Secure Symmetrical Multilevel Diversity Coding

Symmetrical multilevel diversity coding (SMDC) is a network compression problem introduced by Roche (1992) and Yeung (1995). In this setting, a simple separate encoding strategy known as superposition coding was shown to be optimal in terms of achieving the minimum sum rate (Roche-Yeung-Hau 1997) and the entire admissible rate region (Yeung-Zhang 1999) of the general 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 ramp-type secret sharing (Yamamoto 1985 and Blakley-Meadows 1985) 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.