  We transform the way of finding entanglement criterion into two steps: to
obtain necessary criterion of separability by maximizing an algebra function
for a set of characteristic variables of the witness operator and the given
number of partitions, then to obtain the sufficient criterion by minimizing an
algebra function with respect to the characteristic variables for a given
quantum state. Our method avoids the semi-definite program calculation in the
witness operator entanglement detection. The necessary and sufficient criterion
of separability for the three qubit X shaped state is given to illustrate the
procedure of finding the criterion. We give the necessary and sufficient
criteria of the three partite and full separabilities for the four qubit noisy
GHZ state and the four qubit noisy cluster state.
