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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 $5$.
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참고문헌 신청
A. C esmelioglu / 2013 / A construction of bent functions from plateaued functions / Des. Codes Cryptogr 66(1-3):231~242
X. -D. Hou / 2004 / p-ary and q-ary versions of certain results about bent functions and resilient functions / Finite Fields Appl 10(4):566~582
J. Y. Hyun / 2016 / Explicit criteria for construction of plateaued functions / IEEE Trans. Inform. Theory 62(12):7555~7565
X. -D. Hou / 2004 / p-ary and q-ary versions of certain results about bent functions and resilient functions / Finite Fields Appl 10(4):566~582
J. Y. Hyun / 2016 / Explicit criteria for construction of plateaued functions / IEEE Trans. Inform. Theory 62(12):7555~7565
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