3,317 research outputs found
Decoding by Embedding: Correct Decoding Radius and DMT Optimality
The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP)
are the core algorithmic problems on Euclidean lattices. They are central to
the applications of lattices in many problems of communications and
cryptography. Kannan's \emph{embedding technique} is a powerful technique for
solving the approximate CVP, yet its remarkable practical performance is not
well understood. In this paper, the embedding technique is analyzed from a
\emph{bounded distance decoding} (BDD) viewpoint. We present two complementary
analyses of the embedding technique: We establish a reduction from BDD to
Hermite SVP (via unique SVP), which can be used along with any Hermite SVP
solver (including, among others, the Lenstra, Lenstra and Lov\'asz (LLL)
algorithm), and show that, in the special case of LLL, it performs at least as
well as Babai's nearest plane algorithm (LLL-aided SIC). The former analysis
helps to explain the folklore practical observation that unique SVP is easier
than standard approximate SVP. It is proven that when the LLL algorithm is
employed, the embedding technique can solve the CVP provided that the noise
norm is smaller than a decoding radius , where
is the minimum distance of the lattice, and . This
substantially improves the previously best known correct decoding bound . Focusing on the applications of BDD to decoding of
multiple-input multiple-output (MIMO) systems, we also prove that BDD of the
regularized lattice is optimal in terms of the diversity-multiplexing gain
tradeoff (DMT), and propose practical variants of embedding decoding which
require no knowledge of the minimum distance of the lattice and/or further
improve the error performance.Comment: To appear in IEEE Transactions on Information Theor
Local Testing for Membership in Lattices
Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)
Quantum Algorithm for Computing the Period Lattice of an Infrastructure
We present a quantum algorithm for computing the period lattice of
infrastructures of fixed dimension. The algorithm applies to infrastructures
that satisfy certain conditions. The latter are always fulfilled for
infrastructures obtained from global fields, i.e., algebraic number fields and
function fields with finite constant fields.
The first of our main contributions is an exponentially better method for
sampling approximations of vectors of the dual lattice of the period lattice
than the methods outlined in the works of Hallgren and Schmidt and Vollmer.
This new method improves the success probability by a factor of at least
2^{n^2-1} where n is the dimension. The second main contribution is a rigorous
and complete proof that the running time of the algorithm is polynomial in the
logarithm of the determinant of the period lattice and exponential in n. The
third contribution is the determination of an explicit lower bound on the
success probability of our algorithm which greatly improves on the bounds given
in the above works.
The exponential scaling seems inevitable because the best currently known
methods for carrying out fundamental arithmetic operations in infrastructures
obtained from algebraic number fields take exponential time. In contrast, the
problem of computing the period lattice of infrastructures arising from
function fields can be solved without the exponential dependence on the
dimension n since this problem reduces efficiently to the abelian hidden
subgroup problem. This is also true for other important computational problems
in algebraic geometry. The running time of the best classical algorithms for
infrastructures arising from global fields increases subexponentially with the
determinant of the period lattice.Comment: 52 pages, 4 figure
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