  This paper proposes and analyzes a novel fully discrete finite element scheme
with the interpolation operator for stochastic Cahn-Hilliard equations with
functional-type noise. The nonlinear term satisfies a one-side Lipschitz
condition and the diffusion term is globally Lipschitz continuous. The
novelties of this paper are threefold. First, the $L^2$-stability ($L^\infty$
in time) and the discrete $H^2$-stability ($L^2$ in time) are proved for the
proposed scheme. The idea is to utilize the special structure of the matrix
assembled by the nonlinear term. None of these stability results has been
proved for the fully implicit scheme in existing literature due to the
difficulty arising from the interaction of the nonlinearity and the
multiplicative noise. Second, the higher moment stability in $L^2$-norm of the
discrete solution is established based on the previous stability results.
Third, the H\"older continuity in time for the strong solution is established
under the minimum assumption of the strong solution. Based on these, the
discrete $H^{-1}$-norm of the strong convergence is discussed. Several
numerical experiments including stability and convergence are also presented to
validate our theoretical results.
