In this paper, we give first some non-trivial improvements of the well-known bounds of effective Nullstellensätze. Using these bounds, we show that the Gröbner basis (for any monomial ordering) of a zero–dimensional ideal may be computed within a bit complexity which is essentially polynomial in $D^{n^{2}}$ where $n$ is the number of unknowns and $D$ is the mean value of the degrees of input polynomials.