New transference theorems on lattices possessing n∈-unique shortest vectors

We prove three optimal transference theorems on lattices possessing n -unique shortest vectors which relate to the successive minima, the covering radius and the minimal length of generating vectors respectively. The theorems result in reductions between GapSVPγ′ and GapSIVPγ for this class of lattices. Furthermore, we prove a new transference theorem giving an optimal lower bound relating the successive minima of a lattice with its dual. As an application, we compare the respective advantages of current upper bounds on the smoothing parameter of discrete Gaussian measures over lattices and show a more appropriate bound for lattices whose duals possess √ n-unique shortest vectors.