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Existence and Cardinality of k-Normal Elements in Finite Fields

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Arithmetic of Finite Fields (WAIFI 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12542))

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Abstract

Normal bases in finite fields constitute a vast topic of large theoretical and practical interest. Recently, k-normal elements were introduced as a natural extension of normal elements. The existence and the number of k-normal elements in a fixed extension of a finite field are both open problems in full generality, and comprise a promising research avenue. In this paper, we first formulate a general lower bound for the number of k-normal elements, assuming that they exist. We further derive a new existence condition for k-normal elements using the general factorization of the polynomial \(x^m-1\) into cyclotomic polynomials. Finally, we provide an existence condition for normal elements in \({\mathbb {F}_{q^m}}\) with a non-maximal but high multiplicative order in the group of units of the finite field.

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References

  1. Ash, D.W., Blake, I.F., Vanstone, S.A.: Low complexity normal bases. Discret. Appl. Math. 25(3), 191–210 (1989). https://doi.org/10.1016/0166-218X(89)90001-2

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlitz, L.: Primitive roots in a finite field. Trans. Am. Math. Soc. 73, 373–382 (1952). https://doi.org/10.2307/1990797

    Article  MathSciNet  MATH  Google Scholar 

  3. Cohen, S.D., Huczynska, S.: The strong primitive normal basis theorem. Acta Arith. 143(4), 299–332 (2010). https://doi.org/10.4064/aa143-4-1

  4. Davenport, H.: Bases for finite fields. J. Lond. Math. Soc. 43, 21–39 (1968). https://doi.org/10.1112/jlms/s1-43.1.21

    Article  MathSciNet  MATH  Google Scholar 

  5. Frandsen, G.S.: On the density of normal bases in finite fields. Finite Fields Appl. 6(1), 23–38 (2000). https://doi.org/10.1006/ffta.1999.0263

    Article  MathSciNet  MATH  Google Scholar 

  6. Gao, S.: Normal bases over finite fields. ProQuest LLC, Ann Arbor, MI, thesis (Ph.D.)-University of Waterloo (Canada) (1993)

    Google Scholar 

  7. Gao, S., Panario, D.: Density of normal elements. Finite Fields Appl. 3(2), 141–150 (1997). https://doi.org/10.1006/ffta.1996.0177

    Article  MathSciNet  MATH  Google Scholar 

  8. Hensel, K.: Ueber die Darstellung der Zahlen eines Gattungsbereiches für einen beliebigen Primdivisor. J. Reine Angew. Math. 103, 230–237 (1888). https://doi.org/10.1515/crll.1888.103.230

    Article  MathSciNet  MATH  Google Scholar 

  9. Huczynska, S., Mullen, G.L., Panario, D., Thomson, D.: Existence and properties of \(k\)-normal elements over finite fields. Finite Fields Appl. 24, 170–183 (2013). https://doi.org/10.1016/j.ffa.2013.07.004

    Article  MathSciNet  MATH  Google Scholar 

  10. Hyde, T.: Normal elements in finite fields. arXiv preprint arXiv:1809.02155 (2018)

  11. Kapetanakis, G.: Normal bases and primitive elements over finite fields. Finite Fields Appl. 26, 123–143 (2014). https://doi.org/10.1016/j.ffa.2013.12.002

    Article  MathSciNet  MATH  Google Scholar 

  12. Kapetanakis, G., Reis, L.: Variations of the primitive normal basis theorem. Des. Codes Crypt. 87(7), 1459–1480 (2018). https://doi.org/10.1007/s10623-018-0543-9

    Article  MathSciNet  MATH  Google Scholar 

  13. Lenstra Jr., H.W., Schoof, R.J.: Primitive normal bases for finite fields. Math. Comput. 48(177), 217–231 (1987). https://doi.org/10.2307/2007886

    Article  MathSciNet  MATH  Google Scholar 

  14. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications, 2nd edn, vol. 20. Cambridge University Press, Cambridge (1997). With a foreword by P. M. Cohn

    Google Scholar 

  15. Lüneburg, H.: Translation Planes. Springer, Heidelberg (2012). https://books.google.ch/books?id=UuTrCAAAQBAJ

  16. Menezes, A.J., Blake, I.F., Gao, X., Mullin, R.C., Vanstone, S.A., Yaghoobian, T.: Applications of Finite Fields. The Kluwer International Series in Engineering and Computer Science, vol. 199. Kluwer Academic Publishers, Boston (1993). https://doi.org/10.1007/978-1-4757-2226-0

  17. Mullen, G.L.: Some open problems arising from my recent finite research. In: Contemporary Developments in Finite Fields and Applications, pp. 254–269. World Sci. Publ., Hackensack (2016)

    Google Scholar 

  18. Ore, O.: Contributions to the theory of finite fields. Trans. Am. Math. Soc. 36(2), 243–274 (1934). https://doi.org/10.2307/1989836

    Article  MathSciNet  MATH  Google Scholar 

  19. Reis, L.: Existence results on \(k\)-normal elements over finite fields. Rev. Mat. Iberoam. 35(3), 805–822 (2019). https://doi.org/10.4171/rmi/1070

    Article  MathSciNet  MATH  Google Scholar 

  20. Reis, L., Thomson, D.: Existence of primitive 1-normal elements in finite fields. Finite Fields Appl. 51, 238–269 (2018). https://doi.org/10.1016/j.ffa.2018.02.002

    Article  MathSciNet  MATH  Google Scholar 

  21. Saygı, Z., Tilenbaev, E., Ürtiş, C.: On the number of \(k\)-normal elements over finite fields. Turkish J. Math. 43(2), 795–812 (2019). https://doi.org/10.3906/mat-1805-113

    Article  MathSciNet  MATH  Google Scholar 

  22. Sozaya-Chan, J.A., Tapia-Recillas, H.: On k-normal elements over finite fields. Finite Fields Appl. 52, 94–107 (2018). https://doi.org/10.1016/j.ffa.2018.03.006

    Article  MathSciNet  MATH  Google Scholar 

  23. The Sage Developers: SageMath, the Sage Mathematics Software System (Version 8.6) (2020). https://www.sagemath.org

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Acknowledgement

This work was partially supported by Swiss National Science Foundation grant no. 188430. The authors are also greatly thankful to Gianira Alfarano for her thorough proofreading and constructive feedback on this manuscript.

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Authors and Affiliations

  1. University of Zürich, Winterthurerstrasse, 8057, Zurich, Switzerland

    Simran Tinani & Joachim Rosenthal

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  1. Simran Tinani
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  2. Joachim Rosenthal
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Correspondence to Simran Tinani .

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Editors and Affiliations

  1. Sorbonne Paris, Paris, France

    Jean Claude Bajard

  2. Sabancı University, Istanbul, Turkey

    Alev Topuzoğlu

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Tinani, S., Rosenthal, J. (2021). Existence and Cardinality of k-Normal Elements in Finite Fields. In: Bajard, J.C., Topuzoğlu, A. (eds) Arithmetic of Finite Fields. WAIFI 2020. Lecture Notes in Computer Science(), vol 12542. Springer, Cham. https://doi.org/10.1007/978-3-030-68869-1_15

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  • DOI: https://doi.org/10.1007/978-3-030-68869-1_15

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