  In a progress toward searching for the QCD critical point, we study the
finite density phase transition of $N_f = 4$ and 2 lattice QCD at finite
temperature with the canonical ensemble approach. We develop a winding number
expansion method to accurately project out the particle number from the fermion
determinant which greatly extends the applicable range of baryon number sectors
to make the study feasible. Our lattice simulation was carried out with the
clover fermions and improved gauge action. For a given temperature, we
calculate the baryon chemical potential from the canonical approach to look for
the mixed phase as a signal for the first order phase transition. In the case
of $N_f=4$, we observe an "S-shape" structure in the chemical potential-density
plane due to the surface tension of the mixed phase in a finite volume which is
a signal for the first order phase transition. We use the Maxwell construction
to determine the phase boundaries for three temperatures below $T_c$. The
intersecting point of the two extrapolated boundaries turns out to be at the
expected first order transition point at $T_c$ with $\mu = 0$. This serves as a
check for our method of identifying the critical point. We also studied the
$N_f =2$ case, but do not see a signal of the mixed phase for temperature as
low as 0.83 $T_c$.
