  We propose and analyze a linearly stabilized semi-implicit diffusive
Crank--Nicolson scheme for the Cahn--Hilliard gradient flow. In this scheme,
the nonlinear bulk force is treated explicitly with two second-order
stabilization terms. This treatment leads to linear elliptic system with
constant coefficients and provable discrete energy dissipation. Rigorous error
analysis is carried out for the fully discrete scheme. When the time step-size
and the space step-size are small enough, second order accuracy in time is
obtained with a prefactor controlled by some lower degree polynomial of
$1/\varepsilon$. {Here $\varepsilon$ is the thickness of the interface}.
Numerical results together with an adaptive time stepping are presented to
verify the accuracy and efficiency of the proposed scheme.
