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Some semi-bent functions with polynomial trace form

Authors
  • Chen, Hao1
  • Cao, Xiwang1
  • 1 Nanjing University of Aeronautics and Aeronautics, Department of Math, Nanjing, 211106, China , Nanjing (China)
Type
Published Article
Journal
Journal of Systems Science and Complexity
Publisher
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Publication Date
Aug 09, 2014
Volume
27
Issue
4
Pages
777–784
Identifiers
DOI: 10.1007/s11424-014-2090-4
Source
Springer Nature
Keywords
License
Yellow

Abstract

This paper is devoted to the study of semi-bent functions with several parameters flexible on the finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{2^n } $\end{document}. Boolean functions defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{2^n } $\end{document} of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{a,b}^{(r)} (x) = Tr_1^n (ax^{r(2^m - 1)} ) + Tr_1^4 (bx^{\tfrac{{2^n - 1}} {5}} ) $\end{document} and the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{a,b,c,d}^{(r,s)} (x) = Tr_1^n (ax^{r(2^m - 1)} ) + Tr_1^4 (bx^{\tfrac{{2^n - 1}} {5}} ) + Tr_1^n (cx^{(2^m - 1)\tfrac{1} {2} + 1} ) + Tr_1^n (dx^{(2^m - 1)s + 1} ) $\end{document} where n = 2m, m ≡ 2 (mod 4), a, c ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{16} $\end{document}, and b ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_2 $\end{document}, d ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_2 $\end{document}, are investigated in constructing new classes of semi-bent functions. Some characteristic sums such as Kloosterman sums and Weil sums are employed to determine whether the above functions are semi-bent or not.

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