A parametric version of LLL and some consequences: parametric shortest and closest vector problems
Abstract
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.
Keywords
Cite
@article{arxiv.1909.04762,
title = {A parametric version of LLL and some consequences: parametric shortest and closest vector problems},
author = {Tristram Bogart and John Goodrick and Kevin Woods},
journal= {arXiv preprint arXiv:1909.04762},
year = {2020}
}
Comments
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