Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Ideal Lattices and Gr\"obner Bases   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.

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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom