17,093 research outputs found
Low-dimensional lattice basis reduction revisited
International audienceLattice reduction is a geometric generalization of the problem of computing greatest common divisors. Most of the interesting algorithmic problems related to lattice reduction are NP-hard as the lattice dimension increases. This article deals with the low-dimensional case. We study a greedy lattice basis reduction algorithm for the Euclidean norm, which is arguably the most natural lattice basis reduction algorithm, because it is a straightforward generalization of an old two-dimensional algorithm of Lagrange, usually known as Gauss' algorithm, and which is very similar to Euclid's gcd algorithm. Our results are two-fold. From a mathematical point of view, we show that up to dimension four, the output of the greedy algorithm is optimal: the output basis reaches all the successive minima of the lattice. However, as soon as the lattice dimension is strictly higher than four, the output basis may be arbitrarily bad as it may not even reach the first minimum. More importantly, from a computational point of view, we show that up to dimension four, the bit-complexity of the greedy algorithm is quadratic without fast integer arithmetic, just like Euclid's gcd algorithm. This was already proved by Semaev up to dimension three using rather technical means, but it was previously unknown whether or not the algorithm was still polynomial in dimension four. We propose two different analyzes: a global approach based on the geometry of the current basis when the length decrease stalls, and a local approach showing directly that a significant length decrease must occur every O(1) consecutive steps. Our analyzes simplify Semaev's analysis in dimensions two and three, and unify the cases of dimensions two to four. Although the global approach is much simpler, we also present the local approach because it gives further information on the behavior of the algorithm
Seiberg-Witten Curve for E-String Theory Revisited
We discuss various properties of the Seiberg-Witten curve for the E-string
theory which we have obtained recently in hep-th/0203025. Seiberg-Witten curve
for the E-string describes the low-energy dynamics of a six-dimensional (1,0)
SUSY theory when compactified on R^4 x T^2. It has a manifest affine E_8 global
symmetry with modulus \tau and E_8 Wilson line parameters {m_i},i=1,2,...,8
which are associated with the geometry of the rational elliptic surface. When
the radii R_5,R_6 of the torus T^2 degenerate R_5,R_6 --> 0, E-string curve is
reduced to the known Seiberg-Witten curves of four- and five-dimensional gauge
theories. In this paper we first study the geometry of rational elliptic
surface and identify the geometrical significance of the Wilson line
parameters. By fine tuning these parameters we also study degenerations of our
curve corresponding to various unbroken symmetry groups. We also find a new way
of reduction to four-dimensional theories without taking a degenerate limit of
T^2 so that the SL(2,Z) symmetry is left intact. By setting some of the Wilson
line parameters to special values we obtain the four-dimensional SU(2)
Seiberg-Witten theory with 4 flavors and also a curve by Donagi and Witten
describing the dynamics of a perturbed N=4 theory.Comment: 35 pages, 2 figures, LaTeX2
Generalised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and
feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve
cryptography. Their form is such that modular reduction is extremely efficient,
thus making them an attractive choice for modular multiplication
implementation. However, the issue of residue multiplication efficiency seems
to have been overlooked. Asymptotically, using a cyclic rather than a linear
convolution, residue multiplication modulo a Mersenne number is twice as fast
as integer multiplication; this property does not hold for prime GMNs, unless
they are of Mersenne's form. In this work we exploit an alternative
generalisation of Mersenne numbers for which an analogue of the above property
--- and hence the same efficiency ratio --- holds, even at bitlengths for which
schoolbook multiplication is optimal, while also maintaining very efficient
reduction. Moreover, our proposed primes are abundant at any bitlength, whereas
GMNs are extremely rare. Our multiplication and reduction algorithms can also
be easily parallelised, making our arithmetic particularly suitable for
hardware implementation. Furthermore, the field representation we propose also
naturally protects against side-channel attacks, including timing attacks,
simple power analysis and differential power analysis, which is essential in
many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio
Supersymmetric lattices
Discretization of supersymmetric theories is an old problem in lattice field
theory. It has resisted solution until quite recently when new ideas drawn from
orbifold constructions and topological field theory have been brought to bear
on the question. The result has been the creation of a new class of lattice
gauge theory in which the lattice action is invariant under one or more
supersymmetries. The resultant theories are local and free of doublers and in
the case of Yang-Mills theories also possess exact gauge invariance. In
principle they form the basis for a truly non-perturbative definition of the
continuum supersymmetric field theory. In this talk these ideas are reviewed
with particular emphasis being placed on super Yang-Mills theory.Comment: Plenary talk at the symposium Quantum Theory and Symmetries,
Lexington, Kentucky, July 2009. References adde
Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians
The first step in elliptic curve scalar multiplication algorithms based on
scalar decompositions using efficient endomorphisms-including
Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as
well as higher-dimensional and higher-genus constructions-is to produce a short
basis of a certain integer lattice involving the eigenvalues of the
endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar
coefficients, and the faster the resulting scalar multiplication. Typically,
knowledge of the eigenvalues allows us to write down a long basis, which we
then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more
specialized algorithm. In this work, we use elementary facts about quadratic
rings to immediately write down a short basis of the lattice for the GLV, GLS,
GLV+GLS, and Q-curve constructions on elliptic curves, and for genus 2 real
multiplication constructions. We do not pretend that this represents a
significant optimization in scalar multiplication, since the lattice reduction
step is always an offline precomputation---but it does give a better insight
into the structure of scalar decompositions. In any case, it is always more
convenient to use a ready-made short basis than it is to compute a new one
String Islands
We discuss string theories with small numbers of non-compact moduli and
describe constructions of string theories whose low-energy limit is described
by various pure supergravity theories. We also construct a D=4,N=4
compactification of type II string theory with 34 vector fields.Comment: An erroneous example removed. We thank Massimo Bianchi and Cumrun
Vafa for pointing out this erro
On the Proximity Factors of Lattice Reduction-Aided Decoding
Lattice reduction-aided decoding features reduced decoding complexity and
near-optimum performance in multi-input multi-output communications. In this
paper, a quantitative analysis of lattice reduction-aided decoding is
presented. To this aim, the proximity factors are defined to measure the
worst-case losses in distances relative to closest point search (in an infinite
lattice). Upper bounds on the proximity factors are derived, which are
functions of the dimension of the lattice alone. The study is then extended
to the dual-basis reduction. It is found that the bounds for dual basis
reduction may be smaller. Reasonably good bounds are derived in many cases. The
constant bounds on proximity factors not only imply the same diversity order in
fading channels, but also relate the error probabilities of (infinite) lattice
decoding and lattice reduction-aided decoding.Comment: remove redundant figure
- …