Ternary Binomial and Trinomial Bent Functions in the Completed Maiorana-McFarland Class

TL;DR

This paper introduces new classes of ternary bent functions within the completed Maiorana-McFarland class, providing explicit constructions and a novel criterion for verifying bentness based on derivatives.

Contribution

It presents explicit constructions of ternary bent functions of degree four with two or three terms, expanding the known classes within the completed Maiorana-McFarland class, and introduces a new method for checking bentness.

Findings

Explicit binomial and trinomial bent functions are constructed.

A new criterion for bentness based on derivatives is developed.

Particular subclasses can be represented by exceptional polynomials.

Abstract

Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthree$ given by $f(x)=\Tr_{4k}\big(a_1 x^{2(3^k+1)}+a_2 x^{(3^k+1)^2}\big)$, where $a_1$ is a nonsquare in $\F_{3^{4k}}$ and $a_2$ is defined explicitly by $a_1$. Particular subclasses of the binomial bent functions we found can be represented by exceptional polynomials over $\fthreek$. Bent trinomials are mappings $\F_{3^{2k}}\mapsto\fthree$ given by $f(x)=\Tr_n\big(a_1 x^{2\cdot3^k+4} + a_2 x^{3^k+5} + a_3 x^2\big)$ with coefficients explicitly defined by the parity of $k$. The proof is based on a new criterion that allows checking bentness by analyzing first- and second-order derivatives of $f$ in the direction of a chosen $n/2$-dimensional subspace.