Paper 2010/189

New generic algorithms for hard knapsacks

Nick Howgrave-Graham and Antoine Joux

Abstract

In this paper, we study the complexity of solving hard knapsack problems, i.e., knapsacks with a density close to $1$ where lattice-based low density attacks are not an option. For such knapsacks, the current state-of-the-art is a 31-year old algorithm by Schroeppel and Shamir which is based on birthday paradox techniques and yields a running time of $$ for knapsacks of $$ elements and uses $$ storage. We propose here two new algorithms which improve on this bound, finally lowering the running time down to $$ for almost all knapsacks of density $$. We also demonstrate the practicality of these algorithms with an implementation.