Solving Low Density Knapsacks

Abstract

Let al,...,an and s be a set of integers. The knapsack (or subset sum) problem is to find a 0--1 vector ($ε$l,...,$ε$n) such that $Σ$ $ε$iai = s or to show that such a vector does not exist. The integers al,...,an are sometimes referred to as weights. The general knapsack problem is known to be NP complete [5,6]. Several cryptosystems based on the knapsack problem have been designed [9,12,16]. In April, 1982, Adi Shamir [14] announced a method for breaking the Merkle-Hellman cryptosystem. Since that time there has been a flurry of activity to extend his results to include all of the proposed knapsack based cryptosystems [1,2,3,7,13].