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arXiv 1909.04762math.COmath.OC
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LLL的参数化版本及其结果：参数化最短和最近向量问题

A parametric version of LLL and some consequences: parametric shortest and closest vector problems

Tristram Bogart, John Goodrick, Kevin Woods

备注：
20 pages. Accepted for publication in SIAM Journal on Discrete Mathematics. Revised title and opening paragraphs, slightly modified statement of Theorem 1.4, added explanation of some steps in Section 3, and implemented various minor improvements suggested by anonymous referees
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摘要：

给定一个参数格，其基由Z[t]中的多项式给出，我们给出了一个算法来构造一个LLL-约化基，其元素最终是t中的拟多项式：即，它们由t中的分段多项式公式给出（对于足够大的t），使得每一段由模周期的同余类给出。因此，我们证明了最短向量问题（SVP）和最近向量问题（CVP）的参数解最终也是t中的拟多项式。

英文摘要：

Given a parametric lattice with a basis given by polynomials in Z[t], we give an algorithm to construct an LLL-reduced basis whose elements are eventually quasi-polynomial in t: that is, they are given by formulas that are piecewise polynomial in t (for sufficiently large t), such that each piece is given by a congruence class modulo a period. As a consequence, we show that there are parametric solutions of the shortest vector problem (SVP) and closest vector problem (CVP) that are also eventually quasi-polynomial in t.

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