  The one-dimensional overdamped Brownian motion in a symmetric periodic
potential modulated by external time-reversible noise is analyzed. The
calculation of the effective diffusion coefficient is reduced to the mean first
passage time problem. We derive general equations to calculate the effective
diffusion coefficient of Brownian particles moving in arbitrary supersymmetric
potential modulated: (i) by external white Gaussian noise and (ii) by Markovian
dichotomous noise. For both cases the exact expressions for the effective
diffusion coefficient are derived. We obtain acceleration of diffusion in
comparison with the free diffusion case for fast fluctuating potentials with
arbitrary profile and for sawtooth potential in case (ii). In this case the
parameter region where this effect can be observed is given. We obtain also a
finite net diffusion in the absence of thermal noise. For rectangular potential
the diffusion slows down in comparison with the case when particles diffuse
freely, for all parameters of noise and of potential.
