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Existence of rational primitive normal pairs over finite fields | ||
| International Journal of Group Theory | ||
| مقاله 3، دوره 13، شماره 1، خرداد 2024، صفحه 17-30 اصل مقاله (509.33 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 | ||
| 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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[14] Ch. Paar and J. Pelzl, Public-Key Cryptosystems Based on the Discrete Logarithm Problem , Springer Berlin Heidel-berg, Berlin, Heidelberg, (2010) 205-238.
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