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

TL;DR

This paper extends the study of lattice isomorphism problems from codes over finite fields to codes over finite rings, showing reductions to known problems in specific cases, thus broadening the understanding of post-quantum cryptography security.

Contribution

It generalizes the reduction of lattice isomorphism problems from finite field codes to finite ring codes, covering more cases where the ring order is odd or not divisible by 4.

Findings

Reduces lattice isomorphism problem for codes over finite rings to known problems in certain cases.

Extends previous work from finite fields to finite rings, including $ ext{Z}/k ext{Z}$ with specific conditions.

Provides a broader framework for analyzing lattice isomorphism problems in post-quantum cryptography.

Abstract

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 \textit{finite field}, this paper considers lattices constructed by a code over a \textit{finite ring} $\mathbb{Z}/k\mathbb{Z}$, 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.