  Information flow is the branch of security that studies the leakage of
information due to correlation between secrets and observables. Since in
general such correlation cannot be avoided completely, it is important to
quantify the leakage. The most followed approaches to defining appropriate
measures are those based on information theory. In particular, one of the most
successful approaches is the recently proposed $g$-leakage framework, which
encompasses most of the information-theoretic ones. A problem with $g$-leakage,
however, is that it is defined in terms of a minimization problem, which, in
the case of large systems, can be computationally rather heavy. In this paper
we study the case in which the channel associated to the system can be
decomposed into simpler channels, which typically happens when the observables
consist of multiple components. Our main contribution is the derivation of
bounds on the (multiplicative version of) $g$-leakage of the whole system in
terms of the $g$-leakages of its components. We also consider the particular
cases of min-entropy leakage and of parallel channels, generalizing and
systematizing results from the literature. We demonstrate the effectiveness of
our method and evaluate the precision of our bounds using examples.
