  We provide a differential structure on arbitrary cleft extensions
$B:=A^{\mathrm{co}H}\subseteq A$ for an $H$-comodule algebra $A$. This is
achieved by constructing a covariant calculus on the corresponding crossed
product algebra $B\#_\sigma H$ from the data of a bicovariant calculus on the
structure Hopf algebra $H$ and a calculus on the base algebra $B$, which is
compatible with the $2$-cocycle and measure of the crossed product. The result
is a quantum principal bundle with canonical strong connection and we describe
the induced bimodule covariant derivatives on associated bundles of the crossed
product. It is proven that connections of the quantum principal bundle are in
bijection with connection $1$-forms. All results specialize to trivial
extensions and smash product algebras $B\#H$ and we give a characterization of
the smash product calculus in terms of the differentials of the cleaving map
$j\colon H\to A$ and the inclusion $B\hookrightarrow A$. The construction is
exemplified for pointed Hopf algebras. In particular, the case of Radford Hopf
algebras $H_{(r,n,q)}$ is spelled out in detail.
