Lattice coding for the Gaussian wiretap channel : a study of the secrecy gain

This thesis is dedicated to the Gaussian wiretap channel coding problem. In particular, lattice Gaussian wiretap codes are considered with respect to a new lattice coding invariant called secrecy gain defined as the maximum of the secrecy function, which characterizes the amount of confusion a lattice can cause at the eavesdropper. The weak secrecy gain is the value of the secrecy function achieved at its symmetry point, conjectured to be maximum, namely, the secrecy gain. The secrecy gain of unimodular lattices, i.e. lattices that are eqeal to their duals, in dimensions less than 24 are computed and the best lattice in each dimension is classified. The computation relies on the study of modular forms, a classical object in analytic number theory. Contemporary subject such as coding theory is also exploited in the computation. The weak secrecy gain of 2- and 3-modular lattices, lattices that are similar to their duals, in small dimensions are computed to compare with the best unimodular lattices. The data shows that, at least in small dimensions, 2- and 3-modular lattices typically have a bigger secrecy gain and hence give more secure lattice Gaussian wiretap codes. Modular lattices are constructed from the ring of integers of algebraic number fields. The computation in this part involves basic algebraic number theory, on top of heavy modular form theory and coding theory. Bounds on the weak secrecy gain of unimodular lattices are studied and the result shows that to maximize the weak secrecy gain of a unimodular lattice of dimension n, one has only to minimize the number of vectors with norm no bigger than ⌊n/ 8⌋ This facilitates a way to bound the weak secrecy gain of unimodular lattices by counting the number of their small norm vectors.​Doctor of Philosophy (SPMS