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

TL;DR

This paper introduces a parametric version of the LLL lattice reduction algorithm, producing bases that are quasi-polynomial in a parameter, and applies this to solve parametric shortest and closest vector problems with similar properties.

Contribution

It develops an algorithm for parametric LLL reduction with bases that are quasi-polynomial in the parameter, enabling solutions to parametric SVP and CVP.

Findings

Existence of quasi-polynomial parametric solutions for SVP and CVP.

Algorithmic construction of parametric LLL-reduced bases.

Implications for parametric lattice problems in computational number theory.

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.