On the singular cardinal hypothesis

We use core model theory to obtain the following lower bounds to the consistency strength for the failure of the Singular Cardinal Hypothesis: Suppose that $\kappa$ is a singular strong limit cardinal such that ${2^\kappa } > {\kappa ^ + }$. Then there is an inner model $K$ such that $o(\kappa ) = {\kappa ^{ + + }}$ in $K$ if $\kappa$ has uncountable cofinality, and $\forall \alpha < \kappa \exists \nu < \kappa o(\kappa ) \geqslant \nu$ in $K$ otherwise.