3,788 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
Attacks on the Search-RLWE problem with small errors
The Ring Learning-With-Errors (RLWE) problem shows great promise for
post-quantum cryptography and homomorphic encryption. We describe a new attack
on the non-dual search RLWE problem with small error widths, using ring
homomorphisms to finite fields and the chi-squared statistical test. In
particular, we identify a "subfield vulnerability" (Section 5.2) and give a new
attack which finds this vulnerability by mapping to a finite field extension
and detecting non-uniformity with respect to the number of elements in the
subfield. We use this attack to give examples of vulnerable RLWE instances in
Galois number fields. We also extend the well-known search-to-decision
reduction result to Galois fields with any unramified prime modulus q,
regardless of the residue degree f of q, and we use this in our attacks. The
time complexity of our attack is O(nq2f), where n is the degree of K and f is
the residue degree of q in K. We also show an attack on the non-dual (resp.
dual) RLWE problem with narrow error distributions in prime cyclotomic rings
when the modulus is a ramified prime (resp. any integer). We demonstrate the
attacks in practice by finding many vulnerable instances and successfully
attacking them. We include the code for all attacks
Some new formulas for
We show how to find series expansions for of the form
, where S(n) is some
polynomial in (depending on ). We prove that there exist such
expansions for , , , for any , and give explicit
examples for such expansions for small values of , and .Comment: 28 pages, LaTeX; some important references adde
The shapes of an epidemic: using Functional Data Analysis to characterize COVID-19 in Italy
We investigate patterns of COVID-19 mortality across 20 Italian regions and
their association with mobility, positivity, and socio-demographic,
infrastructural and environmental covariates. Notwithstanding limitations in
accuracy and resolution of the data available from public sources, we pinpoint
significant trends exploiting information in curves and shapes with Functional
Data Analysis techniques. These depict two starkly different epidemics; an
"exponential" one unfolding in Lombardia and the worst hit areas of the north,
and a milder, "flat(tened)" one in the rest of the country -- including Veneto,
where cases appeared concurrently with Lombardia but aggressive testing was
implemented early on. We find that mobility and positivity can predict COVID-19
mortality, also when controlling for relevant covariates. Among the latter,
primary care appears to mitigate mortality, and contacts in hospitals, schools
and work places to aggravate it. The techniques we describe could capture
additional and potentially sharper signals if applied to richer data
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