Paper 2025/2201

On $k$-sum algorithms for $\{-1,1\}^m$ vectors

Abstract

We study a new variant of the $k$-sum problem. In the classical formulation, one is given $k$ lists of binary vectors and must find one vector from each list such that their sum is the zero vector. In our variant, vectors are instead chosen from the set $\{-1,1\}^m$. First, we analyse how the complexity of the original problem changes when one use $\pm 1$ vectors. We show that this variant achieves better than square-root complexity when using more than two lists, although its overall complexity for $k \geq 4$ is worse than that for the binary case. As an application of our algorithms, we present a combinatorial attack strategy for a recently proposed variant of the Inhomogeneous Short Integer Solution (ISIS) problem known as the randomised one-more ISIS assumption. Our combinatorial attack is more effective than previous combinatorial attacks on this problem. We further consider a generalised setting of the $k$-sum problem in which the target sum is not the zero vector, but a given integer vector $Y \in \mathbb{Z}_p^m$ that is known to be a sum of $\pm 1$ vectors. We study the complexity of reconstructing such a decomposition of $Y$ using $k$ lists of candidate vectors. Our analysis shows that, as the number of lists increases, the problem becomes solvable in approximately cube-root time.