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 $t$.

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

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