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Existence of rational primitive normal pairs over finite fields | ||
| International Journal of Group Theory | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 13 تیر 1401 اصل مقاله (484 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2022.133016.1784 | ||
| نویسندگان | ||
Rajendra Kumar Sharma   1؛ Soniya Takshak1؛ Ambrish Awasthi2؛ Hariom Sharma3
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| 1Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi, India | ||
| 2Scientific Analysis Group, Defence Research and Development Organisation, Metcalfe House, Delhi, India | ||
| 3Mathematics, Assistant Professor, S.S. Govt. P.G. College, Tigaon, Faridabad, Haryana, India | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Finite Field؛ Primitive Element؛ Normal Element؛ Character | ||
| مراجع | ||
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[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), | ||
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