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

Abstract

Two classes of ternary (vectorial) bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland original class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthreek$ given by $\Tr^{4k}_k\big(a_1 x^{2(3^k+1)}+a_2 x^{(3^k+1)^2}\big)$, where $a_1$ is a nonsquare in $\F_{3^{4k}}$ and $a_2$ is defined explicitly by $a_1$. Particular subclass of such binomial vectorial bent function is represented by the exceptional polynomials over $\fthreek$. Bent trinomials are mappings $\F_{3^{2k}}\mapsto\fthree$ given by $\Tr_{2k}\big(a_1 x^{2(3^k+2)} + a_2 x^{3^k+5} + a_3 x^2\big)$ with $k>1$ not divisible by four and coefficients explicitly defined depending on the parity of $k$. The proof is based on new criterion that allows checking bentness by analyzing the first-order derivatives of $f$ in the direction of a chosen $n/2$-dimensional subspace ($n$ is the degree of the domain of $f$ over $\fp$).