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arXiv · 2003.09748

Existence and Cardinality of $k$-Normal Elements in Finite Fields

Simran Tinani·Joachim Rosenthal
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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 $\fqm$ with a non-maximal but high multiplicative order in the group of units of the finite field.

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Simran Tinani, Joachim Rosenthal. 2020-03-21. Existence and Cardinality of $k$-Normal Elements in Finite Fields. https://doi.org/10.1007/978-3-030-68869-1_15

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