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

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.