International Association
for Cryptologic Research

We prove three optimal transference theorems on lattices possessing $n^{\\epsilon}$-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 GapSVP$_{\\gamma\'}$ and GapSIVP$_\\gamma$ 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 $\\sqrt{n}$-unique shortest vectors.