# Some semi-bent functions with polynomial trace form

Authors
• 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
Publication Date
Aug 09, 2014
Volume
27
Issue
4
Pages
777–784
Identifiers
DOI: 10.1007/s11424-014-2090-4
Source
Springer Nature
Keywords
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.