  The coordinate transformation offers a remarkable way to design cloaks that
can steer electromagnetic fields so as to prevent waves from penetrating into
the {\em cloaked region} (denoted by $\Omega_0$, where the objects inside are
invisible to observers outside). The ideal circular and elliptic cylindrical
cloaked regions are blown up from a point and a line segment, respectively, so
the transformed material parameters and the corresponding coefficients of the
resulted equations are highly singular at the cloaking boundary $\partial
\Omega_0$. The electric field or magnetic field is not continuous across
$\partial\Omega_0.$ The imposition of appropriate {\em cloaking boundary
conditions} (CBCs) to achieve perfect concealment is a crucial but challenging
issue.
  Based upon the principle that finite electromagnetic fields in the original
space must be finite in the transformed space as well, we obtain CBCs that
intrinsically relate to the essential "pole" conditions of a singular
transformation. We also find that for the elliptic cylindrical cloak, the CBCs
should be imposed differently for the cosine-elliptic and sine-elliptic
components of the decomposed fields. With these at our disposal, we can
rigorously show that the governing equation in $\Omega_0$ can be decoupled from
the exterior region $\Omega_0^c$, and the total fields in the cloaked region
vanish. We emphasize that our proposal of CBCs is different from any existing
ones.
