Diffusion based on stochastic chaos in random dynamical systems is studied. There is noise-induced anomalous diffusion and ageing in a spatially extended dynamical systems, where the unperturbed version is a well-established model for deterministic diffusion. The power law exponents for the mean square displacement and the waiting time distribution match to predictions by continuous time random walk theory. When we fix a particular noise realization, we find noise-induced synchronization of jump times and the breaking of self-averaging. Robustness of this phenomenon is also briefly discussed by revealing noise-induced anomalous diffusion in a class of nonlinear dynamical systems.