PreprintCC BY 4.0arXiv
Adelic reduction of module lattices
Abstract
We give a strict generalization of the LLL algorithm over number fields, based on the reduction theory of GL(n) over the adele ring of a number field. Our algorithm is free of heuristics, with rigorous bounds on output quality and complexity. As a consequence, we obtain a hierarchy of reductions from module-(H)SVP to ideal-HSVP, an example of which has runtime and approximation factors subexponential in the field degree. More importantly, we uncover a close connection between structured lattice reduction and a Diophantine approximation over number fields.
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BibTeX
@misc{bambury2026adelic,
title = {{Adelic reduction of module lattices}},
author = {Henry Bambury and Seungki Kim and Changmin Lee and Phong Q. Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32936},
eprint = {2609.32936},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32936},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Adelic reduction of module lattices
AU - Henry Bambury
AU - Seungki Kim
AU - Changmin Lee
AU - Phong Q. Nguyen
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32936
UR - https://arxiv.org/abs/2609.32936
AB - We give a strict generalization of the LLL algorithm over number fields, based on the reduction theory of GL(n) over the adele ring of a number field. Our algorithm is free of heuristics, with rigorous bounds on output quality and complexity. As a consequence, we obtain a hierarchy of reductions from module-(H)SVP to ideal-HSVP, an example of which has runtime and approximation factors subexponential in the field degree. More importantly, we uncover a close connection between structured lattice reduction and a Diophantine approximation over number fields.
ER -