  In literature, different common informations were defined by G\'acs and
K\"orner, by Wyner, and by Kumar, Li, and Gamal, respectively. In this paper,
we define two generalized versions of common informations, named approximate
and exact information-correlation functions, by exploiting the conditional
maximal correlation as a commonness or privacy measure. These two generalized
common informations encompass the notions of G\'acs-K\"orner's, Wyner's, and
Kumar-Li-Gamal's common informations as special cases. Furthermore, to give
operational characterizations of these two generalized common informations, we
also study the problems of private sources synthesis and common information
extraction, and show that the information-correlation functions are equal to
the minimum rates of commonness needed to ensure that some conditional maximal
correlation constraints are satisfied for the centralized setting versions of
these problems. As a byproduct, the conditional maximal correlation has been
studied as well.
