7,537 research outputs found
Fast Quantum Algorithm for Solving Multivariate Quadratic Equations
In August 2015 the cryptographic world was shaken by a sudden and surprising
announcement by the US National Security Agency NSA concerning plans to
transition to post-quantum algorithms. Since this announcement post-quantum
cryptography has become a topic of primary interest for several standardization
bodies. The transition from the currently deployed public-key algorithms to
post-quantum algorithms has been found to be challenging in many aspects. In
particular the problem of evaluating the quantum-bit security of such
post-quantum cryptosystems remains vastly open. Of course this question is of
primarily concern in the process of standardizing the post-quantum
cryptosystems. In this paper we consider the quantum security of the problem of
solving a system of {\it Boolean multivariate quadratic equations in
variables} (\MQb); a central problem in post-quantum cryptography. When ,
under a natural algebraic assumption, we present a Las-Vegas quantum algorithm
solving \MQb{} that requires the evaluation of, on average,
quantum gates. To our knowledge this is the fastest algorithm for solving
\MQb{}
On the Complexity of Solving Quadratic Boolean Systems
A fundamental problem in computer science is to find all the common zeroes of
quadratic polynomials in unknowns over . The
cryptanalysis of several modern ciphers reduces to this problem. Up to now, the
best complexity bound was reached by an exhaustive search in
operations. We give an algorithm that reduces the problem to a combination of
exhaustive search and sparse linear algebra. This algorithm has several
variants depending on the method used for the linear algebra step. Under
precise algebraic assumptions on the input system, we show that the
deterministic variant of our algorithm has complexity bounded by
when , while a probabilistic variant of the Las Vegas type
has expected complexity . Experiments on random systems show
that the algebraic assumptions are satisfied with probability very close to~1.
We also give a rough estimate for the actual threshold between our method and
exhaustive search, which is as low as~200, and thus very relevant for
cryptographic applications.Comment: 25 page
ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra
Background: Many biological systems are modeled qualitatively with discrete
models, such as probabilistic Boolean networks, logical models, Petri nets, and
agent-based models, with the goal to gain a better understanding of the system.
The computational complexity to analyze the complete dynamics of these models
grows exponentially in the number of variables, which impedes working with
complex models. Although there exist sophisticated algorithms to determine the
dynamics of discrete models, their implementations usually require
labor-intensive formatting of the model formulation, and they are oftentimes
not accessible to users without programming skills. Efficient analysis methods
are needed that are accessible to modelers and easy to use. Method: By
converting discrete models into algebraic models, tools from computational
algebra can be used to analyze their dynamics. Specifically, we propose a
method to identify attractors of a discrete model that is equivalent to solving
a system of polynomial equations, a long-studied problem in computer algebra.
Results: A method for efficiently identifying attractors, and the web-based
tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other
analysis methods for discrete models. ADAM converts several discrete model
types automatically into polynomial dynamical systems and analyzes their
dynamics using tools from computer algebra. Based on extensive experimentation
with both discrete models arising in systems biology and randomly generated
networks, we found that the algebraic algorithms presented in this manuscript
are fast for systems with the structure maintained by most biological systems,
namely sparseness, i.e., while the number of nodes in a biological network may
be quite large, each node is affected only by a small number of other nodes,
and robustness, i.e., small number of attractors
New Developments in Quantum Algorithms
In this survey, we describe two recent developments in quantum algorithms.
The first new development is a quantum algorithm for evaluating a Boolean
formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This
provides quantum speedups for any problem that can be expressed via Boolean
formulas. This result can be also extended to span problems, a generalization
of Boolean formulas. This provides an optimal quantum algorithm for any Boolean
function in the black-box query model.
The second new development is a quantum algorithm for solving systems of
linear equations. In contrast with traditional algorithms that run in time
O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in
time O(\log^c N). It outputs a quantum state describing the solution of the
system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201
Dimension Reduction of Large AND-NOT Network Models
Boolean networks have been used successfully in modeling biological networks
and provide a good framework for theoretical analysis. However, the analysis of
large networks is not trivial. In order to simplify the analysis of such
networks, several model reduction algorithms have been proposed; however, it is
not clear if such algorithms scale well with respect to the number of nodes.
The goal of this paper is to propose and implement an algorithm for the
reduction of AND-NOT network models for the purpose of steady state
computation. Our method of network reduction is the use of "steady state
approximations" that do not change the number of steady states. Our algorithm
is designed to work at the wiring diagram level without the need to evaluate or
simplify Boolean functions. Also, our implementation of the algorithm takes
advantage of the sparsity typical of discrete models of biological systems. The
main features of our algorithm are that it works at the wiring diagram level,
it runs in polynomial time, and it preserves the number of steady states. We
used our results to study AND-NOT network models of gene networks and showed
that our algorithm greatly simplifies steady state analysis. Furthermore, our
algorithm can handle sparse AND-NOT networks with up to 1000000 nodes
Spectrum optimization in multi-user multi-carrier systems with iterative convex and nonconvex approximation methods
Several practical multi-user multi-carrier communication systems are
characterized by a multi-carrier interference channel system model where the
interference is treated as noise. For these systems, spectrum optimization is a
promising means to mitigate interference. This however corresponds to a
challenging nonconvex optimization problem. Existing iterative convex
approximation (ICA) methods consist in solving a series of improving convex
approximations and are typically implemented in a per-user iterative approach.
However they do not take this typical iterative implementation into account in
their design. This paper proposes a novel class of iterative approximation
methods that focuses explicitly on the per-user iterative implementation, which
allows to relax the problem significantly, dropping joint convexity and even
convexity requirements for the approximations. A systematic design framework is
proposed to construct instances of this novel class, where several new
iterative approximation methods are developed with improved per-user convex and
nonconvex approximations that are both tighter and simpler to solve (in
closed-form). As a result, these novel methods display a much faster
convergence speed and require a significantly lower computational cost.
Furthermore, a majority of the proposed methods can tackle the issue of getting
stuck in bad locally optimal solutions, and hence improve solution quality
compared to existing ICA methods.Comment: 33 pages, 7 figures. This work has been submitted for possible
publicatio
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