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On Lattice Isomorphism Problems for Lattices from LCD Codes over Finite Rings
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DOI：10.1587/transfun.2025CIP0017.png)
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Abstract
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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 Zn and the graph isomorphism problem. While the previous work by Ducas-Gibbons only considered lattices constructed by a code over a finite field, this paper considers lattices constructed by a code over a finite ring Z/kZ, 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 Zn and the graph isomorphism problem.
Keywords:
post-quantum cryptography
lattice-based cryptography
lattice isomorphism problem
code equivalence problem
graph isomorphism problem
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