A wireless relay network consists of multiple source nodes, multiple destination nodes, and possibly many relay nodes in between to facilitate its transmission. It is clear that the performance of such networks highly depends on information for- warding strategies adopted at the relay nodes. This dissertation studies a particular information forwarding strategy called compute-and-forward. Compute-and-forward is a novel paradigm that tries to incorporate the idea of network coding within the physical layer and hence is often referred to as physical layer network coding. The main idea is to exploit the superposition nature of the wireless medium to directly compute or decode functions of transmitted signals at intermediate relays in a net- work. Thus, the coding performed at the physical layer serves the purpose of error correction as well as permits recovery of functions of transmitted signals. For the bidirectional relaying problem with Gaussian channels, it has been shown by Wilson et al. and Nam et al. that the compute-and-forward paradigm is asymptotically optimal and achieves the capacity region to within 1 bit; however, similar results beyond the memoryless case are still lacking. This is mainly because channels with memory would destroy the lattice structure that is most crucial for the compute-and-forward paradigm. Hence, how to extend compute-and-forward to such channels has been a challenging issue. This motivates this study of the extension of compute-and-forward to channels with memory, such as inter-symbol interference. The bidirectional relaying problem with parallel Gaussian channels is also studied, which is a relevant model for the Gaussian bidirectional channel with inter-symbol interference and that with multiple-input multiple-output channels. Motivated by the recent success of linear finite-field deterministic model, we first investigate the corresponding deterministic parallel bidirectional relay channel and fully characterize its capacity region. Two compute-and-forward schemes are then proposed for the Gaussian model and the capacity region is approximately characterized to within a constant gap. The design of coding schemes for the compute-and-forward paradigm with low decoding complexity is then considered. Based on the separation-based framework proposed previously by Tunali et al., this study proposes a family of constellations that are suitable for the compute-and-forward paradigm. Moreover, by using Chinese remainder theorem, it is shown that the proposed constellations are isomorphic to product fields and therefore can be put into a multilevel coding framework. This study then proposes multilevel coding for the proposed constellations and uses multistage decoding to further reduce decoding complexity.

In this work, we study two problems in information-theoretic security. Firstly, we study a wireless network where two nodes want to securely exchange messages via an honest-but-curious bidirectional relay. There is no direct link between the user nodes, and all communication must take place through the relay. The relay behaves like a passive eavesdropper, but otherwise follows the protocol it is assigned. Our objective is to design a scheme where the user nodes can reliably exchange messages such that the relay gets no information about the individual messages. We first describe a perfectly secure scheme using nested lattices, and show that our scheme achieves secrecy regardless of the distribution of the additive noise, and even if this distribution is unknown to the user nodes. Our scheme is explicit, in the sense that for any pair of nested lattices, we give the distribution used for randomization at the encoders to guarantee security. We then give a strongly secure lattice coding scheme, and we characterize the performance of both these schemes in the presence of Gaussian noise. We then extend our perfectly-secure and strongly-secure schemes to obtain a protocol that guarantees end-to-end secrecy in a multichip line network. We also briefly study the robustness of our bidirectional relaying schemes to channel imperfections. In the second problem, we consider the scenario where multiple terminals have access to private correlated Gaussian sources and a public noiseless communication channel. The objective is to generate a group secret key using their sources and public communication in a way that an eavesdropper having access to the public communication can obtain no information about the key. We give a nested lattice-based protocol for generating strongly secure secret keys from independent and identically distributed copies of the correlated random variables. Under certain assumptions on the joint distribution of the sources, we derive achievable secret key rates. The tools used in designing protocols for both these problems are nested lattice codes, which have been widely used in several problems of communication and security. In this thesis, we also study lattice constructions that permit polynomial-time encoding and decoding. In this regard, we first look at a class of lattices obtained from low-density parity-check (LDPC) codes, called Low-density Construction-A (LDA) lattices. We show that high-dimensional LDA lattices have several “goodness” properties that are desirable in many problems of communication and security. We also present a new class of low-complexity lattice coding schemes that achieve the capacity of the AWGN channel. Codes in this class are obtained by concatenating an inner Construction-A lattice code with an outer Reed-Solomon code or an expander code. We show that this class of codes can achieve the capacity of the AWGN channel with polynomial encoding and decoding complexities. Furthermore, the probability of error decays exponentially in the block length for a fixed transmission rate R that is strictly less than the capacity. To the best of our knowledge, this is the first capacity-achieving coding scheme for the AWGN channel which has an exponentially decaying probability of error and polynomial encoding/decoding complexities.