Advances in Cryptology – EUROCRYPT 2010

Volume 6110 of the series Lecture Notes in Computer Science pp 235-256

New Generic Algorithms for Hard Knapsacks

  • Nick Howgrave-GrahamAffiliated with 
  • , Antoine JouxAffiliated withDGA and Université de Versailles Saint-Quentin-en-Yvelines, UVSQ PRISM


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 \(\tilde{O}(2^{n/2})\) for knapsacks of n elements and uses \(\tilde{O}(2^{n/4})\) storage. We propose here two new algorithms which improve on this bound, finally lowering the running time down to either \(\tilde{O} (2^{0.385\, n})\) or \(\tilde{O} (2^{0.3113\, n})\) under a reasonable heuristic. We also demonstrate the practicality of these algorithms with an implementation.