214 research outputs found
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
Solving polynomial systems via symbolic-numeric reduction to geometric involutive form
AbstractWe briefly survey several existing methods for solving polynomial systems with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method
Moment ideals of local Dirac mixtures
In this paper we study ideals arising from moments of local Dirac measures
and their mixtures. We provide generators for the case of first order local
Diracs and explain how to obtain the moment ideal of the Pareto distribution
from them. We then use elimination theory and Prony's method for parameter
estimation of finite mixtures. Our results are showcased with applications in
signal processing and statistics. We highlight the natural connections to
algebraic statistics, combinatorics and applications in analysis throughout the
paper.Comment: 26 pages, 3 figure
Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases
International audienc
VDOO: A Short, Fast, Post-Quantum Multivariate Digital Signature Scheme
Hard lattice problems are predominant in constructing post-quantum
cryptosystems. However, we need to continue developing post-quantum
cryptosystems based on other quantum hard problems to prevent a complete
collapse of post-quantum cryptography due to a sudden breakthrough in solving
hard lattice problems. Solving large multivariate quadratic systems is one such
quantum hard problem.
Unbalanced Oil-Vinegar is a signature scheme based on the hardness of solving
multivariate equations. In this work, we present a post-quantum digital
signature algorithm VDOO (Vinegar-Diagonal-Oil-Oil) based on solving
multivariate equations. We introduce a new layer called the diagonal layer over
the oil-vinegar-based signature scheme Rainbow. This layer helps to improve the
security of our scheme without increasing the parameters considerably. Due to
this modification, the complexity of the main computational bottleneck of
multivariate quadratic systems i.e. the Gaussian elimination reduces
significantly. Thus making our scheme one of the fastest multivariate quadratic
signature schemes. Further, we show that our carefully chosen parameters can
resist all existing state-of-the-art attacks. The signature sizes of our scheme
for the National Institute of Standards and Technology's security level of I,
III, and V are 96, 226, and 316 bytes, respectively. This is the smallest
signature size among all known post-quantum signature schemes of similar
security
On the Bit Complexity of Sum-of-Squares Proofs
It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm. In a recent paper, Ryan O\u27Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits low-degree proofs of non-negativity, but these proofs necessarily involve numbers with an exponential number of bits, causing the Ellipsoid algorithm to take exponential time. In this paper we obtain both positive and negative results on the bit complexity of SoS proofs.
First, we propose a sufficient condition on a polynomial system that implies a bound on the coefficients in an SoS proof. We demonstrate that this sufficient condition is applicable for common use-cases of the SoS algorithm, such as Max-CSP, Balanced Separator, Max-Clique, Max-Bisection, and Unit-Vector constraints.
On the negative side, O\u27Donnell asked whether every polynomial system containing Boolean constraints admits proofs of polynomial bit complexity. We answer this question in the negative, giving a counterexample system and non-negative polynomial which has degree two SoS proofs, but no SoS proof with small coefficients until degree sqrt(n)
Computational Methods for Computer Vision : Minimal Solvers and Convex Relaxations
Robust fitting of geometric models is a core problem in computer vision. The most common approach is to use a hypothesize-and-test framework, such as RANSAC. In these frameworks the model is estimated from as few measurements as possible, which minimizes the risk of selecting corrupted measurements. These estimation problems are called minimal problems, and they can often be formulated as systems of polynomial equations. In this thesis we present new methods for building so-called minimal solvers or polynomial solvers, which are specialized code for solving such systems. On several minimal problems we improve on the state-of-the-art both with respect to numerical stability and execution time.In many computer vision problems low rank matrices naturally occur. The rank can serve as a measure of model complexity and typically a low rank is desired. Optimization problems containing rank penalties or constraints are in general difficult. Recently convex relaxations, such as the nuclear norm, have been used to make these problems tractable. In this thesis we present new convex relaxations for rank-based optimization which avoid drawbacks of previous approaches and provide tighter relaxations. We evaluate our methods on a number of real and synthetic datasets and show state-of-the-art results
A new approach based on quadratic forms to attack the McEliece cryptosystem
We bring in here a novel algebraic approach for attacking the McEliece
cryptosystem. It consists in introducing a subspace of matrices representing
quadratic forms. Those are associated with quadratic relationships for the
component-wise product in the dual of the code used in the cryptosystem.
Depending on the characteristic of the code field, this space of matrices
consists only of symmetric matrices or skew-symmetric matrices. This matrix
space is shown to contain unusually low-rank matrices (rank or
depending on the characteristic) which reveal the secret polynomial structure
of the code. Finding such matrices can then be used to recover the secret key
of the scheme. We devise a dedicated approach in characteristic consisting
in using a Gr\"obner basis modeling that a skew-symmetric matrix is of rank
. This allows to analyze the complexity of solving the corresponding
algebraic system with Gr\"obner bases techniques. This computation behaves
differently when applied to the skew-symmetric matrix space associated with a
random code rather than with a Goppa or an alternant code. This gives a
distinguisher of the latter code family. We give a bound on its complexity
which turns out to interpolate nicely between polynomial and exponential
depending on the code parameters. A distinguisher for alternant/Goppa codes was
already known [FGO+11]. It is of polynomial complexity but works only in a
narrow parameter regime. This new distinguisher is also polynomial for the
parameter regime necessary for [FGO+11] but contrarily to the previous one is
able to operate for virtually all code parameters relevant to cryptography.
Moreover, we use this matrix space to find a polynomial time attack of the
McEliece cryptosystem provided that the Goppa code is distinguishable by the
method of [FGO+11] and its degree is less than , where is the alphabet
size of the code.Comment: 61 page
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