Using atomistic computer simulations, we calculate the diffusion parameters of an Ll2-ordered Ni3Al-alloy. A conjugate gradient algorithm under constant zero pressure and a semi-empirical N-body potential are applied to evaluate potential barrier heights and pre-exponential factors. We focus our investigation on those diffusion mechanisms that have been proposed to account for the experimentally observed but theoretically still disputed fast self-diffusion of Al in Ni3Al: antisite bridge mechanisms and correlated six-jump cycles. Our results demonstrate that the most competitive jumps or jump sequences involve AlNJ-antistructure atoms which are shown to play a key role in the diffusion process by either actively or passively lowering the activation energy of migration.