238 research outputs found
On formal verification of arithmetic-based cryptographic primitives
Cryptographic primitives are fundamental for information security: they are
used as basic components for cryptographic protocols or public-key
cryptosystems. In many cases, their security proofs consist in showing that
they are reducible to computationally hard problems. Those reductions can be
subtle and tedious, and thus not easily checkable. On top of the proof
assistant Coq, we had implemented in previous work a toolbox for writing and
checking game-based security proofs of cryptographic primitives. In this paper
we describe its extension with number-theoretic capabilities so that it is now
possible to write and check arithmetic-based cryptographic primitives in our
toolbox. We illustrate our work by machine checking the game-based proofs of
unpredictability of the pseudo-random bit generator of Blum, Blum and Shub, and
semantic security of the public-key cryptographic scheme of Goldwasser and
Micali.Comment: 13 page
The Range of Topological Effects on Communication
We continue the study of communication cost of computing functions when
inputs are distributed among processors, each of which is located at one
vertex of a network/graph called a terminal. Every other node of the network
also has a processor, with no input. The communication is point-to-point and
the cost is the total number of bits exchanged by the protocol, in the worst
case, on all edges.
Chattopadhyay, Radhakrishnan and Rudra (FOCS'14) recently initiated a study
of the effect of topology of the network on the total communication cost using
tools from embeddings. Their techniques provided tight bounds for simple
functions like Element-Distinctness (ED), which depend on the 1-median of the
graph. This work addresses two other kinds of natural functions. We show that
for a large class of natural functions like Set-Disjointness the communication
cost is essentially times the cost of the optimal Steiner tree connecting
the terminals. Further, we show for natural composed functions like and , the naive protocols
suggested by their definition is optimal for general networks. Interestingly,
the bounds for these functions depend on more involved topological parameters
that are a combination of Steiner tree and 1-median costs.
To obtain our results, we use some new tools in addition to ones used in
Chattopadhyay et. al. These include (i) viewing the communication constraints
via a linear program; (ii) using tools from the theory of tree embeddings to
prove topology sensitive direct sum results that handle the case of composed
functions and (iii) representing the communication constraints of certain
problems as a family of collection of multiway cuts, where each multiway cut
simulates the hardness of computing the function on the star topology
Online unit clustering in higher dimensions
We revisit the online Unit Clustering and Unit Covering problems in higher
dimensions: Given a set of points in a metric space, that arrive one by
one, Unit Clustering asks to partition the points into the minimum number of
clusters (subsets) of diameter at most one; while Unit Covering asks to cover
all points by the minimum number of balls of unit radius. In this paper, we
work in using the norm.
We show that the competitive ratio of any online algorithm (deterministic or
randomized) for Unit Clustering must depend on the dimension . We also give
a randomized online algorithm with competitive ratio for Unit
Clustering}of integer points (i.e., points in , , under norm). We show that the competitive ratio of
any deterministic online algorithm for Unit Covering is at least . This
ratio is the best possible, as it can be attained by a simple deterministic
algorithm that assigns points to a predefined set of unit cubes. We complement
these results with some additional lower bounds for related problems in higher
dimensions.Comment: 15 pages, 4 figures. A preliminary version appeared in the
Proceedings of the 15th Workshop on Approximation and Online Algorithms (WAOA
2017
Recurrence of biased quantum walks on a line
The Polya number of a classical random walk on a regular lattice is known to
depend solely on the dimension of the lattice. For one and two dimensions it
equals one, meaning unit probability to return to the origin. This result is
extremely sensitive to the directional symmetry, any deviation from the equal
probability to travel in each direction results in a change of the character of
the walk from recurrent to transient. Applying our definition of the Polya
number to quantum walks on a line we show that the recurrence character of
quantum walks is more stable against bias. We determine the range of parameters
for which biased quantum walks remain recurrent. We find that there exist
genuine biased quantum walks which are recurrent.Comment: Journal reference added, minor corrections in the tex
Stochastic Budget Optimization in Internet Advertising
Internet advertising is a sophisticated game in which the many advertisers
"play" to optimize their return on investment. There are many "targets" for the
advertisements, and each "target" has a collection of games with a potentially
different set of players involved. In this paper, we study the problem of how
advertisers allocate their budget across these "targets". In particular, we
focus on formulating their best response strategy as an optimization problem.
Advertisers have a set of keywords ("targets") and some stochastic information
about the future, namely a probability distribution over scenarios of cost vs
click combinations. This summarizes the potential states of the world assuming
that the strategies of other players are fixed. Then, the best response can be
abstracted as stochastic budget optimization problems to figure out how to
spread a given budget across these keywords to maximize the expected number of
clicks.
We present the first known non-trivial poly-logarithmic approximation for
these problems as well as the first known hardness results of getting better
than logarithmic approximation ratios in the various parameters involved. We
also identify several special cases of these problems of practical interest,
such as with fixed number of scenarios or with polynomial-sized parameters
related to cost, which are solvable either in polynomial time or with improved
approximation ratios. Stochastic budget optimization with scenarios has
sophisticated technical structure. Our approximation and hardness results come
from relating these problems to a special type of (0/1, bipartite) quadratic
programs inherent in them. Our research answers some open problems raised by
the authors in (Stochastic Models for Budget Optimization in Search-Based
Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio
Quantum transport on two-dimensional regular graphs
We study the quantum-mechanical transport on two-dimensional graphs by means
of continuous-time quantum walks and analyse the effect of different boundary
conditions (BCs). For periodic BCs in both directions, i.e., for tori, the
problem can be treated in a large measure analytically. Some of these results
carry over to graphs which obey open boundary conditions (OBCs), such as
cylinders or rectangles. Under OBCs the long time transition probabilities
(LPs) also display asymmetries for certain graphs, as a function of their
particular sizes. Interestingly, these effects do not show up in the marginal
distributions, obtained by summing the LPs along one direction.Comment: 22 pages, 11 figure, acceted for publication in J.Phys.
The effect of large-decoherence on mixing-time in Continuous-time quantum walks on long-range interacting cycles
In this paper, we consider decoherence in continuous-time quantum walks on
long-range interacting cycles (LRICs), which are the extensions of the cycle
graphs. For this purpose, we use Gurvitz's model and assume that every node is
monitored by the corresponding point contact induced the decoherence process.
Then, we focus on large rates of decoherence and calculate the probability
distribution analytically and obtain the lower and upper bounds of the mixing
time. Our results prove that the mixing time is proportional to the rate of
decoherence and the inverse of the distance parameter (\emph{m}) squared.
This shows that the mixing time decreases with increasing the range of
interaction. Also, what we obtain for \emph{m}=0 is in agreement with
Fedichkin, Solenov and Tamon's results \cite{FST} for cycle, and see that the
mixing time of CTQWs on cycle improves with adding interacting edges.Comment: 16 Pages, 2 Figure
Statistical Mechanics of maximal independent sets
The graph theoretic concept of maximal independent set arises in several
practical problems in computer science as well as in game theory. A maximal
independent set is defined by the set of occupied nodes that satisfy some
packing and covering constraints. It is known that finding minimum and
maximum-density maximal independent sets are hard optimization problems. In
this paper, we use cavity method of statistical physics and Monte Carlo
simulations to study the corresponding constraint satisfaction problem on
random graphs. We obtain the entropy of maximal independent sets within the
replica symmetric and one-step replica symmetry breaking frameworks, shedding
light on the metric structure of the landscape of solutions and suggesting a
class of possible algorithms. This is of particular relevance for the
application to the study of strategic interactions in social and economic
networks, where maximal independent sets correspond to pure Nash equilibria of
a graphical game of public goods allocation
Terrestrial Implications of Cosmological Gamma-Ray Burst Models
The observation by the BATSE instrument on the Compton Gamma Ray Observatory
that gamma-ray bursts (GRBs) are distributed isotropically around the Earth but
nonuniformly in distance has led to the widespread conclusion that GRBs are
most likely to be at cosmological distances, making them the most luminous
sources known in the Universe. If bursts arise from events that occur in normal
galaxies, such as neutron star binary inspirals, then they will also occur in
our Galaxy about every hundred thousand to million years. The gamma-ray flux at
the Earth due to a Galactic GRB would far exceed that from even the largest
solar flares. The absorption of this radiation in the atmosphere would
substantially increase the stratospheric nitric oxide concentration through
photodissociation of N, greatly reducing the ozone concentration for
several years through NO catalysis, with important biospheric effects due
to increased solar ultraviolet flux. A nearby GRB may also leave traces in
anomalous radionuclide abundances.Comment: uuencoded, gzip-ed postscript; 6 pages; submitted to ApJ Letter
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
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