Can one use lattice basis reduction algorithms, such as LLL over (low-rank) module lattices over rings of number fields of degree greater than 1? Is there any work on lattice reductions over Euclidean rings (for example, using BKZ, lattice enumeraiton, etc)?
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$\begingroup$ There is a lot of research on ideal lattices in cryptography, see en.wikipedia.org/wiki/Ideal_lattice_cryptography and eprint.iacr.org $\endgroup$– Martin SeysenCommented Feb 21, 2018 at 22:27
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$\begingroup$ @MartinSeysen I have, I just haven’t found something for this specific instance. $\endgroup$– terettCommented Feb 21, 2018 at 22:32
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Apparently, this was studied in the following work: https://eprint.iacr.org/2019/1035
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2$\begingroup$ Note the work here is relevant as well. In both cases, while the algorithms are often called "LLL", they do require (approximate) SVP oracles for lattices in the underlying number field. This is perhaps closer to a BKZ-type algorithm when the number field degree is large. $\endgroup$ Commented Jul 12, 2023 at 18:24