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.

2022