#### Title

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 $\Bbb{Z}[t]$, 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 and closest vector problem that are also eventually quasi-polynomial in $t$.

#### Repository Citation

Bogart, Tristram, John Goodrick, and Kevin Woods. 2020. “A Parametric Version of LLL and Some Consequences: Parametric Shortest and Closest Vector Problems.” SIAM Journal on Discrete Mathematics 34(4): 2363–2387.

#### Publisher

Society for Industrial and Applied Mathematics

#### Publication Date

9-1-2020

#### Publication Title

SIAM Journal on Discrete Mathematics

#### Department

Mathematics

#### Document Type

Article

#### DOI

10.1137/20M1327422

#### Keywords

Lattices, LLL algorithm, Shortest vector problem, Parametric lattices

#### Language

English

#### Format

text