On Ideal Lattices and Gröbner Bases

In this paper, we draw a connection between ideal lattices and Gröbner bases in the multivariate polynomial rings over integers. We study extension of ideal lattices in Z[x]/〈f〉 (Lyubashevsky & Micciancio, 2006) to ideal lattices in Z[x1, . . . , xn]/a, the multivariate case, where f is a polynomial in Z[X] and a is an ideal in Z[x1, . . . , xn]. 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 Gröbner basis (Francis & Dukkipati, 2014) of a is monic in multivariate case. We, thereby, give a necessary and sufficient condition for residue class polynomial rings over Z to have ideal lattices. We also characterize ideals in Z[x1, . . . , xn] that give rise to full rank lattices.