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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Christopoulou, M., Garefalakis, T., Panario, D. et al. The trace of an optimal normal element and low complexity normal bases. Des. Codes Cryptogr. 49, 199–215 (2008). https://doi.org/10.1007/s10623-008-9195-5
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DOI: https://doi.org/10.1007/s10623-008-9195-5