Existence of rational primitive normal pairs over finite fields

Document Type : Research Paper


1 Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi, India

2 Scientific Analysis Group, Defence Research and Development Organisation, Metcalfe House, Delhi, India

3 Mathematics, Assistant Professor, S.S. Govt. P.G. College, Tigaon, Faridabad, Haryana, India


For a finite field $𝔽_{q^n}$ and a rational function $f=\frac{f_1}{f_2} \in 𝔽_{q^n}(x)$, we present a sufficient condition for the existence of a primitive normal element $\alpha \in 𝔽_{q^n}$ in such a way $f(\alpha)$ is also primitive in $𝔽_{q^n}$, where $f(x)$ is a rational function in $𝔽_{q^n}(x)$ of degree sum $m$ (degree sum of $f(x)=\frac{f_1(x)}{f_2(x)}$ is defined to be the sum of the degrees of $f_1(x)$ and $f_2(x)$). Additionally, for rational functions of degree sum 4, we proved that there are only $37$ and $16$ exceptional values of $(q,n)$ when $q=2^k$ and $q=3^k$ respectively.


Main Subjects

[1] Anju and R. K. Sharma, Existence of some special primitive normal elements over  nite  elds, Finite Fields Appl. ,
46 (2017) 280-303.
[2] C. Carvalho, J. P. Guardieiro, Victor G. L. Neumann and G. Tizziotti, On the existence of pairs of primitive and
normal elements over  nite  elds, Bul l. Braz. Math. Soc . (N.S.) , 53 (2022) 677-699.
[3] T. Cochrane and Ch. Pinner, Using stepanov's method for exponential sums involving rational functions, J. Number
Theory , 116 (2006) 270-292.
[4] S. D. Cohen, Pair of primitive elements in  elds of even order, Finite Fields Appl . , 28 (2014) 22-42.
[5] S. D. Cohen and A. Gupta, Primitive element pairs with a prescribed trace in the quartic extension of a  nite  eld,
J. Algebra Appl. , 20 (2021) 14 pp.
[6] S. D. Cohen and S. Huczynska, The primitive normal basis theorem-without a computer, J. London Math. Soc. (2) ,
67 (2003) 41-56.
[7] S. D. Cohen and S. Huczynska, The strong primitive normal basis theorem, Acta Ari th. , 143 (2010) 299-332.
[8] S. D. Cohen, H. Sharma and R. Sharma, Primitive values of rational functions at primitive elements of a  nite  eld,
J. Number Theory , 219 (2021) 237-246.
[9] L. Fu and D. Q.Wan, A class of incomplete character sums, Quart. J. Math. , 65 (2014) 1195-1211.
[10] A. Gupta, R. K. Sharma and S. D. Cohen, Some special primitive elements with prescribed trace over  nite  elds,
Finite Fields Appl. (2018) 54 1-18.
[11] G. Kapetanakis, Normal bases and primitive elements over  nite  elds, Finite Fields Appl . , 26 (2014) 123-143.
[12] H. W. Lenstra and R. J. Schoof, Primitive normal bases for  nite  elds, Mathematics of Compu tation , 48 no. 177
(1987) 217-231.
[13] R. Lidl and H. Niederreiter, Finite Fields , With a foreword by P. M. Cohn. Second edition, Encyclopedia of Math-ematics and its Applications, 20 , Cambridge University Press, Cambridge, 1997.
[14] Ch. Paar and J. Pelzl, Public-Key Cryptosystems Based on the Discrete Logarithm Problem , Springer Berlin Heidel-berg, Berlin, Heidelberg, (2010) 205-238.
[15] A. K. Sharma, M. Rani and Sh. K. Tiwari, Primitive normal values of rational functions over  nite  elds. (2021).
[16] H. Sharma and R. K. Sharma, Existence of primitive normal pairs with one prescribed trace over  nite
 elds, Des. Codes Cryptogr. , 89 (2021) 2841-2855.
[17] R. K. Sharma, A. Awasthi and A. Gupta, Existence of pair of primitive elements over  nite  elds of characteristic
2, J. Number Theory , 193 (2018) 386-394.
[18] SageMath., the Sage Mathematics Software System (Version 9.2), The Sage Dev elopers , (2020),