Well-Rounded Lattices and Applications to Physical Layer Security

This thesis is devoted to the analysis and construction of well-rounded lattices. The main motivation for studying these objects comes from problems in wireless communications. In particular, we study lattice codes for wiretap channels in the context of physical layer security. The set of well-rounded lattices has been investigated from topological, geometric and arithmetic point of views. We contribute to this study by first answering various questions about well-rounded lattices arising from real quadratic fields. Then, motivated by number theoretic constructions, we derive a more general generic construction of lattices generated by their minimal vectors in every dimension. The last part of the thesis is dedicated to analysing the performance of lattice codes carved from well-rounded lattices. We then motivate the use of such codes over fading channels to achieve both reliable and secure communication. We show that these codes provide nearly optimal solutions in terms of reliability for both additive white Gaussian noisy channels and Rayleigh fading channels. For the security part, we led an experimental and analytical study of well-rounded lattice codes on the wiretap channel. We conclude that, similarly as well-rounded lattices provide maximisers of lattice sphere packings, they also provide excellent candidates for minimising the eavesdroppers' chances of intercepting a transmitted message.