Approximate common divisors via lattices

TL;DR

This paper extends the analysis of lattice-based algorithms for the approximate common divisor problem to multiple variables, improving error tolerance bounds and connecting cryptanalysis with coding theory, including list decoding of advanced codes.

Contribution

It provides a multivariate generalization of Howgrave-Graham's algorithm, offering improved error bounds and a novel link between cryptanalysis and list decoding of multivariate codes.

Findings

Improved error tolerance bounds for multivariate approximate common divisors.

Established a lattice-based list decoding approach for multivariate codes.

Demonstrated practical performance surpassing theoretical expectations.

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 N^beta, this improves the size of the error tolerated from N^(beta^2) to N^(beta^((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 2^(n^epsilon) approximation algorithm with epsilon<2/3 for lattice basis reduction in n dimensions could be used to break these parameters. We have implemented our 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 Parvaresh-Vardy 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.