1,080 research outputs found
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
The F5 Criterion revised
The purpose of this work is to generalize part of the theory behind Faugere's
"F5" algorithm. This is one of the fastest known algorithms to compute a
Groebner basis of a polynomial ideal I generated by polynomials
f_{1},...,f_{m}. A major reason for this is what Faugere called the algorithm's
"new" criterion, and we call "the F5 criterion"; it provides a sufficient
condition for a set of polynomials G to be a Groebner basis. However, the F5
algorithm is difficult to grasp, and there are unresolved questions regarding
its termination.
This paper introduces some new concepts that place the criterion in a more
general setting: S-Groebner bases and primitive S-irreducible polynomials. We
use these to propose a new, simple algorithm based on a revised F5 criterion.
The new concepts also enable us to remove various restrictions, such as proving
termination without the requirement that f_{1},...,f_{m} be a regular sequence.Comment: Originally submitted by Arri in 2009, with material added by Perry
since 2010. The 2016 editions correct typographical issues not caught in
previous editions bring the theory of the body into conformity with the
published version of the pape
A survey on signature-based Gr\"obner basis computations
This paper is a survey on the area of signature-based Gr\"obner basis
algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain
the general ideas behind the usage of signatures. We show how to classify the
various known variants by 3 different orderings. For this we give translations
between different notations and show that besides notations many approaches are
just the same. Moreover, we give a general description of how the idea of
signatures is quite natural when performing the reduction process using linear
algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table
On formulas for decoding binary cyclic codes
We adress the problem of the algebraic decoding of any cyclic code up to the
true minimum distance. For this, we use the classical formulation of the
problem, which is to find the error locator polynomial in terms of the syndroms
of the received word. This is usually done with the Berlekamp-Massey algorithm
in the case of BCH codes and related codes, but for the general case, there is
no generic algorithm to decode cyclic codes. Even in the case of the quadratic
residue codes, which are good codes with a very strong algebraic structure,
there is no available general decoding algorithm. For this particular case of
quadratic residue codes, several authors have worked out, by hand, formulas for
the coefficients of the locator polynomial in terms of the syndroms, using the
Newton identities. This work has to be done for each particular quadratic
residue code, and is more and more difficult as the length is growing.
Furthermore, it is error-prone. We propose to automate these computations,
using elimination theory and Grbner bases. We prove that, by computing
appropriate Grbner bases, one automatically recovers formulas for the
coefficients of the locator polynomial, in terms of the syndroms
On the Complexity of the F5 Gr\"obner basis Algorithm
We study the complexity of Gr\"obner bases computation, in particular in the
generic situation where the variables are in simultaneous Noether position with
respect to the system.
We give a bound on the number of polynomials of degree in a Gr\"obner
basis computed by Faug\`ere's algorithm~(Fau02) in this generic case for
the grevlex ordering (which is also a bound on the number of polynomials for a
reduced Gr\"obner basis, independently of the algorithm used). Next, we analyse
more precisely the structure of the polynomials in the Gr\"obner bases with
signatures that computes and use it to bound the complexity of the
algorithm.
Our estimates show that the version of~ we analyse, which uses only
standard Gaussian elimination techniques, outperforms row reduction of the
Macaulay matrix with the best known algorithms for moderate degrees, and even
for degrees up to the thousands if Strassen's multiplication is used. The
degree being fixed, the factor of improvement grows exponentially with the
number of variables.Comment: 24 page
Moment Varieties of Gaussian Mixtures
The points of a moment variety are the vectors of all moments up to some
order of a family of probability distributions. We study this variety for
mixtures of Gaussians. Following up on Pearson's classical work from 1894, we
apply current tools from computational algebra to recover the parameters from
the moments. Our moment varieties extend objects familiar to algebraic
geometers. For instance, the secant varieties of Veronese varieties are the
loci obtained by setting all covariance matrices to zero. We compute the ideals
of the 5-dimensional moment varieties representing mixtures of two univariate
Gaussians, and we offer a comparison to the maximum likelihood approach.Comment: 17 pages, 2 figure
Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case
Let and be
two sets of nonlinear polynomials over
( being a field). We consider the computational problem of finding
-- if any -- an invertible transformation on the variables mapping
to . The corresponding equivalence problem is known as {\tt
Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental
problem in multivariate cryptography. The main result is a randomized
polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a
particular case of importance in cryptography and somewhat justifying {\it a
posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic
instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt
IP1S} for quadratic polynomials can be reduced to a variant of the classical
module isomorphism problem in representation theory, which involves to test the
orthogonal simultaneous conjugacy of symmetric matrices. We show that we can
essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to
test the orthogonal simultaneous similarity of symmetric matrices; this latter
problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding
an invertible matrix in the linear space of matrices over and to compute the square root in a matrix
algebra. While computing square roots of matrices can be done efficiently using
numerical methods, it seems difficult to control the bit complexity of such
methods. However, we present exact and polynomial-time algorithms for computing
the square root in for various fields (including
finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt
IP1S} for quadratic instances. In particular, we provide a (complete)
characterization of the automorphism group of homogeneous quadratic
polynomials. Finally, we also consider the more general {\it Isomorphism of
Polynomials} ({\tt IP}) problem where we allow an invertible linear
transformation on the variables \emph{and} on the set of polynomials. A
randomized polynomial-time algorithm for solving {\tt IP} when
is presented. From an algorithmic point
of view, the problem boils down to factoring the determinant of a linear matrix
(\emph{i.e.}\ a matrix whose components are linear polynomials). This extends
to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3
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