Division algorithms for the fixed weight subset sum problem

TL;DR

This paper introduces new exponential algorithms for the fixed weight subset sum problem that do not depend on lattice methods, using splitting systems and a k-set birthday approach, with efficient space and parallelization features.

Contribution

The paper presents lattice-free exponential algorithms for the fixed weight subset sum problem based on splitting systems and a k-set birthday method, applicable when lattice techniques fail.

Findings

Algorithms have time and space complexity satisfying T * S^{log k} = O(n choose l).

The algorithms are highly parallelizable.

Applicable to cryptographic problems where lattice methods are ineffective.

Abstract

Given positive integers $a_1,..., a_n, t$, the fixed weight subset sum problem is to find a subset of the $a_i$ that sum to $t$, where the subset has a prescribed number of elements. It is this problem that underlies the security of modern knapsack cryptosystems, and solving the problem results directly in a message attack. We present new exponential algorithms that do not rely on lattices, and hence will be applicable when lattice basis reduction algorithms fail. These algorithms rely on a generalization of the notion of splitting system given by Stinson. In particular, if the problem has length $n$ and weight $\ell$ then for constant $k$ a power of two less than $n$ we apply a $k$-set birthday algorithm to the splitting system of the problem. This randomized algorithm has time and space complexity that satisfies $T \cdot S^{\log{k}} = O({n \choose \ell})$ (where the constant depends uniformly on $k$). In addition to using space efficiently, the algorithm is highly parallelizable.