We show that finding small solutions to random modular linear equation
is at least as hard as approximating several lattice problems in the
worst case within a factor almost linear in the dimension of the
lattice. The lattice problems we consider are the shortest vector
problem, the shortest independent vectors problem, the covering radius
problem, and the guaranteed distance decoding problem (a variant of
the well known closest vector problem). The approximation factor we
obtain is
*n log ^{O(1)}(n)*
for all three problems. This greatly improves on all previous work on
the subject starting from Ajtai's seminal paper
[Quad. Mat., 13 (2004), pp. 1-32],
up to the strongest previously known results by Micciancio
[SIAM J. Comput., 34 (2004), pp. 118-169].
Our results also bring us closer to the limit where the problems are
no longer known to be in NP intersected coNP.
Our main tools are Gaussian measures on lattices and the
high-dimensional Fourier transform. We start by defining a new lattice
parameter which determines the amount of Gaussian noise that one has
to add to a lattice in order to get close to a uniform distribution.
In addition to yielding quantitatively much stronger results, the use
of this parameter allows us to simplify many of the complications in
previous work.
Our technical contributions are two-fold. First, we show tight
connections between this new parameter and existing lattice parameters.
One such important connection is between this parameter and the length
of the shortest set of linearly independent vectors. Second, we prove
that the distribution that one obtains after adding Gaussian noise to
the lattice has the following interesting property: the distribution
of the noise vector when conditioning on the final value behaves in
many respects like the original Gaussian noise vector. In particular,
its moments remain essentially unchanged.