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 xm−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.
Tinani, Simran