An Improved Multi-set Algorithm for the Dense Subset Sum Problem

Given sets L1, . . . , Lk of elements from Z/mZ, the k-setbirthday problem is to find an element from each list such that theirsum is 0 modulo m. We give a new analysis of the algorithm in [16],proving that it returns a solution with high probability. By the workof Lyubashevsky [10], we get as an immediate corollary an improvedalgorithm for the random modular subset sum problem. Assuming themodulus m = 2nƐ for Ɛ < 1, this problem is now solvable using time andspace O(2nƐ/(1-Ɛ)log n).

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