On (Non-)Isomorphism of Self-Dual Lattices and Codes

2026-06-17

AI Summary

This work investigates the computational complexity of the Lattice Isomorphism Problem (LIP) for self-dual lattices and the Permutation Code Equivalence (PCE) problem for self-dual codes. By leveraging the decomposition structure of self-dual lattices, properties of the mod-2 inner product of shortest characteristic vectors, and randomized algorithmic techniques, the study extends the known algorithms and coNP verification protocols for LIP to a broader class of self-dual lattices. The main contributions include a randomized algorithm for LIP with runtime $2^{n/2 + o(n)}$, a coNP verification protocol for LIP over general self-dual lattices, and a polynomial-time algorithm for PCE on a class of self-dual codes possessing large kernels.

Abstract

A recent line of work motivated by cryptographic applications has studied the complexity of the Lattice Isomorphism Problem (LIP). In this work, we study LIP on self-dual lattices $\cal{L} \subset \mathbb{R}^n$, which appear naturally in many applications. Our main results are a $2^{n/2 + o(n)}$-time randomized algorithm for LIP and a $\mathsf{coNP}$ protocol for LIP on a broad class of self-dual lattices. These results extend recent work on ZLIP, the problem of deciding whether a lattice is isomorphic to $\mathbb{Z}^n$. In particular, the former result extends the $2^{n/2 + o(n)}$-time algorithms for ZLIP of Bennett, Ganju, Peetathawachai, and Stephens-Davidowitz (Eurocrypt, 2023) and of Ducas (Des. Codes Cryptogr., 2024). The latter result extends the $\mathrm{ZLIP} \in \mathsf{coNP}$ result of Hunkenschröder (Math. Prog. Series A, 2024). Our results leverage two key structural properties of self-dual lattices $\cal{L} \subset \mathbb{R}^n$: (1) every such lattice $\cal{L}$ is isomorphic to $\cal{L}_0 \oplus \mathbb{Z}^r$ for some self-dual lattice $\cal{L}_0$ with $λ_1(\cal{L}_0)^2 \geq 2$, and (2) every such lattice $\cal{L}$ has \emph{characteristic vectors}, i.e., there exist vectors $\mathbf{w} \in \cal{L}$ such that for every $\mathbf{v} \in \cal{L}$, $\langle\mathbf{v}, \mathbf{w}\rangle \equiv \langle\mathbf{v}, \mathbf{v}\rangle \pmod{2}$. Our results use a line of work by Elkies and Gaulter on lattices with long shortest characteristic vectors, and can be strengthened assuming a positive answer to a related question of Elkies (Math. Res. Lett., 1995). We also study Permutation Code Equivalence (PCE) on self-dual codes, and we observe that similar structural properties imply a polynomial-time algorithm for PCE on certain such codes. This gives a natural class of codes with large hull for which PCE is easy.

Problem

• Lattice Isomorphism Problem
• Self-dual lattices
• Permutation Code Equivalence
• Self-dual codes
• Isomorphism

Innovation

• Self-dual lattices
• Lattice Isomorphism Problem
• Characteristic vectors
• coNP protocol
• Permutation Code Equivalence