New transference theorems on lattices possessing nϵnϵ-unique shortest vectors

In this paper, we first discuss lattices possessing nϵnϵ-unique shortest vectors. We obtain three optimal transference theorems by establishing close relationships among successive minima, the covering radius and the minimal length of generating vectors. These results can be used to get finer reductions between GapSV Pγ′ and GapSIV Pγ for this class of lattices. Our work improves related results in the literature. In the second part of this paper, we prove a new transference theorem for general lattices where an optimal lower bound relating the successive minima of a lattice with its dual is given. As an application, we compare the respective advantages of current upper bounds on the smoothing parameters related to discrete Gaussian measures on lattices and give a more appropriate bound for lattices with duals possessing n-unique shortest vectors.