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- W1419434701 abstract "In this paper, we draw a connection between ideal lattices and Grobner 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 Grobner 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." @default.
- W1419434701 created "2016-06-24" @default.
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- W1419434701 date "2014-09-27" @default.
- W1419434701 modified "2023-09-27" @default.
- W1419434701 title "On Ideal Lattices and Gröbner Bases" @default.
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