  Symmetrical Multilevel Diversity Coding (SMDC) is a network compression
problem introduced by Roche (1992) and Yeung (1995). In this setting, a simple
separate coding strategy known as superposition coding was shown to be optimal
in terms of achieving the minimum sum rate (Roche, Yeung, and Hau, 1997) and
the entire admissible rate region (Yeung and Zhang, 1999) of the problem. This
paper considers a natural generalization of SMDC to the secure communication
setting with an additional eavesdropper. It is required that all sources need
to be kept perfectly secret from the eavesdropper as long as the number of
encoder outputs available at the eavesdropper is no more than a given
threshold. First, the problem of encoding individual sources is studied. A
precise characterization of the entire admissible rate region is established
via a connection to the problem of secure coding over a three-layer wiretap
network and utilizing some basic polyhedral structure of the admissible rate
region. Building on this result, it is then shown that superposition coding
remains optimal in terms of achieving the minimum sum rate for the general
secure SMDC problem.
