  The Systematic Normal Form (SysNF) is a canonical form of lattices introduced
in [Eldar,Shor '16], in which the basis entries satisfy a certain co-primality
condition. Using a "smooth" analysis of lattices by SysNF lattices we design a
quantum algorithm that can efficiently solve the following variant of the
bounded-distance-decoding problem: given a lattice L, a vector v, and numbers b
= {\lambda}_1(L)/n^{17}, a = {\lambda}_1(L)/n^{13} decide if v's distance from
L is in the range [a/2, a] or at most b, where {\lambda}_1(L) is the length of
L's shortest non-zero vector. Improving these parameters to a = b =
{\lambda}_1(L)/\sqrt{n} would invalidate one of the security assumptions of the
Learning-with-Errors (LWE) cryptosystem against quantum attacks.
