Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter   Cyclic lattices and ideal lattices were introduced by Micciancio in
\cite{D2}, Lyubashevsky and Micciancio in \cite{L1} respectively, which play an
efficient role in Ajtai's construction of a collision resistant Hash function
(see \cite{M1} and \cite{M2}) and in Gentry's construction of fully homomorphic
encryption (see \cite{G}). Let $R=Z[x]/\langle \phi(x)\rangle$ be a quotient
ring of the integer coefficients polynomials ring, Lyubashevsky and Micciancio
regarded an ideal lattice as the correspondence of an ideal of $R$, but they
neither explain how to extend this definition to whole Euclidean space
$\mathbb{R}^n$, nor exhibit the relationship of cyclic lattices and ideal
lattices.
  In this paper, we regard the cyclic lattices and ideal lattices as the
correspondences of finitely generated $R$-modules, so that we may show that
ideal lattices are actually a special subclass of cyclic lattices, namely,
cyclic integer lattices. In fact, there is a one to one correspondence between
cyclic lattices in $\mathbb{R}^n$ and finitely generated $R$-modules (see
Theorem \ref{th4} below). On the other hand, since $R$ is a Noether ring, each
ideal of $R$ is a finitely generated $R$-module, so it is natural and
reasonable to regard ideal lattices as a special subclass of cyclic lattices
(see corollary \ref{co3.4} below). It is worth noting that we use more general
rotation matrix here, so our definition and results on cyclic lattices and
ideal lattices are more general forms. As application, we provide cyclic
lattice with an explicit and countable upper bound for the smoothing parameter
(see Theorem \ref{th5} below). It is an open problem that is the shortest
vector problem on cyclic lattice NP-hard? (see \cite{D2}). Our results may be
viewed as a substantial progress in this direction.

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The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: econ