12 research outputs found
Algebraic Relation of Three MinRank Algebraic Modelings
We give algebraic relations among equations of three algebraic modelings for MinRank problem: support minors modeling, Kipnis–Shamir modeling and minors modeling
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
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
Improvement of algebraic attacks for solving superdetermined MinRank instances
The MinRank (MR) problem is a computational problem that arises in many
cryptographic applications. In Verbel et al. (PQCrypto 2019), the authors
introduced a new way to solve superdetermined instances of the MinRank problem,
starting from the bilinear Kipnis-Shamir (KS) modeling. They use linear algebra
on specific Macaulay matrices, considering only multiples of the initial
equations by one block of variables, the so called ''kernel'' variables. Later,
Bardet et al. (Asiacrypt 2020) introduced a new Support Minors modeling (SM),
that consider the Pl{\"u}cker coordinates associated to the kernel variables,
i.e. the maximal minors of the Kernel matrix in the KS modeling. In this paper,
we give a complete algebraic explanation of the link between the (KS) and (SM)
modelings (for any instance). We then show that superdetermined MinRank
instances can be seen as easy instances of the SM modeling. In particular, we
show that performing computation at the smallest possible degree (the ''first
degree fall'') and the smallest possible number of variables is not always the
best strategy. We give complexity estimates of the attack for generic random
instances.We apply those results to the DAGS cryptosystem, that was submitted
to the first round of the NIST standardization process. We show that the
algebraic attack from Barelli and Couvreur (Asiacrypt 2018), improved in Bardet
et al. (CBC 2019), is a particular superdetermined MinRank instance.Here, the
instances are not generic, but we show that it is possible to analyse the
particular instances from DAGS and provide a way toselect the optimal
parameters (number of shortened positions) to solve a particular instance
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
Homotopy algorithms for solving structured determinantal systems
Multivariate polynomial systems arising in numerous applications have special structures. In particular, determinantal structures and invariant systems appear in a wide range of applications such as in polynomial optimization and related questions in real algebraic geometry. The goal of this thesis is to provide efficient algorithms to solve such structured systems.
In order to solve the first kind of systems, we design efficient algorithms by using the symbolic homotopy continuation techniques. While the homotopy methods, in both numeric and symbolic, are well-understood and widely used in polynomial system solving for square systems, the use of these methods to solve over-detemined systems is not so clear. Meanwhile, determinantal systems are over-determined with more equations than unknowns. We provide probabilistic homotopy algorithms which take advantage of the determinantal structure to compute isolated points in the zero-sets of determinantal systems. The runtimes of our algorithms are polynomial in the sum of the multiplicities of isolated points and the degree of the homotopy curve. We also give the bounds on the
number of isolated points that we have to compute in three contexts: all entries of the input are in classical polynomial rings, all these polynomials are sparse, and they are weighted polynomials.
In the second half of the thesis, we deal with the problem of finding critical points of a symmetric polynomial map on an invariant algebraic set. We exploit the invariance properties of the input to split the solution space according to the orbits of the symmetric group. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in the number of points that we have to compute. Our results are illustrated by applications in studying real algebraic sets defined by invariant polynomial systems by the means of the critical point method
A survey on signature-based algorithms for computing Gröbner basis computations
International audienceThis paper is a survey on the area of signature-based Gröbner basis algorithms that was initiated by Faugère'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