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Distributed computing and cryptography with general weak random sources
The use of randomness in computer science is ubiquitous. Randomized protocols have turned out to be much more efficient than their deterministic counterparts. In addition, many problems in distributed computing and cryptography are impossible to solve without randomness. However, these applications typically require uniform random bits, while in practice almost all natural random phenomena are biased. Moreover, even originally uniform random bits can be damaged if an adversary learns some partial information about these bits. In this thesis, we study how to run randomized protocols in distributed computing and cryptography with imperfect randomness. We use the most general model for imperfect randomness where the weak random source is only required to have a certain amount of min-entropy. One important tool here is the randomness extractor. A randomness extractor is a function that takes as input one or more weak random sources, and outputs a distribution that is close to uniform in statistical distance. Randomness extractors are interesting in their own right and are closely related to many other problems in computer science. Giving efficient constructions of randomness extractors with optimal parameters is one of the major open problems in the area of pseudorandomness. We construct network extractor protocols that extract private random bits for parties in a communication network, assuming that they each start with an independent weak random source, and some parties are corrupted by an adversary who sees all communications in the network. These protocols imply fault-tolerant distributed computing protocols and secure multi-party computation protocols where only imperfect randomness is available. The probabilistic method shows that there exists an extractor for two independent sources with logarithmic min-entropy, while known constructions are far from achieving these parameters. In this thesis we construct extractors for two independent sources with any linear min-entropy, based on a computational assumption. We also construct the best known extractors for three independent sources and affine sources. Finally we study the problem of privacy amplification. In this model, two parties share a private weak random source and they wish to agree on a private uniform random string through communications in a channel controlled by an adversary, who has unlimited computational power and can change the messages in arbitrary ways. All previous results assume that the two parties have local uniform random bits. We show that this problem can be solved even if the two parties only have local weak random sources. We also improve previous results in various aspects by constructing the first explicit non-malleable extractor and giving protocols based on this extractor.Computer Science
Three-Source Extractors for Polylogarithmic Min-Entropy
We continue the study of constructing explicit extractors for independent
general weak random sources. The ultimate goal is to give a construction that
matches what is given by the probabilistic method --- an extractor for two
independent -bit weak random sources with min-entropy as small as . Previously, the best known result in the two-source case is an
extractor by Bourgain \cite{Bourgain05}, which works for min-entropy ;
and the best known result in the general case is an earlier work of the author
\cite{Li13b}, which gives an extractor for a constant number of independent
sources with min-entropy . However, the constant in the
construction of \cite{Li13b} depends on the hidden constant in the best known
seeded extractor, and can be large; moreover the error in that construction is
only .
In this paper, we make two important improvements over the result in
\cite{Li13b}. First, we construct an explicit extractor for \emph{three}
independent sources on bits with min-entropy .
In fact, our extractor works for one independent source with poly-logarithmic
min-entropy and another independent block source with two blocks each having
poly-logarithmic min-entropy. Thus, our result is nearly optimal, and the next
step would be to break the barrier in two-source extractors. Second, we
improve the error of the extractor from to
, which is almost optimal and crucial for cryptographic
applications. Some of the techniques developed here may be of independent
interests
Coherence effects in disordered geometries with a field-theory dual
We investigate the holographic dual of a probe scalar in an asymptotically
Anti-de-Sitter (AdS) disordered background which is an exact solution of
Einstein's equations in three bulk dimensions. Unlike other approaches to model
disorder in holography, we are able to explore quantum wave-like interference
effects between an oscillating or random source and the geometry. In the
weak-disorder limit, we compute analytically and numerically the one-point
correlation function of the dual field theory for different choices of sources
and backgrounds. The most interesting feature is the suppression of the
one-point function in the presence of an oscillating source and weak random
background. We have also computed analytically and numerically the two-point
function in the weak disorder limit. We have found that, in general, the
perturbative contribution induces an additional power-law decay whose exponent
depends on the distribution of disorder. For certain choices of the gravity
background, this contribution becomes dominant for large separations which
indicates breaking of perturbation theory and the possible existence of a phase
transition induced by disorder.Comment: 36 pages, 19 figs, v3 accepted versio
Security of quantum key distribution with imperfect devices
We prove the security of the Bennett-Brassard (BB84) quantum key distribution
protocol in the case where the source and detector are under the limited
control of an adversary. Our proof applies when both the source and the
detector have small basis-dependent flaws, as is typical in practical
implementations of the protocol. We derive a general lower bound on the
asymptotic key generation rate for weakly basis-dependent eavesdropping
attacks, and also estimate the rate in some special cases: sources that emit
weak coherent states with random phases, detectors with basis-dependent
efficiency, and misaligned sources and detectors.Comment: 22 pages. (v3): Minor changes. (v2): Extensively revised and
expanded. New results include a security proof for generic small flaws in the
source and the detecto
Moment-Based Ellipticity Measurement as a Statistical Parameter Estimation Problem
We show that galaxy ellipticity estimation for weak gravitational lensing
with unweighted image moments reduces to the problem of measuring a combination
of the means of three independent normal random variables. Under very general
assumptions, the intrinsic image moments of sources can be recovered from
observations including effects such as the point-spread function and
pixellation. Gaussian pixel noise turns these into three jointly normal random
variables, the means of which are algebraically related to the ellipticity. We
show that the random variables are approximately independent with known
variances, and provide an algorithm for making them exactly independent. Once
the framework is developed, we derive general properties of the ellipticity
estimation problem, such as the signal-to-noise ratio, a generic form of an
ellipticity estimator, and Cram\'er-Rao lower bounds for an unbiased estimator.
We then derive the unbiased ellipticity estimator using unweighted image
moments. We find that this unbiased estimator has a poorly behaved distribution
and does not converge in practical applications, but demonstrates how to derive
and understand the behaviour of new moment-based ellipticity estimators.Comment: 11 pages, 7 figures; v2 matches accepted version with minor change
Finite Device-Independent Extraction of a Block Min-Entropy Source against Quantum Adversaries
The extraction of randomness from weakly random seeds is a problem of central
importance with multiple applications. In the device-independent setting, this
problem of quantum randomness amplification has been mainly restricted to
specific weak sources of Santha-Vazirani type, while extraction from the
general min-entropy sources has required a large number of separated devices
which is impractical. In this paper, we present a device-independent protocol
for amplification of a single min-entropy source (consisting of two blocks of
sufficiently high min-entropy) using a device consisting of two spatially
separated components and show a proof of its security against general quantum
adversaries.Comment: 17 page
Chaos and localization in the wavefunctions of complex atoms NdI, PmI and SmI
Wavefunctions of complex lanthanide atoms NdI, PmI and SmI, obtained via
multi-configuration Dirac-Fock method, are analyzed for density of states in
terms of partial densities, strength functions (), number of principal
components () and occupancies (\lan n_\alpha \ran^E) of single
particle orbits using embedded Gaussian orthogonal ensemble of one plus
two-body random matrix ensembles [EGOE(1+2)]. It is seen that density of states
are in general multi-modal, 's exhibit variations as function of the
basis states energy and 's show structures arising from localized
states. The sources of these departures from EGOE(1+2) are investigated by
examining the partial densities, correlations between , and
\lan n_\alpha \ran^E and also by studying the structure of the Hamiltonian
matrices. These studies point out the operation of EGOE(1+2) but at the same
time suggest that weak admixing between well separated configurations should be
incorporated into EGOE(1+2) for more quantitative description of chaos and
localization in NdI, PmI and SmI.Comment: There are 9 figure
Randomness Extraction in AC0 and with Small Locality
Randomness extractors, which extract high quality (almost-uniform) random
bits from biased random sources, are important objects both in theory and in
practice. While there have been significant progress in obtaining near optimal
constructions of randomness extractors in various settings, the computational
complexity of randomness extractors is still much less studied. In particular,
it is not clear whether randomness extractors with good parameters can be
computed in several interesting complexity classes that are much weaker than P.
In this paper we study randomness extractors in the following two models of
computation: (1) constant-depth circuits (AC0), and (2) the local computation
model. Previous work in these models, such as [Vio05a], [GVW15] and [BG13],
only achieve constructions with weak parameters. In this work we give explicit
constructions of randomness extractors with much better parameters. As an
application, we use our AC0 extractors to study pseudorandom generators in AC0,
and show that we can construct both cryptographic pseudorandom generators
(under reasonable computational assumptions) and unconditional pseudorandom
generators for space bounded computation with very good parameters.
Our constructions combine several previous techniques in randomness
extractors, as well as introduce new techniques to reduce or preserve the
complexity of extractors, which may be of independent interest. These include
(1) a general way to reduce the error of strong seeded extractors while
preserving the AC0 property and small locality, and (2) a seeded randomness
condenser with small locality.Comment: 62 page
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