Paper 2018/741 Original: eprint.iacr.org/2018/741 ↗

LP Solutions of Vectorial Integer Subset Sums - Cryptanalysis of Galbraith's Binary Matrix LWE

Abstract

We consider Galbraith's space efficient LWE variant, where the $(m \times n)$-matrix $A$ is binary. In this binary case, solving a vectorial subset sum problem over the integers allows for decryption. We show how to solve this problem using (Integer) Linear Programming. Our attack requires only a fraction of a second for all instances in a regime for $m$ that cannot be attacked by current lattice algorithms. E.g.\ we are able to solve 100 instances of Galbraith's small LWE challenge $(n,m) = (256, 400)$ all in a fraction of a second. We also show under a mild assumption that instances with $m \leq 2n$ can be broken in polynomial time via LP relaxation. Moreover, we develop a method that identifies weak instances for Galbraith's large LWE challenge $(n,m)=(256, 640)$.