2 research outputs found
The Systematic Normal Form of Lattices
We introduce a new canonical form of lattices called the systematic normal
form (SNF). We show that for every lattice there is an efficiently computable
"nearby" SNF lattice, such that for any lattice one can solve lattice problems
on its "nearby" SNF lattice, and translate the solutions back efficiently to
the original lattice. The SNF provides direct connections between arbitrary
lattices, and various lattice related problems like the
Shortest-Integer-Solution, Approximate Greatest Common Divisor. As our main
application of SNF we derive a new set of worst-to-average case lattice
reductions that deviate significantly from the template of Ajtai and improve
upon previous reductions in terms of simplicity
A Discrete Fourier Transform on Lattices with Quantum Applications
In this work, we introduce a definition of the Discrete Fourier Transform
(DFT) on Euclidean lattices in , that generalizes the -th fold DFT of
the integer lattice to arbitrary lattices. This definition is not
applicable for every lattice, but can be defined on lattices known as
Systematic Normal Form (SysNF) introduced in \cite{ES16}. Systematic Normal
Form lattices are sets of integer vectors that satisfy a single homogeneous
modular equation, which itself satisfies a certain number-theoretic property.
Such lattices form a dense set in the space of -dimensional lattices, and
can be used to approximate efficiently any lattice. This implies that for every
lattice a DFT can be computed efficiently on a lattice near .
Our proof of the statement above uses arguments from quantum computing, and
as an application of our definition we show a quantum algorithm for sampling
from discrete distributions on lattices, that extends our ability to sample
efficiently from the discrete Gaussian distribution \cite{GPV08} to any
distribution that is sufficiently "smooth". We conjecture that studying the
eigenvectors of the newly-defined lattice DFT may provide new insights into the
structure of lattices, especially regarding hard computational problems, like
the shortest vector problem.Comment: Modified introduction and reference