Lossless Secure Source Coding: Yamamoto's Setting

Given a private source of information, $X^{n}$ and a public correlated source, $Y^{n}$, we study the problem of encoding the two-dimensional source $(X^{n}, Y^{n})$ into an index $J$ such that a remote party, knowing $J$ and some external side information $Z^{n}$, can losslessly recover $Y^{n}$ while any eavesdropper knowing $J$ and possibly a correlated side information $E^{n}$ can retrieve very little information about $X^{n}$. We give general converse results for the amount of information about $X^{n}$ that might be leaked in such systems and and also achievability results that are optimal in some special cases.