On Random High Density Subset Sums

In the Subset Sum problem, we are given n integers a1, . . . , an and a target number t, and are asked to find the subset of the ai’s such that the sum is t. A version of the subset sum problem is the Random Modular Subset Sum problem. In this version, the ai’s are generated randomly in the range [0,M), and we are asked to produce a subset of them such that the sum is t(modM). The hardness of RMSS depends on the relationship between the parameters M and n. When M = 2 2), RMSS can be solved in polynomial time by a reduction to the shortest vector problem. When M = 2 , the problem can be solved in polynomial time by dynamic programming, and recently an algorithm was proposed that solves the problem in polynomial time for M = 2 2 . In this work, we present an algorithm that solves the Random Modular Subset Sum problem for parameter M = 2 for < 1 in time (and space) 2 n log n . As far as we know, this is the first algorithm that performs in time better than 2 ) for arbitrary < 1.