  Strong external difference families (SEDFs) were introduced by Paterson and
Stinson as a more restrictive version of external difference families. SEDFs
can be used to produce optimal strong algebraic manipulation detection codes.
We characterize the parameters $(v, m, k, \lambda)$ of a nontrivial SEDF that
is near-complete (satisfying $v=km+1$). We construct the first known nontrivial
example of a $(v, m, k, \lambda)$ SEDF having $m > 2$. The parameters of this
example are $(243,11,22,20)$, giving a near-complete SEDF, and its group is
$\mathbb{Z}_3^5$. We provide a comprehensive framework for the study of SEDFs
using character theory and algebraic number theory, showing that the cases
$m=2$ and $m>2$ are fundamentally different. We prove a range of nonexistence
results, greatly narrowing the scope of possible parameters of SEDFs.
