An Efficient Quantum Algorithm for a Variant of the Closest Lattice-Vector Problem

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 = λ_1(L)/n^{17}, a = λ_1(L)/n^{13} decide if v's distance from L is in the range [a/2, a] or at most b, where λ_1(L) is the length of L's shortest non-zero vector. Improving these parameters to a = b = λ_1(L)/\sqrt{n} would invalidate one of the security assumptions of the Learning-with-Errors (LWE) cryptosystem against quantum attacks.