On étudie le problème de la transmission de l'information à travers le canal AWGN en utilisant des réseaux. On commence par considérer des constellations infinies. Une nouvelle famille de réseaux obtenus par Construction A à partir de codes linéaires non binaires est proposée. Ces réseaux sont appelés LDA ("Low-Density Construction A") et sont caractérisés par des matrices de parité p-aires creuses, qui les mettent en relation directe avec les codes LPDC. Deux résultats sur leur possibilité d'atteindre la capacité de Poltyrev sont provés ; cela est d'abord démontré pour des poids des lignes logarithmiques des matrices de parité associées, puis pour des poids constants. Le deuxième résultat est basé sur certaines propriétés d'expansion des graphes de Tanner correspondants à ces matrices. Un autre sujet de ce travail concerne les constellations finies de réseaux. une nouvelle preuve est donnée du fait que des réseaux aléatoires obtenus par Construction A generale atteignent la capacité avec décodage de type "lattice decoding". Cela prolonge et améliore le travail de Erez et Zamir (2004), Ordentlich et Edrez (2012) Ling et Belfiore (2013). Cette preuve est basée sur les constellations de Coronoï et la multiplication par le coefficient de Wiener ("MMSE scaling") du siganl en sortie du canal. Finalement, ce résultat est adapté au cas des réseaux LDA, qui eux aussi atteignent la capacité avec le même procédé de transmission. Encore une fois, il est nécessaire d'exploiter les propriétés d'expansion des graphes de Tanner. A la fin de la dissertation, on présente un algorithme de décodage itératif et de type "message-passing" approprié au décodage des LDA en grandes dimensions. / The probleme of transmission of information over the AWGN channel using lattices is addressed. Firstly, infinite constellations are considered. A nex family of integer lattices built by means of construction A with non-binary linear condes is introduced. These lattices are called LPA (Low-Density Construction A) and are characterised by sparse p-ary parity-chedk matrices, that put them in direct relation with LPDC codes. Two results about the Poltyrev-capacity-archieving qualities of this family are proved, respectively for logarithmic row degree and constant row degree of the associated parity-check matrices. The second result is based on some expansion poperties of the Tanner graphs related to these matrices. Another topic of this work concerns finite lattice constellations. A new proff that heneral random Construction A lattices achieve capacity under lattice deconding is provided, continuing and pimproving the work of Erez and Zamir (2004), Ordentlich an Erez (2012), and Ling and Belfiore (2013). This proof is based on Voronoi lattice constellations and MMSE scaling of the channel output. Finally, this approach is adapted to the LDA case abd ut us scgiwn tgat LDA lattices achive capacity with the ame transmission scheme, too. Once again, it is necessary to exploit the expansion properties of the Tanner graphs. At he end of the dissertation, an iterative message-passing algorithm suitable for decoding LDA lattices in high dimensions is presented.

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.