To my knowledge, there are at least three published methods for killing the
$GCH$ everywhere:

1) forcing with supercompact Radin forcing (by Foreman-Woodin) . In fact Foreman and Woodin introduced an intermediate submodel of the supercompact Radin forcing in which the $GCH$ fails everywhere. They used a supercompact cardinal with infinitely many inaccessibles above it.

2) forcing with ordinary Radin forcing using guiding generics. See for example Cummings paper "A model in which $GCH$ holds at successors but fails at limits". The proof uses a strong cardinal (in fact a ($\kappa+3)-$strong cardinal $\kappa$ is sufficient)

3) forcing with extender based Radin forcing (Merimovich). The large cardinal assumption is the existence of a measurable cardinal $\kappa$ of Mitchell order $O(\kappa)=\kappa^{++}+\kappa^+.$

numberof known methods hardly makes sense. -- Voted to close. $\endgroup$ – Stefan Kohl Jan 28 '14 at 17:11