We consider a communication system where two transmitters wish to exchange information through a central relay. The data is assumed to be transmitted over synchronized, average power constrained additive white Gaussian noise channels with a real input with signal-to-noise ratio (SNR) of snr. It has been shown that using lattice codes and lattice decoding, a rate of 1/2 log_2(1/2 plus snr) can be obtained asymptotically, which is essentially optimal at high SNR. However, there has been a lack of practical encoding/decoding schemes for the above mentioned system. We address this issue in this thesis by developing encoding/decoding strategies for the bidirectional relaying system using low-dimensional lattice codes. Our efforts are aimed at developing coding schemes which possess low computational complexity while at the same time providing good performance. We demonstrate two schemes using low-dimensional lattice codes. Both these schemes have their own advantages and are suitable for different classes of lattice codes. The two schemes are tested with different lattices and their performance is compared to that of other schemes for bidirectional relays. The first scheme is termed as demodulate and forward and it essentially consists of performing optimal estimation at the relay. It is primarily implemented with lattice codes of low rate and possesses low decoding complexity. When used with a two-dimensional hexagonal lattice, it achieves a gain of around 3.5 dB in comparison to other schemes like Analog network coding. The second scheme is the sphere decoding scheme which has been implemented with high-rate lattice codes. The sphere decoder is a low-complexity decoder which is used for decoding to a lattice point at the relay. We observe that as the dimensionality of the lattice code is increased, the performance of the sphere decoder for the bidirectional relay gets consistently better. The sphere decoder is also used at high SNR for those instances in which the low density lattice code(LDLC) decoder makes an error and it is found that the sphere decoder can correct around 90 percent of these errors at an SNR of 9.75 dB.

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.