A new construction of rotation symmetric bent functions with maximal algebraic degree
A new construction of rotation symmetric bent functions with maximal algebraic degree
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DOI：
10.3934/amc.2019017
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发表时间：
2019-02
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通讯作者：
Sihong Su
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文献类型：
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作者：
Sihong Su
对任意偶数的Begin{Document}$n=2m\ge4$\end{Document}，给出了具有最大代数度的Begin{Document}$n$\end{Document}-变量旋转对称Bent函数的一个新构造：Begin{Document}$f(x0，x1\cdots，X_{n-1})=\Bigoplus\Limits_{i=0}^{m-1}(x_ix_{m+i})\Oplus\Bigoplus\Limits_{i=0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2}\overline{x_{i+m}})，$\end{Document}，其双重功能为\Begin{Document}$\Widdetilde{f}(x_0，x_1\cdots，X_{n-1})=\Bigoplus\Limits_{i=0}^{m-1}(x_ix_{m+i})\Oplus\Bigoplus\Limits_{i=0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2}\overline{x_{i+n-2}})，$\end{Document}其中，\Begin{Document}$\Overline{x_{i}}=x_{i}\O加1$\end{Document}，且\Begin{Document}$x$\end{Document}的下标为模\Begin{Document}$n$\end{Document}。
In this paper, for any even integer \begin{document}$ n = 2m\ge4 $\end{document} , a new construction of \begin{document}$ n $\end{document} -variable rotation symmetric bent function with maximal algebraic degree \begin{document}$ m $\end{document} is given as \begin{document}$ f(x_0,x_1\cdots,x_{n-1}) = \bigoplus\limits_{i = 0}^{m-1}(x_ix_{m+i})\oplus \bigoplus\limits_{i = 0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2} \overline{x_{i+m}} ), $\end{document} whose dual function is \begin{document}$ \widetilde{f}(x_0,x_1\cdots,x_{n-1}) = \bigoplus\limits_{i = 0}^{m-1}(x_ix_{m+i})\oplus \bigoplus\limits_{i = 0}^{n-1}(x_ix_{i+1}\cdots x_{i+m-2} \overline{x_{i+n-2}} ), $\end{document} where \begin{document}$ \overline{x_{i}} = x_{i}\oplus 1 $\end{document} and the subscript of \begin{document}$ x $\end{document} is modulo \begin{document}$ n $\end{document} .