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

TL;DR

This paper determines the duals of Coulter-Matthews bent functions over finite fields and introduces new classes of ternary bent functions with specific trace term counts, expanding the known landscape of bent functions.

Contribution

It completely characterizes the dual functions of Coulter-Matthews bent functions and constructs new classes of ternary bent functions with a limited number of trace terms.

Findings

Identified the dual functions of Coulter-Matthews bent functions.

Constructed new ternary bent functions with 8 or 21 trace terms.

Discovered classes of bent functions not previously reported.

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.