The trace of an optimal normal element and low complexity normal bases |
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Authors: | Maria Christopoulou Theo Garefalakis Daniel Panario David Thomson |
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Institution: | 1. Department of Mathematics, University of Crete, 714 09, Heraklion, Crete, Greece 2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Dr., Ottawa, ON, Canada, K1S 5B6 3. Department of Electrical and Computer Engineering, University of Waterloo, 200 University Ave. West, Waterloo, ON, Canada, N2L 3G1
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Abstract: | Let ${\mathbb{F}}_{q}$ be a finite field and consider an extension ${\mathbb{F}}_{q^{n}}$ where an optimal normal element exists. Using the trace of an optimal normal element in ${\mathbb{F}}_{q^{n}}$ , we provide low complexity normal elements in ${\mathbb{F}}_{q^{m}}$ , 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 ${\mathbb{F}}_{q^{m}}$ ; otherwise we give low complexity normal elements. Since optimal normal elements do not exist for every extension degree m of every finite field ${\mathbb{F}}_{q}$ , our results could have a practical impact in expanding the available extension degrees for fast arithmetic using normal bases. |
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