Approximate common divisors via lattices

Abstract

We analyze the multivariate generalization of Howgrave-Graham’s algorithm for the approximate common divisor problem. In the m-variable case with modulus N and approximate common divisor of size Nβ , this improves the size of the error tolerated from Nβ 2 to Nβ (m+1)/m , under a commonly used heuristic assumption. This gives a more detailed analysis of the hardness assumption underlying the recent fully homomorphic cryptosystem of van Dijk, Gentry, Halevi, and Vaikuntanathan. While these results do not challenge the suggested parameters, a 2n e approximation algorithm with e < 2/3 for lattice basis reduction in n dimensions could be used to break these parameters. We have implemented the algorithm, and it performs better in practice than the theoretical analysis suggests. Our results fit into a broader context of analogies between cryptanalysis and coding theory. The multivariate approximate common divisor problem is the number-theoretic analogue of multivariate polynomial reconstruction, and we develop a corresponding lattice-based algorithm for the latter problem. In particular, it specializes to a lattice-based list decoding algorithm for ParvareshVardy and Guruswami-Rudra codes, which are multivariate extensions of Reed-Solomon codes. This yields a new proof of the list decoding radii for these codes.