New Classes of Ternary Bent Functions from the Coulter-Matthews Bent Functions

Abstract

It has been an active research issue for many years to construct new bent functions. For $k$ odd with $\gcd(n, k)=1$, and $a\in\mathbb{F}_{3^n}^{*}$, the function $f(x)=Tr(ax^{\frac{3^k+1}{2}})$ is weakly regular bent over $\mathbb{F}_{3^n}$, where $Tr(\cdot):\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3$ is the trace function. This is the well-known Coulter-Matthews bent function. In this paper, we determine the dual function of $f(x)$ completely. As a consequence, we find many classes of ternary bent functions not reported in the literature previously. Such bent functions are not quadratic if $k>1$, and have $\left(\left(\frac{1+\sqrt{5}}{2}\right)^{w+1}-\right.$ $\left.\left(\frac{1-\sqrt{5}}{2}\right)^{w+1}\right)/\sqrt{5}$ or $\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-w+1}-\right.$ $\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-w+1}\right)/\sqrt{5}$ trace terms, where $0<w<n$ and $wk\equiv 1\ (\bmod\;n)$. Among them, five special cases are especially interesting: for the case of $k=(n+1)/2$, the number of trace terms is $\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\right.$ $\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right)/\sqrt{5}$; for the case of $k=n-1$, the number of trace terms is $\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\right.$ $\left.\left(\frac{1-\sqrt{5}}{2}\right)^n\right)/\sqrt{5}$; for the case of $k=(n-1)/2$, the number of trace terms is $\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n-1}-\right.$ $\left.\left(\frac{1-\sqrt{5}}{2}\right)^{n-1}\right)/\sqrt{5}$; for the case of $(n, k)=(5t+4, 4t+3)$ or $(5t+1, 4t+1)$ with $t\geq 1$, the number of trace terms is 8; and for the case of $(n, k)=(7t+6, 6t+5)$ or $(7t+1, 6t+1)$ with $t\geq 1$, the number of trace terms is 21. As a byproduct, we find new classes of ternary bent functions with only 8 or 21 trace terms.