Characterization of certain types of $r$-plateaued functions
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초록
We study a subclass of $p$-ary functions in $n$ variables, denoted by ${\mathcal A}_n$, which is a collection of $p$-ary functions in $n$ variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all $p$-ary $(n-1)$-plateaued functions in $n$ variables by proving that every $(n-1)$-plateaued function should be contained in $\mathcal{A}_n$. Secondly, we prove that if $f$ is a $p$-ary $r$-plateaued function contained in ${\mathcal A}_n$ with $\deg{f} > 1+\frac{n-r}{4}(p-1)$, then the highest degree term of $f$ is only a single term. Furthermore, we prove that there is no $p$-ary $r$-plateaued function in ${\mathcal A}_n$ with maximum degree $(p-1)\frac{n-r}{2}+1$. As application, we partially classify all $(n-2)$-plateaued functions in ${\mathcal A}_n$ when $p=3,5,$ and $7$, and $p$-ary bent functions in ${\mathcal A}_2$ are completely classified for the cases $p=3$ and $5$.
Abstract
We study a subclass of $p$-ary functions in $n$ variables, denoted by ${\mathcal A}_n$, which is a collection of $p$-ary functions in $n$ variables satisfying a certain condition on the exponents of its monomial terms. Firstly, we completely classify all $p$-ary $(n-1)$-plateaued functions in $n$ variables by proving that every $(n-1)$-plateaued function should be contained in $\mathcal{A}_n$. Secondly, we prove that if $f$ is a $p$-ary $r$-plateaued function contained in ${\mathcal A}_n$ with $\deg{f} > 1+\frac{n-r}{4}(p-1)$, then the highest degree term of $f$ is only a single term. Furthermore, we prove that there is no $p$-ary $r$-plateaued function in ${\mathcal A}_n$ with maximum degree $(p-1)\frac{n-r}{2}+1$. As application, we partially classify all $(n-2)$-plateaued functions in ${\mathcal A}_n$ when $p=3,5,$ and $7$, and $p$-ary bent functions in ${\mathcal A}_2$ are completely classified for the cases $p=3$ and $5$.
참고문헌 (12)
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