Jean-Charles Faugère
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Jean-Charles Faugère has a total of 54 co-authors. Here are the top co-authors by number of shared publications:
- Ludovic Perret
Laboratoire d'Informatique de Paris 6 LIP6 Sorbonne Université CNRS INRIA Équipe PolSys 4 place Jussieu F-75005 Paris France
12 shared publications - Jérémy Berthomieu
Sorbonne Université CNRS LIP6 Paris France
4 shared publications - Carlos Cid
Royal Holloway, University of London
3 shared publications - Guénaël Renault
INRIA, POLSYS, Centre Paris-Rocquencourt
3 shared publications - Mohab Safey El Din
Sorbonne Université, CNRS, LIP6, Paris, France
3 shared publications - Ayoub Otmani
University of Rouen
2 shared publications
Latest Publications
- 18 April 2021
- Published by Springer Nature in Designs, Codes and Cryptography
- Vol. 89 (6) , 1335-1364
Cryptanalysis of the extension field cancellation cryptosystem
- 18 April 2021
- Published by Springer Nature in Designs, Codes and Cryptography
- Vol. 89 (6) , 1335-1364
Cryptanalysis of the extension field cancellation cryptosystem
- Open Access
A nearly optimal algorithm to decompose binary forms
Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous polynomials, that is to write a homogeneous polynomial in n variables of degree D as a sum of D-th powers of linear forms, using the minimal number of summands. This minimal number is called the rank of the polynomial/tensor. We focus on decomposing binary forms, a problem that corresponds to the decomposition of symmetric tensors of dimension 2 and order D. Under this formulation, the problem finds its roots in invariant theory where the decompositions are known as canonical forms. In this context many different algorithms were proposed. We introduce a superfast algorithm that improves the previous approaches with results from structured linear algebra. It achieves a softly linear arithmetic complexity bound. To the best of our knowledge, the previously known algorithms have at least quadratic complexity bounds. Our algorithm computes a symbolic decomposition in $O(M(D) log(D))$ arithmetic operations, where $M(D)$ is the complexity of multiplying two polynomials of degree D. It is deterministic when the decomposition is unique. When the decomposition is not unique, our algorithm is randomized. We present a Monte Carlo version of it and we show how to modify it to a Las Vegas one, within the same complexity. From the symbolic decomposition, we approximate the terms of the decomposition with an error of $2^{--$\epsilon$}$ , in $O(D log^2(D) (log^2(D) + log($\epsilon$)))$ arithmetic operations. We use results from Kaltofen and Yagati (1989) to bound the size of the representation of the coefficients involved in the decomposition and we bound the algebraic degree of the problem by min(rank, D -- rank + 1). We show that this bound can be tight. When the input polynomial has integer coefficients, our algorithm performs, up to poly-logarithmic factors, $O\_{bit} (D{\ell} + D^4 + D^3 $\tau$)$ bit operations, where $$\tau$$ is the maximum bitsize of the coefficients and $2^{--{\ell}}$ is the relative error of the terms in the decomposition.Open AccessA nearly optimal algorithm to decompose binary forms
Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous polynomials, that is to write a homogeneous polynomial in n variables of degree D as a sum of D-th powers of linear forms, using the minimal number of summands. This minimal number is called the rank of the polynomial/tensor. We focus on decomposing binary forms, a problem that corresponds to the decomposition of symmetric tensors of dimension 2 and order D. Under this formulation, the problem finds its roots in invariant theory where the decompositions are known as canonical forms. In this context many different algorithms were proposed. We introduce a superfast algorithm that improves the previous approaches with results from structured linear algebra. It achieves a softly linear arithmetic complexity bound. To the best of our knowledge, the previously known algorithms have at least quadratic complexity bounds. Our algorithm computes a symbolic decomposition in $O(M(D) log(D))$ arithmetic operations, where $M(D)$ is the complexity of multiplying two polynomials of degree D. It is deterministic when the decomposition is unique. When the decomposition is not unique, our algorithm is randomized. We present a Monte Carlo version of it and we show how to modify it to a Las Vegas one, within the same complexity. From the symbolic decomposition, we approximate the terms of the decomposition with an error of $2^{--$\epsilon$}$ , in $O(D log^2(D) (log^2(D) + log($\epsilon$)))$ arithmetic operations. We use results from Kaltofen and Yagati (1989) to bound the size of the representation of the coefficients involved in the decomposition and we bound the algebraic degree of the problem by min(rank, D -- rank + 1). We show that this bound can be tight. When the input polynomial has integer coefficients, our algorithm performs, up to poly-logarithmic factors, $O\_{bit} (D{\ell} + D^4 + D^3 $\tau$)$ bit operations, where $$\tau$$ is the maximum bitsize of the coefficients and $2^{--{\ell}}$ is the relative error of the terms in the decomposition. - 1 October 2018
- Published by Springer Nature in Designs, Codes and Cryptography
- Vol. 86 (10) , 2279-2314
The point decomposition problem over hyperelliptic curves
- 1 October 2018
- Published by Springer Nature in Designs, Codes and Cryptography
- Vol. 86 (10) , 2279-2314
The point decomposition problem over hyperelliptic curves
- Open Access
In-depth comparison of the Berlekamp--Massey--Sakata and the Scalar-FGLM algorithms: the adaptive variants
The Berlekamp--Massey--Sakata algorithm and the Scalar-FGLM algorithm both compute the ideal of relations of a multidimensional linear recurrent sequence.Whenever quering a single sequence element is prohibitive, the bottleneck of these algorithms becomes the computation of all the needed sequence terms. As such, having adaptive variants of these algorithms, reducing the number of sequence queries, becomes mandatory.A native adaptive variant of the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive Scalar-FGLM algorithm.In this paper, our first contribution is to make the Berlekamp--Massey--Sakata algorithm more efficient by making it adaptive to avoid some useless relation test-ings. This variant allows us to divide by four in dimension 2 and by seven in dimension 3 the number of basic operations performed on some sequence family.Then, we compare the two adaptive algorithms. We show that their behaviors differ in a way that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other. We detail precisely the differences and the similarities of both algorithms and conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries and performs fewer basic operations than the Adaptive Berlekamp--Massey--Sakata algorithm.We also show that these variants are always more efficient than the original algorithms.
Open AccessIn-depth comparison of the Berlekamp--Massey--Sakata and the Scalar-FGLM algorithms: the adaptive variants
The Berlekamp--Massey--Sakata algorithm and the Scalar-FGLM algorithm both compute the ideal of relations of a multidimensional linear recurrent sequence.Whenever quering a single sequence element is prohibitive, the bottleneck of these algorithms becomes the computation of all the needed sequence terms. As such, having adaptive variants of these algorithms, reducing the number of sequence queries, becomes mandatory.A native adaptive variant of the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive Scalar-FGLM algorithm.In this paper, our first contribution is to make the Berlekamp--Massey--Sakata algorithm more efficient by making it adaptive to avoid some useless relation test-ings. This variant allows us to divide by four in dimension 2 and by seven in dimension 3 the number of basic operations performed on some sequence family.Then, we compare the two adaptive algorithms. We show that their behaviors differ in a way that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other. We detail precisely the differences and the similarities of both algorithms and conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries and performs fewer basic operations than the Adaptive Berlekamp--Massey--Sakata algorithm.We also show that these variants are always more efficient than the original algorithms.
- Open Access
In-depth comparison of the Berlekamp -- Massey -- Sakata and the Scalar-FGLM algorithms: the non adaptive variants
We compare thoroughly the Berlekamp -- Massey -- Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multi-dimensional linear recurrent sequence. Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.
Open AccessIn-depth comparison of the Berlekamp -- Massey -- Sakata and the Scalar-FGLM algorithms: the non adaptive variants
We compare thoroughly the Berlekamp -- Massey -- Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multi-dimensional linear recurrent sequence. Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.
- Open Access
In-depth comparison of the Berlekamp – Massey – Sakata and the Scalar-FGLM algorithms: the non adaptive variants
We compare thoroughly the Berlekamp – Massey – Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multi-dimensional linear recurrent sequence. Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.
Open AccessIn-depth comparison of the Berlekamp – Massey – Sakata and the Scalar-FGLM algorithms: the non adaptive variants
We compare thoroughly the Berlekamp – Massey – Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multi-dimensional linear recurrent sequence. Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.
- Open Access
Algebraic-geometric techniques for the feedback classification and robustness of the optimal control of a pair of Bloch equations with application to Magnetic Resonance Imaging
The aim of this article is to classify the singular trajectories associated with the optimal control problems of a pair of controlled Bloch equations. The motivation is to analyze the robustness of the optimal solutions to the contrast and the time-minimal saturation problem, in magnetic resonance imaging, with respect to the parameters and B1-inhomogeneity. For this purpose, we use various computer algebra algorithms and methods to study solutions of polynomial systems of equations and inequalities which are used for classification issues: Gröbner basis, cylindrical algebraic decomposition of semi-algebraic sets, Thom's isotopy lemma.
Open AccessAlgebraic-geometric techniques for the feedback classification and robustness of the optimal control of a pair of Bloch equations with application to Magnetic Resonance Imaging
The aim of this article is to classify the singular trajectories associated with the optimal control problems of a pair of controlled Bloch equations. The motivation is to analyze the robustness of the optimal solutions to the contrast and the time-minimal saturation problem, in magnetic resonance imaging, with respect to the parameters and B1-inhomogeneity. For this purpose, we use various computer algebra algorithms and methods to study solutions of polynomial systems of equations and inequalities which are used for classification issues: Gröbner basis, cylindrical algebraic decomposition of semi-algebraic sets, Thom's isotopy lemma.
- Open Access
Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems
B{\'e}zout 's theorem states that dense generic systems of n multivariate quadratic equations in n variables have 2 n solutions over algebraically closed fields. When only a small subset M of monomials appear in the equations (fewnomial systems), the number of solutions may decrease dramatically. We focus in this work on subsets of quadratic monomials M such that generic systems with support M do not admit any solution at all. For these systems, Hilbert's Nullstellensatz ensures the existence of algebraic certificates of inconsistency. However, up to our knowledge all known bounds on the sizes of such certificates -including those which take into account the Newton polytopes of the polynomials- are exponential in n. Our main results show that if the inequality 2|M| -- 2n $\le$ $\sqrt$ 1 + 8{\nu} -- 1 holds for a quadratic fewnomial system -- where {\nu} is the matching number of a graph associated with M, and |M| is the cardinality of M -- then there exists generically a certificate of inconsistency of linear size (measured as the number of coefficients in the ground field K). Moreover this certificate can be computed within a polynomial number of arithmetic operations. Next, we evaluate how often this inequality holds, and we give evidence that the probability that the inequality is satisfied depends strongly on the number of squares. More precisely, we show that if M is picked uniformly at random among the subsets of n + k + 1 quadratic monomials containing at least $\Omega$(n 1/2+$\epsilon$) squares, then the probability that the inequality holds tends to 1 as n grows. Interestingly, this phenomenon is related with the matching number of random graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results showing that certificates in inconsistency can be computed for systems with more than 10000 variables and equations.Open AccessComputing Small Certificates of Inconsistency of Quadratic Fewnomial Systems
B{\'e}zout 's theorem states that dense generic systems of n multivariate quadratic equations in n variables have 2 n solutions over algebraically closed fields. When only a small subset M of monomials appear in the equations (fewnomial systems), the number of solutions may decrease dramatically. We focus in this work on subsets of quadratic monomials M such that generic systems with support M do not admit any solution at all. For these systems, Hilbert's Nullstellensatz ensures the existence of algebraic certificates of inconsistency. However, up to our knowledge all known bounds on the sizes of such certificates -including those which take into account the Newton polytopes of the polynomials- are exponential in n. Our main results show that if the inequality 2|M| -- 2n $\le$ $\sqrt$ 1 + 8{\nu} -- 1 holds for a quadratic fewnomial system -- where {\nu} is the matching number of a graph associated with M, and |M| is the cardinality of M -- then there exists generically a certificate of inconsistency of linear size (measured as the number of coefficients in the ground field K). Moreover this certificate can be computed within a polynomial number of arithmetic operations. Next, we evaluate how often this inequality holds, and we give evidence that the probability that the inequality is satisfied depends strongly on the number of squares. More precisely, we show that if M is picked uniformly at random among the subsets of n + k + 1 quadratic monomials containing at least $\Omega$(n 1/2+$\epsilon$) squares, then the probability that the inequality holds tends to 1 as n grows. Interestingly, this phenomenon is related with the matching number of random graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results showing that certificates in inconsistency can be computed for systems with more than 10000 variables and equations. - Open Access
GBLA -- Gröbner Basis Linear Algebra Package
This is a system paper about a new GPLv2 open source C library GBLA implementing and improving the idea of Faug\`ere and Lachartre (GB reduction). We further exploit underlying structures in matrices generated during Gr\"obner basis computations in algorithms like F4 or F5 taking advantage of block patterns by using a special data structure called multilines. Moreover, we discuss a new order of operations for the reduction process. In various different experimental results we show that GBLA performs better than GB reduction or Magma in sequential computations (up to 40% faster) and scales much better than GB reduction for a higher number of cores: On 32 cores we reach a scaling of up to 26. GBLA is up to 7 times faster than GB reduction. Further, we compare different parallel schedulers GBLA can be used with. We also developed a new advanced storage format that exploits the fact that our matrices are coming from Gr\"obner basis computations, shrinking storage by a factor of up to 4. A huge database of our matrices is freely available with GBLA.
Open AccessGBLA -- Gröbner Basis Linear Algebra Package
This is a system paper about a new GPLv2 open source C library GBLA implementing and improving the idea of Faug\`ere and Lachartre (GB reduction). We further exploit underlying structures in matrices generated during Gr\"obner basis computations in algorithms like F4 or F5 taking advantage of block patterns by using a special data structure called multilines. Moreover, we discuss a new order of operations for the reduction process. In various different experimental results we show that GBLA performs better than GB reduction or Magma in sequential computations (up to 40% faster) and scales much better than GB reduction for a higher number of cores: On 32 cores we reach a scaling of up to 26. GBLA is up to 7 times faster than GB reduction. Further, we compare different parallel schedulers GBLA can be used with. We also developed a new advanced storage format that exploits the fact that our matrices are coming from Gr\"obner basis computations, shrinking storage by a factor of up to 4. A huge database of our matrices is freely available with GBLA.