On Lattice Isomorphism Problems for Lattices from LCD Codes over Finite Rings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918124760072192 |
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| author | Nishimura, Yusaku Takashima, Katsuyuki Miezaki, Tsuyoshi |
| author_facet | Nishimura, Yusaku Takashima, Katsuyuki Miezaki, Tsuyoshi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Lattice Isomorphism Problems for Lattices from LCD Codes over Finite Rings Nishimura, Yusaku Takashima, Katsuyuki Miezaki, Tsuyoshi Information Theory Combinatorics Primary 11T71, Secondary 14G50 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. |
| title | On Lattice Isomorphism Problems for Lattices from LCD Codes over Finite Rings |
| topic | Information Theory Combinatorics Primary 11T71, Secondary 14G50 |
| url | https://arxiv.org/abs/2507.09257 |