# 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

https://dx.doi.org/10.1137/20M1327422

## Keywords

Lattices, LLL algorithm, Shortest vector problem, Parametric lattices

## Language

English

## Format

text