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

Lior Eldar; Peter W. Shor

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arXiv · 1611.06999

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

Lior Eldar·Peter W. Shor

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Abstract

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.

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Lior Eldar, Peter W. Shor. 2016-11-21. An Efficient Quantum Algorithm for a Variant of the Closest Lattice-Vector Problem. https://arxiv.org/abs/1611.06999

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