关于Coulter-Matthews Bent函数的对偶
On the Dual of the Coulter-Matthews Bent Functions
摘要：
对于任何bent函数，确定它的对偶函数是非常有趣的，因为对偶函数在某些情况下也是bent的。对于$k$奇数和$\gcd(n, k)=1$，已知Coulter-Matthews弯曲函数$f(x)=Tr(ax^{\frac{3^k+1}{2}})$是弱正则弯曲于$\mathbb{F}_{3^n}$，其中$a\in\mathbb{F}_{3^n}^{*}$，并且$Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3$是迹函数。本文研究了$f(x)$的对偶功能，并挖掘出一个普适公式。特别是，对于两种情况下，我们明确地确定公式：对于$n=3t+1$和$k=2t+1$与$t\geq 2$的情况下，对偶函数由$$Tr\left(-\frac{x^{3^{2t+1}+3^{t+1}+2}}{a^{3^{2t+1}+3^{t+1}+1}}-\frac{x^{3^{2t}+1}}{a^{-3^{2t}+3^{t}+1}}+\frac{x^{2}}{a^{-3^{2t+1}+3^{t+1}+1}}\right);$$给出，对于$n=3t+2$和$k=2t+1$与$t\geq 2$的情况下，对偶函数由$$Tr\left(-\frac{x^{3^{2t+2}+1}}{a^{3^{2t+2}-3^{t+1}+3}}-\frac{x^{2\cdot3^{2t+1}+3^{t+1}+1}}{a^{3^{2t+2}+3^{t+1}+1}}+\frac{x^2}{a^{-3^{2t+2}+3^{t+1}+3}}\right).$$给出。此外，我们还证明了在某些情况下$f(x)$是规则弯曲的。
英文摘要：
For any bent function, it is very interesting to determine its dual function because the dual function is also bent in certain cases. For $k$ odd and $\gcd(n, k)=1$, it is known that the Coulter-Matthews bent function $f(x)=Tr(ax^{\frac{3^k+1}{2}})$ is weakly regular bent over $\mathbb{F}_{3^n}$, where $a\in\mathbb{F}_{3^n}^{*}$, and $Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3$ is the trace function. In this paper, we investigate the dual function of $f(x)$, and dig out an universal formula. In particular, for two cases, we determine the formula explicitly: for the case of $n=3t+1$ and $k=2t+1$ with $t\geq 2$, the dual function is given by $$Tr\left(-\frac{x^{3^{2t+1}+3^{t+1}+2}}{a^{3^{2t+1}+3^{t+1}+1}}-\frac{x^{3^{2t}+1}}{a^{-3^{2t}+3^{t}+1}}+\frac{x^{2}}{a^{-3^{2t+1}+3^{t+1}+1}}\right);$$ and for the case of $n=3t+2$ and $k=2t+1$ with $t\geq 2$, the dual function is given by $$Tr\left(-\frac{x^{3^{2t+2}+1}}{a^{3^{2t+2}-3^{t+1}+3}}-\frac{x^{2\cdot3^{2t+1}+3^{t+1}+1}}{a^{3^{2t+2}+3^{t+1}+1}}+\frac{x^2}{a^{-3^{2t+2}+3^{t+1}+3}}\right).$$ As a byproduct, we find two new classes of ternary bent functions with only three terms. Moreover, we also prove that in certain cases $f(x)$ is regular bent.
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