Provably Solving the Hidden Subset Sum Problem via Statistical Learning

At Crypto ’99, Nguyen and Stern described a lattice based algorithm for solving the hidden subset sum problem, a variant of the classical subset sum problem where the n weights are also hidden. As an application, they showed how to break the Boyko et al. fast generator of random pairs (x, g x(mod p)). The Nguyen-Stern algorithm works quite well in practice for moderate values of n, but its complexity is exponential in n. A polynomial-time variant was recently described at Crypto 2020, based on a multivariate technique, but the approach is heuristic only. In this paper, we describe a proven polynomial-time algorithm for solving the hidden subset-sum problem, based on statistical learning. In addition, we show that the statistical approach is also quite efficient in practice: using the FastICA algorithm, we can reach n = 250 in reasonable time.