Solving Subset Sum Problems of Densioty close to 1 by "randomized" BKZ-reduction.
Claus-Peter Schnorr, Taras Shevchenko · IACR Cryptology ePrint Archive · 2012
Subset sum or Knapsack problems of dimension n are known to be hardest for knapsacks of density close to 1. These problems are NP-hard for arbitrary n. One can solve such problems either by lattice basis reduction or by optimized birthday algorithms. Recently Becker, Coron, Joux [BCJ10] present a birthday algorithm that follows Schroeppel, Shamir [SS81], and HowgraveGraham, Joux [HJ10]. This algorithm solves 50 random knapsacks of dimension 80 and density close to 1 in roughly 15 hours on a 2.67 GHz PC. We present an optimized lattice basis reduction algorithm that follows Schnorr, Euchner [SE03] using pruning of Schnorr, Horner [SH95] that solves such random knapsacks of dimension 80 on average in less than a minute, and 50 such problems all together about 9.4 times faster with less space than [BCJ10] on another 2.67 GHz PC.