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In this paper, we consider the characterization of the bentness of quadraticBoolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1}Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$is even and $c_i\in GF(2^e)$. For a general $m$, it is difficult to determinethe bentness of these functions. We present the bentness of quadratic Booleanfunction for two cases: $m=2^vp^r$ and $m=2^vpq$, where $p$ and $q$ are twodistinct primes. Further, we give the enumeration of quadratic bent functionsfor the case $m=2^vpq$.

Effective Construction of a Class of Bent Quadratic Boolean Functions