On Lattice Isomorphism Problems for Lattices from LCD Codes over Finite Rings

These days, post-quantum cryptography based on the lattice isomorphism problem has been proposed. Ducas-Gibbons introduced the hull attack, which solves the lattice isomorphism problem for lattices obtained by Construction A from an LCD code over a finite field. Using this attack, they showed that the lattice isomorphism problem for such lattices can be reduced to the lattice isomorphism problem with the trivial lattice ${\mathbb{Z}}^n$ and the graph isomorphism problem. While the previous work by Ducas-Gibbons only considered lattices constructed by a code over a $\mathit{\text{finite}}\mathit{\text{field}}$, this paper considers lattices constructed by a code over a $\mathit{\text{finite}}\mathit{\text{ring}}$ $\mathbb{Z}\mathrm{/}k\mathbb{Z}$, which is a more general case. In particular, when $k$ is odd, an odd prime power, or not divisible by $4$, we show that the lattice isomorphism problem can be reduced to the lattice isomorphism problem for ${\mathbb{Z}}^n$ and the graph isomorphism problem.