Effective Construction of a Class of Bent Quadratic Boolean Functions

Abstract

In this paper, we consider the characterization of the bentness of quadratic Boolean functions of the form $f(x)={\sum }_{i=1}^{\frac{m}{2}-1}T{r}_1^n(c_i{x}^{1+2^{ei}})+T{r}_1^{n/2}(c_{m/2}{x}^{1+2^{n/2}}),$ where $n=me$, $m$ is even and $c_i\in GF(2^e)$. For a general $m$, it is difficult to determine the bentness of these functions. We present the bentness of quadratic Boolean function for two cases: $m=2^v{p}^r$ and $m=2^{v}pq$, where $p$ and $q$ are two distinct primes. Further, we give the enumeration of quadratic bent functions for the case $m=2^{v}pq$.