Lossless secure source coding: Yamamoto´s setting

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