Solving $x+x^{2^{l}}+\cdots +x^{2^{ml}}=a$ over $\mathbb {F}_{2^{n}}$

Cryptography and Communications - Tập 12 - Trang 809-817 - 2020
Sihem Mesnager1,2,3, Kwang Ho Kim4,5, Jong Hyok Choe4, Dok Nam Lee4, Dae Song Go6
1Department of Mathematics, University of Paris VIII, Saint-Denis, France
2University of Paris XIII, CNRS, LAGA UMR 7539, Sorbonne Paris Cité, Villetaneuse, France
3Telecom ParisTech, Palaiseau, France
4Institute of Mathematics, State Academy of Sciences, Pyongyang, Democratic People’s Republic of Korea
5PGItech Corp, Pyongyang, Democratic People’s Republic of Korea
6Master School, University of Natural Science, Pyongyang, Democratic People’s Republic of Korea

Tóm tắt

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 $\mathbb {F}_{2^{n}}$ .

Tài liệu tham khảo

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