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Erschienen in:
11.02.2020
Solving \(x+x^{2^{l}}+\cdots +x^{2^{ml}}=a\) over \(\mathbb {F}_{2^{n}}\)
Erschienen in: Cryptography and Communications | Ausgabe 4/2020
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Abstract
This paper presents an explicit representation for the solutions of the equation \({\sum }_{i=0}^{\frac kl-1}x^{2^{li}} = a \in \mathbb {F}_{2^{n}}\) for any given positive integers k, l with l|k and n, in the closed field \({\overline {\mathbb {F}_{2}}}\) and in the finite field \(\mathbb {F}_{2^{n}}\). As a by-product of our study, we are able to completely characterize the a’s for which this equation has solutions in \(\mathbb {F}_{2^{n}}\).