Lattices that Admit Logarithmic Worst-Case to Average-Case Connection Factors
Chris Peikert and Alon Rosen
Abstract
We demonstrate an \emph{average-case} problem which is as hard as
finding -approximate shortest vectors in certain
-dimensional lattices in the \emph{worst case}, where . The previously best known factor for any class
of lattices was .
To obtain our results, we focus on families of lattices having
special algebraic structure. Specifically, we consider lattices
that correspond to \emph{ideals} in the ring of integers of an
algebraic number field. The worst-case assumption we rely on is
that in some length, it is hard to find approximate
shortest vectors in these lattices, under an appropriate form of
preprocessing of the number field. Our results build upon prior
works by Micciancio (FOCS 2002), Peikert and Rosen (TCC 2006), and
Lyubashevsky and Micciancio (ICALP 2006).
For the connection factors we achieve, the corresponding
\emph{decisional} promise problems on ideal lattices are \emph{not}
known to be NP-hard; in fact, they are in P. However, the
\emph{search} approximation problems still appear to be very hard.
Indeed, ideal lattices are well-studied objects in computational
number theory, and the best known algorithms for them seem to
perform \emph{no better} than the best known algorithms for general
lattices.
To obtain the best possible connection factor, we instantiate our
constructions with infinite families of number fields having
constant \emph{root discriminant}. Such families are known to exist
and are computable, though no efficient construction is yet known.
Our work motivates the search for such constructions. Even
constructions of number fields having root discriminant up to
would yield connection factors better than the
current best of~.