shparlinski, I.E. On the distribution of primitive roots in finite fields, Uspechi Matem. Nauk. 45 (1990), 185–186.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Donald Mills
    • 1
  • Gavin McNay
    • 2
  1. 1.Department of Mathematical SciencesU.S. Military AcademyWest PointUSA
  2. 2.Tottenham LondonEngland, UK

Personalised recommendations