Paper 2025/1376

On Hull Attacks on the Module Lattice Isomorphism Problem

Abstract

The Lattice Isomorphism Problem (LIP) is a relatively recent cryptographic assumption whose precise hardness remains not fully understood. Certain weak instances have been identified through hull attacks on $p$-ary lattices constructed via Construction A using linear codes with trivial hulls. In this work, we generalize the notion of the hull by introducing ideal-based hulls for Hermitian lattices. We propose a new hull attack targeting lattices derived from Generalized Construction A over number fields, under specific structural conditions. Furthermore, we show that modular lattices offer intrinsic resistance to hull attacks: the hull introduces only a limited variation in the lattice gap, bounded by a factor depending on the root discriminant of the number field. In particular, for modular $\mathbb{Z}$-lattices, the hull gap coincides exactly with the original lattice gap. As a concrete example, we show that the family of Barnes-Wall lattices, which are alternatively unimodular and 2-modular over $\mathbb{Z}$, are resistant to hull attacks.