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New Generic Algorithms for Hard Knapsacks

  • Nick Howgrave-Graham
  • Antoine Joux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6110)

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 \(\tilde{O}(2^{n/2})\) 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.

Keywords

Knapsack Problem Priority Queue Lattice Reduction Memory Complexity Early Abort 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nick Howgrave-Graham
    • 1
  • Antoine Joux
    • 2
  1. 1. Arlington
  2. 2.DGA and Université de Versailles Saint-Quentin-en-Yvelines, UVSQ PRISMVersailles cedexFrance

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