Solving random subset sum problem by lp -norm SVP oracle

Abstract

It is well known that almost all random subset sum instances with density less than 0.6463. can be solved with an l2-norm SVP oracle by Lagarias and Odlyzko. Later, Coster et al. improved the bound to 0.9408. by using a different lattice. In this paper, we generalize this classical result to lp-norm. More precisely, we show that for p ∈ ℤ+, an lp-norm SVP oracle can be used to solve almost all random subset sum instances with density bounded by δp, where δ1=0.5761 and for δp=1/(1/2p log (2 p+1-2)+log2(1+1/2(2p-1)(1-(1/2p+1-2)(2p-1))))) for p ≥ 3 (asymptotically, δ p ≈2 p /(p+2)). Since δp goes increasingly to infinity when p tends to infinity, it can be concluded that an lp -norm SVP oracle with bigger p can solve more subset sum instances. An interesting phenomenon is that an lp-norm SVP oracle with ≥3 can help solve almost all random subset sum instances with density one, which are thought to be the most difficult instances. © 2014 Springer-Verlag Berlin Heidelberg.