The trace of an optimal normal element and low complexity normal bases

Let Fq be a finite field and consider an extension Fqn where an optimal normal element exists. Using the trace of an optimal normal element in Fqn , we provide low complexity normal elements in Fqm , with m = n/k. We give theorems for Type I and Type II optimal normal elements. When Type I normal elements are used with m = n/2, m odd and q even, our construction gives Type II optimal normal elements in Fqm ; otherwise we give low complexity normal elements. Since optimal normal elements do not exist for every extension degree m of every finite field Fq, our results could have a practical impact in expanding the available extension degrees for fast arithmetic using normal bases.