Secure symmetrical multilevel diversity coding (S-SMDC) is a source coding problem, where a total of L - N discrete memoryless sources (S1,...,S_L-N) are to be encoded by a total of L encoders. This thesis considers a natural generalization of SMDC to the secure communication setting with an additional eavesdropper. In a general S-SMDC system, a legitimate receiver and an eavesdropper have access to a subset U and A of the encoder outputs, respectively. Which subsets U and A will materialize are unknown a priori at the encoders. No matter which subsets U and A actually occur, the sources (S1,...,Sk) need to be perfectly reconstructable at the legitimate receiver whenever |U| = N +k, and all sources (S1,...,S_L-N) need to be kept perfectly secure from the eavesdropper as long as |A| <= N. 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 properties of 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.

Symmetrical multilevel diversity coding (SMDC) is a classical model for coding over distributed storage. 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 and the entire admissible rate region of the problem in the literature. The proofs utilized carefully constructed induction arguments, for which the classical subset entropy inequality of Han played a key role. This thesis includes two parts. In the first part the existing optimality proofs for classical SMDC are revisited, with a focus on their connections to subset entropy inequalities. First, a new sliding-window subset entropy inequality is introduced and then used to establish the optimality of superposition coding for achieving the minimum sum rate under a weaker source-reconstruction requirement. Second, a subset entropy inequality recently proved by Madiman and Tetali is used to develop a new structural understanding to the proof of Yeung and Zhang on the optimality of superposition coding for achieving the entire admissible rate region. Building on the connections between classical SMDC and the subset entropy inequalities developed in the first part, in the second part the optimality of superposition coding is further extended to the cases where there is an additional all-access encoder, an additional secrecy constraint or an encoder hierarchy.

Symmetrical Multilevel Diversity Coding (SMDC) is a network compression problem for which a simple separate coding strategy known as superposition coding is optimal in terms of achieving the entire admissible rate region. Carefully constructed induction argument along with the classical subset entropy inequality of Han played a key role in proving the optimality. This thesis considers a generalization of SMDC for which, in addition to the randomly accessible encoders, there is also an all-access encoder. It is shown that superposition coding remains optimal in terms of achieving the entire admissible rate region of the problem. Key to our proof is to identify the supporting hyperplanes that define the boundary of the admissible rate region and then build on a generalization of Han's subset inequality. As a special case, the (R0,Rs) admissible rate region, which captures all possible tradeoffs between the encoding rate, R0, of the all-access encoder and the sum encoding rate, Rs, of the randomly accessible encoders, is explicitly characterized. To provide explicit proof of the optimality of superposition coding in this case, a new sliding-window subset entropy inequality is introduced and is shown to directly imply the classical subset entropy inequality of Han.

Secure communication under channel uncertainty is an important and challenging problem in physical-layer security and cryptography. In this dissertation, we take a fundamental information-theoretic view at three concrete settings and use them to shed insight into efficient secure communication techniques for different scenarios under channel uncertainty. First, a multi-input multi-output (MIMO) Gaussian broadcast channel with two receivers and two messages: a common message intended for both receivers (i.e., channel uncertainty for decoding the common message at the receivers) and a confidential message intended for one of the receivers but needing to be kept asymptotically perfectly secret from the other is considered. A matrix characterization of the secrecy capacity region is established via a channel-enhancement argument and an extremal entropy inequality previously established for characterizing the capacity region of a degraded compound MIMO Gaussian broadcast channel. Second, a multilevel security wiretap channel where there is one possible realization for the legitimate receiver channel but multiple possible realizations for the eavesdropper channel (i.e., channel uncertainty at the eavesdropper) is considered. A coding scheme is designed such that the number of secure bits delivered to the legitimate receiver depends on the actual realization of the eavesdropper channel. More specifically, when the eavesdropper channel realization is weak, all bits delivered to the legitimate receiver need to be secure. In addition, when the eavesdropper channel realization is strong, a prescribed part of the bits needs to remain secure. We call such codes security embedding codes, referring to the fact that high-security bits are now embedded into the low-security ones. We show that the key to achieving efficient security embedding is to jointly encode the low-security and high-security bits. In particular, the low-security bits can be used as (part of) the transmitter randomness to protect the high-security ones. Finally, motivated by the recent interest in building secure, robust and efficient distributed information storage systems, the problem of secure symmetrical multilevel diversity coding (S-SMDC) is considered. This is a setting where there are channel uncertainties at both the legitimate receiver and the eavesdropper. The problem of encoding individual sources is first 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 the simple coding strategy of separately encoding individual sources at the encoders can achieve the minimum sum rate for the general S-SMDC problem.