Paper 2025/1852 Original: eprint.iacr.org/2025/1852 ↗

A Gaussian Leftover Hash Lemma for Modules over Number Fields

Abstract

Leftover Hash Lemma (LHL) states that \(\mathbf{X} \cdot \mathbf{v}\) for a Gaussian \(\mathbf{v}\) is an essentially independent Gaussian sample. It has seen numerous applications in cryptography for hiding sensitive distributions of \(\mathbf{v}\). We generalise the Gaussian LHL initially stated over \(\mathbb{Z}\) by Agrawal, Gentry, Halevi, and Sahai (2013) to modules over number fields. Our results have a sub-linear dependency on the degree of the number field and require only polynomial norm growth: \(\lVert\mathbf{v}\rVert/\lVert\mathbf{X}\rVert\). To this end, we also prove when \(\mathbf{X}\) is surjective (assuming the Generalised Riemann Hypothesis) and give bounds on the smoothing parameter of the kernel of \(\mathbf{X}\). We also establish when the resulting distribution is independent of the geometry of \(\mathbf{X}\) and establish the hardness of the \(k\)-SIS and \(k\)-LWE problems over modules (\(k\)-MSIS/\(k\)-MLWE) based on the hardness of SIS and LWE over modules (MSIS/MLWE) respectively, which was assumed without proof in prior works.