On the Dual of the Coulter-Matthews Bent Functions

TL;DR

This paper investigates the dual functions of Coulter-Matthews bent functions over finite fields, providing explicit formulas for specific cases, discovering new ternary bent functions, and establishing conditions for regularity.

Contribution

It derives explicit dual formulas for Coulter-Matthews bent functions in particular cases, introduces two new classes of ternary bent functions, and proves regularity conditions.

Findings

Explicit dual formulas for specific cases of Coulter-Matthews bent functions.

Discovery of two new classes of ternary bent functions with three terms.

Proof that certain Coulter-Matthews bent functions are regular bent.

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 $f(x)=Tr(ax^{\frac{3^k+1}{2}})$ 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.