20,266 research outputs found
Security of almost ALL discrete log bits
Let G be a finite cyclic group with generator \alpha and with an encoding so that multiplication is computable in polynomial time. We study the security of bits of the discrete log x when given \exp_{\alpha}(x), assuming that the exponentiation function \exp_{\alpha}(x) = \alpha^x is one-way. We reduce he general problem to the case that G has odd order q. If G has odd order q the security of the least-significant bits of x and of the most significant bits of the rational number \frac{x}{q} \in [0,1) follows from the work of Peralta [P85] and Long and Wigderson [LW88]. We generalize these bits and study the security of consecutive shift bits lsb(2^{-i}x mod q) for i=k+1,...,k+j. When we restrict \exp_{\alpha} to arguments x such that some sequence of j consecutive shift bits of x is constant (i.e., not depending on x) we call it a 2^{-j}-fraction of \exp_{\alpha}. For groups of odd group order q we show that every two 2^{-j}-fractions of \exp_{\alpha} are equally one-way by a polynomial time transformation: Either they are all one-way or none of them. Our key theorem shows that arbitrary j consecutive shift bits of x are simultaneously secure when given \exp_{\alpha}(x) iff the 2^{-j}-fractions of \exp_{\alpha} are one-way. In particular this applies to the j least-significant bits of x and to the j most-significant bits of \frac{x}{q} \in [0,1). For one-way \exp_{\alpha} the individual bits of x are secure when given \exp_{\alpha}(x) by the method of Hastad, N\"aslund [HN98]. For groups of even order 2^{s}q we show that the j least-significant bits of \lfloor x/2^s\rfloor, as well as the j most-significant bits of \frac{x}{q} \in [0,1), are simultaneously secure iff the 2^{-j}-fractions of \exp_{\alpha'} are one-way for \alpha' := \alpha^{2^s}. We use and extend the models of generic algorithms of Nechaev (1994) and Shoup (1997). We determine the generic complexity of inverting fractions of \exp_{\alpha} for the case that \alpha has prime order q. As a consequence, arbitrary segments of (1-\varepsilon)\lg q consecutive shift bits of random x are for constant \varepsilon >0 simultaneously secure against generic attacks. Every generic algorithm using generic steps (group operations) for distinguishing bit strings of j consecutive shift bits of x from random bit strings has at most advantage O((\lg q) j\sqrt{t} (2^j/q)^{\frac14})
Statistically Motivated Second Order Pooling
Second-order pooling, a.k.a.~bilinear pooling, has proven effective for deep
learning based visual recognition. However, the resulting second-order networks
yield a final representation that is orders of magnitude larger than that of
standard, first-order ones, making them memory-intensive and cumbersome to
deploy. Here, we introduce a general, parametric compression strategy that can
produce more compact representations than existing compression techniques, yet
outperform both compressed and uncompressed second-order models. Our approach
is motivated by a statistical analysis of the network's activations, relying on
operations that lead to a Gaussian-distributed final representation, as
inherently used by first-order deep networks. As evidenced by our experiments,
this lets us outperform the state-of-the-art first-order and second-order
models on several benchmark recognition datasets.Comment: Accepted to ECCV 2018. Camera ready version. 14 page, 5 figures, 3
table
On the Generalized Harmonic Polylogarithms of One Complex Variable
We describe how to compute numerically in the complex plain a set of
Generalized Harmonic Polylogarithms (GHPLs) with square roots in the weights,
using the C++/GiNaC numerical routines of Vollinga and Weinzierl. As an
example, we provide the numerical values of the NLO electroweak light-fermion
corrections to the Higgs boson production in gluon fusion in the case of
complex W and Z masses.Comment: 25 pages, 1 figur
Universal geometric approach to uncertainty, entropy and information
It is shown that for any ensemble, whether classical or quantum, continuous
or discrete, there is only one measure of the "volume" of the ensemble that is
compatible with several basic geometric postulates. This volume measure is thus
a preferred and universal choice for characterising the inherent spread,
dispersion, localisation, etc, of the ensemble. Remarkably, this unique
"ensemble volume" is a simple function of the ensemble entropy, and hence
provides a new geometric characterisation of the latter quantity. Applications
include unified, volume-based derivations of the Holevo and Shannon bounds in
quantum and classical information theory; a precise geometric interpretation of
thermodynamic entropy for equilibrium ensembles; a geometric derivation of
semi-classical uncertainty relations; a new means for defining classical and
quantum localization for arbitrary evolution processes; a geometric
interpretation of relative entropy; and a new proposed definition for the
spot-size of an optical beam. Advantages of the ensemble volume over other
measures of localization (root-mean-square deviation, Renyi entropies, and
inverse participation ratio) are discussed.Comment: Latex, 38 pages + 2 figures; p(\alpha)->1/|T| in Eq. (72) [Eq. (A10)
of published version
The invariances of power law size distributions
Size varies. Small things are typically more frequent than large things. The
logarithm of frequency often declines linearly with the logarithm of size. That
power law relation forms one of the common patterns of nature. Why does the
complexity of nature reduce to such a simple pattern? Why do things as
different as tree size and enzyme rate follow similarly simple patterns? Here I
analyze such patterns by their invariant properties. For example, a common
pattern should not change when adding a constant value to all observations.
That shift is essentially the renumbering of the points on a ruler without
changing the metric information provided by the ruler. A ruler is shift
invariant only when its scale is properly calibrated to the pattern being
measured. Stretch invariance corresponds to the conservation of the total
amount of something, such as the total biomass and consequently the average
size. Rotational invariance corresponds to pattern that does not depend on the
order in which underlying processes occur, for example, a scale that additively
combines the component processes leading to observed values. I use tree size as
an example to illustrate how the key invariances shape pattern. A simple
interpretation of common pattern follows. That simple interpretation connects
the normal distribution to a wide variety of other common patterns through the
transformations of scale set by the fundamental invariances.Comment: Added appendix discussing the lognormal distribution, updated to
match version 2 of published version at F1000Researc
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