Abstract
For the Galois field extension \(\mathbb {F}_{q^n}\) over \(\mathbb {F}_q\) we let \(PN_n(q)\) denote the number of primitive elements of \(\mathbb {F}_{q^n}\) which are normal over \(\mathbb {F}_q\). We derive lower bounds for \(PN_3(q)\) and \(PN_4(q)\), the number of primitive normal elements in cubic and quartic extensions. Our reasoning relies on basic projective geometry. A comparision with the exact numbers for \(PN_3(q)\) and \(PN_4(q)\) where \(q\le 32\) (altogether 36 instances) indicates that these bounds are very good; we even achieve equality in 14 cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carlitz L.: Primitive roots in a finite field. Trans. Am. Math. Soc. 73, 373–382 (1952).
Cohen S.D.: Gauss sums and a sieve for generators of Galois fields. Publ. Math. Debr. 56, 293–312 (2000).
Cohen S.D., Hachenberger D.: Primitivity, freeness, norm and trace. Discret. Math. 214, 135–144 (2000).
Cohen S.D., Huczynska S.: Primitive free quartics with specified norm and trace. Acta Arith. 109, 359–385 (2003).
Cohen S.D., Huczynska S.: The strong primitive normal basis theorem. Acta Arith. 143, 299–332 (2010).
Davenport H.: Bases for finite fields. J. Lond. Math. Soc. 43, 21–49 (1968).
Hachenberger D.: Finite Fields: Normal Bases and Completely Free Elements. Kluwer Academic Publishers, Boston (1997).
Hachenberger D.: Completely normal bases. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, Sect. 5.4, pp. 128–138. CRC Press, Boca Raton (2013).
Hachenberger D.: Asymptotic existence results for primitive completely normal elements in extensions of Galois fields (submitted).
Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Clarendon Press, Oxford (1998).
Huczynska S.: Existence results for finite field polynomials with specified properties. In: Charpin P., Pott A., Winterhof A. (eds.) Finite Fields and Their Applications: Character Sums and Polynomials. De Gruyter, Berlin (2013).
Huczynska S., Cohen S.D.: Primitive free cubics with specified norm and trace. Trans. Am. Math. Soc. 355, 3099–3116 (2003).
Lenstra Jr., H.W., Schoof R.J.: Primitive normal bases for finite fields. Math. Comput. 48, 217–231 (1987).
Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley, Reading (1983).
Ribenboim P.: Die Welt der Primzahlen. Springer, Berlin (2006).
Sage, free open-source mathematics software system. http://www.sagemath.org/. Accessed 18 Aug 2014.
Acknowledgments
This paper is dedicated to the memory of Scott Vanstone. I thank Scott for many discussions on finite fields and finite geometries, which once have raised my interest in studying specific normal bases. I am also very grateful to Scott for once supporting my Habilitation at the University of Augsburg, published as [7]. Finally, I wish to thank Thomas Gruber for supporting me with the computational results included in Tables 1 and 2.
Author information
Authors and Affiliations
Corresponding author
Additional information
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Cryptography, Codes, Designs and Finite Fields: In Memory of Scott A. Vanstone”.
Rights and permissions
About this article
Cite this article
Hachenberger, D. Primitive normal bases for quartic and cubic extensions: a geometric approach. Des. Codes Cryptogr. 77, 335–350 (2015). https://doi.org/10.1007/s10623-015-0051-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-015-0051-0