Tackling A Class of Hard Subset-Sum Problems: Integration of Lattice Attacks with Disaggregation Techniques

Abstract: Subset-sum problems belong to the NP class and play an important role in both complexity theory and knapsack-based cryptosystems, which have been proved in the literature to become hardest when the so-called density approaches one. Lattice attacks, which are acknowledged in the literature as the most effective methods, fail occasionally even when the number of unknown variables is of medium size. In this paper we propose a modular disaggregation technique and a simplified lattice formulation based on which two lattice attack algorithms are further designed. We introduce the new concept "jump points" in our disaggregation technique, and derive inequality conditions to identify superior jump points which can more easily cut-off non-desirable short integer solutions. Empirical tests have been conducted to show that integrating the disaggregation technique with lattice attacks can effectively raise success ratios to 100% for randomly generated problems with density one and of dimensions up to 100. Finally, statistical regressions are conducted to test significant features, thus revealing reasonable factors behind the empirical success of our algorithms and techniques proposed in this paper.