82,755 research outputs found
New minimal weight representations for left-to-right window methods
Abstract. For an integer w ≥ 2, a radix 2 representation is called a width-w nonadjacent form (w-NAF, for short) if each nonzero digit is an odd integer with absolute value less than 2 w−1, and of any w consecutive digits, at most one is nonzero. In elliptic curve cryptography, the w-NAF window method is used to efficiently compute nP where n is an integer and P is an elliptic curve point. We introduce a new family of radix 2 representations which use the same digits as the w-NAF but have the advantage that they result in a window method which uses less memory. This memory savings results from the fact that these new representations can be deduced using a very simple left-to-right algorithm. Further, we show that like the w-NAF, these new representations have a minimal number of nonzero digits. 1 Window Methods An operation fundamental to elliptic curve cryptography is scalar multiplication; that is, computing nP for an integer, n, and an elliptic curve point, P. A number of different algorithms have been proposed to perform this operation efficiently (see Ch. 3 of [4] for a recent survey). A variety of these algorithms, known as window methods, use the approach described in Algorithm 1.1. For example, suppose D = {0, 1, 3, 5, 7}. Using this digit set, Algorithm 1.1 first computes and stores P, 3P, 5P and 7P. After a D-radix 2 representation of n is computed its digits are read from left to right by the “for ” loop and nP is computed using doubling and addition operations (and no subtractions). One way to compute a D-radix 2 representation of n is to slide a 3-digit window from right to left across the {0, 1}-radix 2 representation of n (see Section 4). Using negative digits takes advantage of the fact that subtracting an elliptic curve point can be done just as efficiently as adding it. Suppose now that D
The Most Irrational Rational Theories
We propose a two-parameter family of modular invariant partition functions of
two-dimensional conformal field theories (CFTs) holographically dual to pure
three-dimensional gravity in anti de Sitter space. Our two parameters control
the central charge, and the representation of . At large
central charge, the partition function has a gap to the first nontrivial
primary state of . As the representation
dimension gets large, the partition function exhibits some of the qualitative
features of an irrational CFT. This, for instance, is captured in the behavior
of the spectral form factor. As part of these analyses, we find similar
behavior in the minimal model spectral form factor as approaches .Comment: 25 pages plus appendices, 11 figure
Memory-Based Shallow Parsing
We present memory-based learning approaches to shallow parsing and apply
these to five tasks: base noun phrase identification, arbitrary base phrase
recognition, clause detection, noun phrase parsing and full parsing. We use
feature selection techniques and system combination methods for improving the
performance of the memory-based learner. Our approach is evaluated on standard
data sets and the results are compared with that of other systems. This reveals
that our approach works well for base phrase identification while its
application towards recognizing embedded structures leaves some room for
improvement
Sparse Coding on Stereo Video for Object Detection
Deep Convolutional Neural Networks (DCNN) require millions of labeled
training examples for image classification and object detection tasks, which
restrict these models to domains where such datasets are available. In this
paper, we explore the use of unsupervised sparse coding applied to stereo-video
data to help alleviate the need for large amounts of labeled data. We show that
replacing a typical supervised convolutional layer with an unsupervised
sparse-coding layer within a DCNN allows for better performance on a car
detection task when only a limited number of labeled training examples is
available. Furthermore, the network that incorporates sparse coding allows for
more consistent performance over varying initializations and ordering of
training examples when compared to a fully supervised DCNN. Finally, we compare
activations between the unsupervised sparse-coding layer and the supervised
convolutional layer, and show that the sparse representation exhibits an
encoding that is depth selective, whereas encodings from the convolutional
layer do not exhibit such selectivity. These result indicates promise for using
unsupervised sparse-coding approaches in real-world computer vision tasks in
domains with limited labeled training data
Optimality of the Width- Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases
Efficient scalar multiplication in Abelian groups (which is an important
operation in public key cryptography) can be performed using digital
expansions. Apart from rational integer bases (double-and-add algorithm),
imaginary quadratic integer bases are of interest for elliptic curve
cryptography, because the Frobenius endomorphism fulfils a quadratic equation.
One strategy for improving the efficiency is to increase the digit set (at the
prize of additional precomputations). A common choice is the width\nbd-
non-adjacent form (\wNAF): each block of consecutive digits contains at
most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number
of non-zero digits, which translates in few costly curve operations. This paper
investigates the following question: Is the \wNAF{}-expansion optimal, where
optimality means minimising the weight over all possible expansions with the
same digit set?
The main characterisation of optimality of \wNAF{}s can be formulated in the
following more general setting: We consider an Abelian group together with an
endomorphism (e.g., multiplication by a base element in a ring) and a finite
digit set. We show that each group element has an optimal \wNAF{}-expansion if
and only if this is the case for each sum of two expansions of weight 1. This
leads both to an algorithmic criterion and to generic answers for various
cases.
Imaginary quadratic integers of trace at least 3 (in absolute value) have
optimal \wNAF{}s for . The same holds for the special case of base
and , which corresponds to Koblitz curves in
characteristic three. In the case of , optimality depends on
the parity of . Computational results for small trace are given
Double Trace Interfaces
We introduce and study renormalization group interfaces between two
holographic conformal theories which are related by deformation by a scalar
double trace operator. At leading order in the 1/N expansion, we derive
expressions for the two point correlation functions of the scalar, as well as
the spectrum of operators living on the interface. We also compute the
interface contribution to the sphere partition function, which in two
dimensions gives the boundary g factor. Checks of our proposal include
reproducing the g factor and some defect overlap coefficients of Gaiotto's RG
interfaces at large N, and the two-point correlation function whenever
conformal perturbation theory is valid.Comment: 59 pages, 2 figure
Quantum theta functions and Gabor frames for modulation spaces
Representations of the celebrated Heisenberg commutation relations in quantum
mechanics and their exponentiated versions form the starting point for a number
of basic constructions, both in mathematics and mathematical physics (geometric
quantization, quantum tori, classical and quantum theta functions) and signal
analysis (Gabor analysis).
In this paper we try to bridge the two communities, represented by the two
co--authors: that of noncommutative geometry and that of signal analysis. After
providing a brief comparative dictionary of the two languages, we will show
e.g. that the Janssen representation of Gabor frames with generalized Gaussians
as Gabor atoms yields in a natural way quantum theta functions, and that the
Rieffel scalar product and associativity relations underlie both the functional
equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change
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