20,081 research outputs found
Recursive lattice reduction -- A framework for finding short lattice vectors
We propose a new framework called recursive lattice reduction for finding
short non-zero vectors in a lattice or for finding dense sublattices of a
lattice. At a high level, the framework works by recursively searching for
dense sublattices of dense sublattices (or their duals). Eventually, the
procedure encounters a recursive call on a lattice with
relatively low rank , at which point we simply use a known algorithm to find
a short non-zero vector in . We view our framework as
complementary to basis reduction algorithms, which similarly work to reduce an
-dimensional lattice problem with some approximation factor to an
exact lattice problem in dimension , with a tradeoff between ,
, and . Our framework provides an alternative and arguably simpler
perspective, which in particular can be described without explicitly
referencing any specific basis of the lattice, Gram-Schmidt vectors, or even
projection (though implementations of algorithms in this framework will likely
make use of such things). We present a number of specific instantiations of our
framework. Our main concrete result is a reduction that matches the tradeoff
between , , and achieved by the best-known basis reduction
algorithms (in terms of the Hermite factor, up to low-order terms) across all
parameter regimes. In fact, this reduction also can be used to find dense
sublattices with any rank satisfying ,
using only an oracle for SVP (or even just Hermite SVP) in dimensions,
which is itself a novel result (as far as the authors know). We also show a
very simple reduction that achieves the same tradeoff in quasipolynomial time.
Finally, we present an automated approach for searching for algorithms in this
framework that (provably) achieve better approximations with fewer oracle
calls
Solving the Closest Vector Problem in Time--- The Discrete Gaussian Strikes Again!
We give a -time and space randomized algorithm for solving the
exact Closest Vector Problem (CVP) on -dimensional Euclidean lattices. This
improves on the previous fastest algorithm, the deterministic
-time and -space algorithm of
Micciancio and Voulgaris.
We achieve our main result in three steps. First, we show how to modify the
sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian
sampling over lattice shifts, , with very low parameters. While the
actual algorithm is a natural generalization of [ADRS15], the analysis uses
substantial new ideas. This yields a -time algorithm for
approximate CVP for any approximation factor .
Second, we show that the approximate closest vectors to a target vector can
be grouped into "lower-dimensional clusters," and we use this to obtain a
recursive reduction from exact CVP to a variant of approximate CVP that
"behaves well with these clusters." Third, we show that our discrete Gaussian
sampling algorithm can be used to solve this variant of approximate CVP.
The analysis depends crucially on some new properties of the discrete
Gaussian distribution and approximate closest vectors, which might be of
independent interest
Theoretical Exploration on the Magnetic Properties of Ferromagnetic Metallic Glass: An Ising Model on Random Recursive Lattice
The ferromagnetic Ising spins are modeled on a recursive lattice constructed
from random-angled rhombus units with stochastic configurations, to study the
magnetic properties of the bulk Fe-based metallic glass. The integration of
spins on the structural glass model well represents the magnetic moments in the
glassy metal. The model is exactly solved by the recursive calculation
technique. The magnetization of the amorphous Ising spins, i.e. the glassy
metallic magnet is investigated by our modeling and calculation on a
theoretical base. The results show that the glassy metallic magnets has a lower
Curie temperature, weaker magnetization, and higher entropy comparing to the
regular ferromagnet in crystal form. These findings can be understood with the
randomness of the amorphous system, and agrees well with others' experimental
observations.Comment: 11 pages, 5 figure
Linear degree growth in lattice equations
We conjecture recurrence relations satisfied by the degrees of some
linearizable lattice equations. This helps to prove linear growth of these
equations. We then use these recurrences to search for lattice equations that
have linear growth and hence are linearizable
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