On the Dual of the Coulter-Matthews Bent Functions

Abstract

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 $f(x)=Tr(a{x}^{\frac{3^k+1}{2}})$ is weakly regular bent over ${\mathbb{F}}_{3^n}$, where $a\in {\mathbb{F}}_{3^n}^{\ast }$, and $Tr(\cdot ):{\mathbb{F}}_{3^n}\to {\mathbb{F}}_3$ 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 $t\ge 2$, 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 $t\ge 2$, the dual function is given by $$Tr\left(-\frac{x^{3^{2t+2}+1}}{a^{3^{2t+2}-3^{t+1}+3}}-\frac{x^{2\cdot 3^{2t+1}+3^{t+1}+1}}{a^{3^{2t+2}+3^{t+1}+1}}+\frac{x^2}{a^{-3^{2t+2}+3^{t+1}+3}}\right).$$ 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.