模格的Adelic约化
Adelic reduction of module lattices
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中文总结 AI 辅助
本文提出数域上LLL算法的严格推广，基于adele环上的约化理论，无启发式假设，获得模-(H)SVP到理想-HSVP的约化层级，并揭示结构化格约化与数域丢番图逼近的紧密联系。
中文摘要 AI 辅助
我们基于数域上$\text{GL}(n)$在adele环上的约化理论，给出了数域上LLL算法的严格推广。我们的算法无启发式假设，对输出质量和复杂度具有严格界。作为推论，我们获得了从模-(H)SVP到理想-HSVP的约化层级，其中一个实例的运行时间和近似因子在域次数上为次指数。更重要的是，我们揭示了结构化格约化与数域上的丢番图逼近之间的紧密联系。
英文摘要
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.
