On Ideal Lattices and Gr\"obner Bases

TL;DR

This paper explores the relationship between ideal lattices and Gr"obner bases in multivariate polynomial rings over integers, extending univariate concepts to multivariate cases and characterizing conditions for their existence.

Contribution

It introduces multivariate cyclic lattices, links ideal lattices with Gr"obner bases, and provides criteria for when residue class rings over integers contain ideal lattices.

Findings

Multivariate ideal lattices generalize cyclic lattices.

Existence of ideal lattices relates to monic Gr"obner bases.

Characterization of ideals yielding full rank lattices.

Abstract

In this paper, we draw a connection between ideal lattices and Gr\"{o}bner bases in the multivariate polynomial rings over integers. We study extension of ideal lattices in $\mathbb{Z}[x]/\langle f \rangle$ (Lyubashevsky \& Micciancio, 2006) to ideal lattices in $\mathbb{Z}[x_1,\ldots,x_n]/\mathfrak{a}$, the multivariate case, where $f$ is a polynomial in $\mathbb{Z}[X]$ and $\mathfrak{a}$ is an ideal in $\mathbb{Z}[x_1,\ldots,x_n]$. Ideal lattices in univariate case are interpreted as generalizations of cyclic lattices. We introduce a notion of multivariate cyclic lattices and we show that multivariate ideal lattices are indeed a generalization of them. We show that the fact that existence of ideal lattice in univariate case if and only if $f$ is monic translates to short reduced Gr\"obner basis (Francis \& Dukkipati, 2014) of $\mathfrak{a}$ is monic in multivariate case. We, thereby, give a necessary and sufficient condition for residue class polynomial rings over $\mathbb{Z}$ to have ideal lattices. We also characterize ideals in $\mathbb{Z}[x_1,\ldots,x_n]$ that give rise to full rank lattices.