Adding a lot of Cohen reals by adding a few I

In this paper we produce models $V_1\subseteq V_2$ of set theory such that adding $\kappa$-many Cohen reals to $V_2$ adds $\lambda$-many Cohen reals to $V_1$, for some $\lambda >\kappa$. We deal mainly with the case when $V_1$ and $V_2$ have the same cardinals.