arXiv subjects

Magali Bardet

Publications and source records attributed to Magali Bardet.

18 recordsLinked to original sources

arXiv2025

The Matrix Subcode Equivalence problem and its application to signature with MPC-in-the-Head

Nowadays, equivalence problems are widely used in cryptography, most notably to establish cryptosystems such as digital signatures, with MEDS, LESS, PERK as the most recent ones. However, in the context of matrix codes, only the code equivalence problem has been studied, while the subcode equivalence is well-defined in the Hamming metric. In this work, we introduce two new problems: the Matrix Subcode Equivalence Problem and the Matrix Code Permuted Kernel Problem, to which we apply the MPCitH paradigm to build a signature scheme. These new problems, closely related to the Matrix Code Equivalence problem, ask to find an isometry given a code $C$ and a subcode $D$. Furthermore, we prove that the Matrix Subcode Equivalence problem reduces to the Hamming Subcode Equivalence problem, which is known to be NP-Complete, thus introducing the matrix code version of the Permuted Kernel Problem. We also adapt the combinatorial and algebraic algorithms for the Matrix Code Equivalence problem to the subcode case, and we analyze their complexities. We find with this analysis that the algorithms perform much worse than in the code equivalence case, which is the same as what happens in the Hamming metric. Finally, our analysis of the attacks allows us to take parameters much smaller than in the Matrix Code Equivalence case. Coupled with the effectiveness of \textit{Threshold-Computation-in-the-Head} or \textit{VOLE-in-the-Head}, we obtain a signature size of $\approx$ 4 800 Bytes, with a public key of $\approx$ 275 Bytes. We thus obtain a reasonable signature size, which brings diversity in the landscape of post-quantum signature schemes, by relying on a new hard problem. In particular, this new signature scheme performs better than SPHINCS+, with a smaller size of public key + signature. Our signature compares also well with other signature schemes: compared to MEDS, the signature is smaller, and we reduced the size of the sum of signature and public key by a factor close to 5. We also obtain a signature size that is almost half the size of the CROSS signature scheme.

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arXiv2025

Computation of the Hilbert Series for the Support-Minors Modeling of the MinRank Problem

The MinRank problem is a simple linear algebra problem: given matrices with coefficients in a field, find a non trivial linear combination of the matrices that has a small rank. There are several algebraic modeling of the problem. The main ones are: the Kipnis-Shamir modeling, the Minors modeling and the Support-Minors modeling. The Minors modeling has been studied by Faug{\`e}re et al. in 2010, where the authors provide an analysis of the complexity of computing a Gr{\"o}bner basis of the modeling, through the computation of the exact Hilbert Series for a generic instance. For the Support-Minors modeling, the first terms of the Hilbert Series are given by Bardet et al. in 2020 based on an heuristic and experimental work. In this work, we provide a formula and a proof for the complete Hilbert Series of the Support Minors modeling for generic instances. This is done by adapting well known results on determinantal ideals to an ideal generated by a particular subset of the set of all minors of a matrix of variables. We then show that this ideal is generated by

cs.CR↗

arXiv2023

Revisiting Algebraic Attacks on MinRank and on the Rank Decoding Problem

The Rank Decoding problem (RD) is at the core of rank-based cryptography. This problem can also be seen as a structured version of MinRank, which is ubiquitous in multivariate cryptography. Recently, \cite{BBBGNRT20,BBCGPSTV20} proposed attacks based on two new algebraic modelings, namely the MaxMinors modeling which is specific to RD and the Support-Minors modeling which applies to MinRank in general. Both improved significantly the complexity of algebraic attacks on these two problems. In the case of RD and contrarily to what was believed up to now, these new attacks were shown to be able to outperform combinatorial attacks and this even for very small field sizes. However, we prove here that the analysis performed in \cite{BBCGPSTV20} for one of these attacks which consists in mixing the MaxMinors modeling with the Support-Minors modeling to solve RD is too optimistic and leads to underestimate the overall complexity. This is done by exhibiting linear dependencies between these equations and by considering an $\fqm$ version of these modelings which turns out to be instrumental for getting a better understanding of both systems. Moreover, by working over $\Fqm$ rather than over $\ff{q}$, we are able to drastically reduce the number of variables in the system and we (i) still keep enough algebraic equations to be able to solve the system, (ii) are able to analyze rigorously the complexity of our approach. This new approach may improve the older MaxMinors approach on RD from \cite{BBBGNRT20,BBCGPSTV20} for certain parameters. We also introduce a new hybrid approach on the Support-Minors system whose impact is much more general since it applies to any MinRank problem. This technique improves significantly the complexity of the Support-Minors approach for small to moderate field sizes.

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arXiv2023

Polynomial time key-recovery attack on high rate random alternant codes

A long standing open question is whether the distinguisher of high rate alternant codes or Goppa codes \cite{FGOPT11} can be turned into an algorithm recovering the algebraic structure of such codes from the mere knowledge of an arbitrary generator matrix of it. This would allow to break the McEliece scheme as soon as the code rate is large enough and would break all instances of the CFS signature scheme. We give for the first time a positive answer for this problem when the code is {\em a generic alternant code} and when the code field size $q$ is small : $q \in \{2,3\}$ and for {\em all} regime of other parameters for which the aforementioned distinguisher works. This breakthrough has been obtained by two different ingredients : (i) a way of using code shortening and the component-wise product of codes to derive from the original alternant code a sequence of alternant codes of decreasing degree up to getting an alternant code of degree $3$ (with a multiplier and support related to those of the original alternant code); (ii) an original Gröbner basis approach which takes into account the non standard constraints on the multiplier and support of an alternant code which recovers in polynomial time the relevant algebraic structure of an alternant code of degree $3$ from the mere knowledge of a basis for it.

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arXiv2022

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{ü}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.

cs.CR↗

arXiv2021

Decoding Reed-Solomon codes by solving a bilinear system with a Gröbner basis approach

Decoding a Reed-Solomon code can be modeled by a bilinear system which can be solved by Gröbner basis techniques. We will show that in this particular case, these techniques are much more efficient than for generic bilinear systems with the same number of unknowns and equations (where these techniques have exponential complexity). Here we show that they are able to solve the problem in polynomial time up to the Sudan radius. Moreover, beyond this radius these techniques recover automatically polynomial identities that are at the heart of improvements of the power decoding approach for reaching the Johnson decoding radius. They also allow to derive new polynomial identities that can be used to derive new algebraic decoding algorithms for Reed-Solomon codes. We provide numerical evidence that this sometimes allows to correct efficiently slightly more errors than the Johnson radius.

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arXiv2021

An algebraic approach to the Rank Support Learning problem

Rank-metric code-based cryptography relies on the hardness of decoding a random linear code in the rank metric. The Rank Support Learning problem (RSL) is a variant where an attacker has access to N decoding instances whose errors have the same support and wants to solve one of them. This problem is for instance used in the Durandal signature scheme. In this paper, we propose an algebraic attack on RSL which clearly outperforms the previous attacks to solve this problem. We build upon Bardet et al., Asiacrypt 2020, where similar techniques are used to solve MinRank and RD. However, our analysis is simpler and overall our attack relies on very elementary assumptions compared to standard Gr{ö}bner bases attacks. In particular, our results show that key recovery attacks on Durandal are more efficient than was previously thought.

cs.CR↗

arXiv2021

Improvements of Algebraic Attacks for solving the Rank Decoding and MinRank problems

Rank Decoding (RD) is the main underlying problem in rank-based cryptography. Based on this problem and quasi-cyclic versions of it, very efficient schemes have been proposed recently, such as those in the ROLLO and RQC submissions, which have reached the second round of the NIST Post-Quantum competition. Two main approaches have been studied to solve RD: combinatorial ones and algebraic ones. While the former has been studied extensively, a better understanding of the latter was recently obtained by Bardet et al. (EUROCRYPT20) where it appeared that algebraic attacks can often be more efficient than combinatorial ones for cryptographic parameters. This paper gives substantial improvements upon this attack in terms both of complexity and of the assumptions required by the cryptanalysis. We present attacks for ROLLO-I-128, 192, and 256 with bit complexity respectively in 70, 86, and 158, to be compared to 117, 144, and 197 for the aforementionned previous attack. Moreover, unlike this previous attack, ours does not need generic Gröbner basis algorithms since it only requires to solve a linear system. For a case called overdetermined, this modeling allows us to avoid Gröbner basis computations by going directly to solving a linear system. For the other case, called underdetermined, we also improve the results from the previous attack by combining the Ourivski-Johansson modeling together with a new modeling for a generic MinRank instance; the latter modeling allows us to refine the analysis of MinRank's complexity given in the paper by Verbel et al. (PQC19). Finally, since the proposed parameters of ROLLO and RQC are completely broken by our new attack, we give examples of new parameters for ROLLO and RQC that make them resistant to our attacks. These new parameters show that these systems remain attractive, with a loss of only about 50\% in terms of key size for ROLLO-I.

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arXiv2020

An Algebraic Attack on Rank Metric Code-Based Cryptosystems

The Rank metric decoding problem is the main problem considered in cryptography based on codes in the rank metric. Very efficient schemes based on this problem or quasi-cyclic versions of it have been proposed recently, such as those in the submissions ROLLO and RQC currently at the second round of the NIST Post-Quantum Cryptography Standardization Process. While combinatorial attacks on this problem have been extensively studied and seem now well understood, the situation is not as satisfactory for algebraic attacks, for which previous work essentially suggested that they were ineffective for cryptographic parameters. In this paper, starting from Ourivski and Johansson's algebraic modelling of the problem into a system of polynomial equations, we show how to augment this system with easily computed equations so that the augmented system is solved much faster via Groebner bases. This happens because the augmented system has solving degree $r$, $r+1$ or $r+2$ depending on the parameters, where $r$ is the rank weight, which we show by extending results from Verbel et al. (PQCrypto 2019) on systems arising from the MinRank problem; with target rank $r$, Verbel et al. lower the solving degree to $r+2$, and even less for some favorable instances that they call superdetermined. We give complexity bounds for this approach as well as practical timings of an implementation using Magma. This improves upon the previously known complexity estimates for both Groebner basis and (non-quantum) combinatorial approaches, and for example leads to an attack in 200 bits on ROLLO-I-256 whose claimed security was 256 bits.

cs.CR↗

arXiv2019

Practical Algebraic Attack on DAGS

DAGS scheme is a key encapsulation mechanism (KEM) based on quasi-dyadic alternant codes that was submitted to NIST standardization process for a quantum resistant public key algorithm. Recently an algebraic attack was devised by Barelli and Couvreur (Asiacrypt 2018) that efficiently recovers the private key. It shows that DAGS can be totally cryptanalysed by solving a system of bilinear polynomial equations. However, some sets of DAGS parameters were not broken in practice. In this paper we improve the algebraic attack by showing that the original approach was not optimal in terms of the ratio of the number of equations to the number of variables. Contrary to the common belief that reducing at any cost the number of variables in a polynomial system is always beneficial, we actually observed that, provided that the ratio is increased and up to a threshold, the solving can be heavily improved by adding variables to the polynomial system. This enables us to recover the private keys in a few seconds. Furthermore, our experimentations also show that the maximum degree reached during the computation of the Gröbner basis is an important parameter that explains the efficiency of the attack. Finally, the authors of DAGS updated the parameters to take into account the algebraic cryptanalysis of Barelli and Couvreur. In the present article, we propose a hybrid approach that performs an exhaustive search on some variables and computes a Gröbner basis on the polynomial system involving the remaining variables. We then show that the updated set of parameters corresponding to 128-bit security can be broken with 2^83 operations.

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arXiv2019

Permutation Code Equivalence is not Harder than Graph Isomorphism when Hulls are Trivial

The paper deals with the problem of deciding if two finite-dimensional linear subspaces over an arbitrary field are identical up to a permutation of the coordinates. This problem is referred to as the permutation code equivalence. We show that given access to a subroutine that decides if two weighted undirected graphs are isomorphic, one may deterministically decide the permutation code equivalence provided that the underlying vector spaces intersect trivially with their orthogonal complement with respect to an arbitrary inner product. Such a class of vector spaces is usually called linear codes with trivial hulls. The reduction is efficient because it essentially boils down to computing the inverse of a square matrix of order the length of the involved codes. Experimental results obtained with randomly drawn binary codes having trivial hulls show that permutation code equivalence can be decided in a few minutes for lengths up to 50,000.

cs.DS↗

arXiv2016

Algebraic Properties of Polar Codes From a New Polynomial Formalism

Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attain many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a hard task and channel dependent. The purpose of this paper is to reveal some universal properties of these monomials. We will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised fo a binary-input symmetric channel always form a decreasing set. This property turns out to have rather deep consequences on the structure of the polar code. Indeed, the permutation group of a decreasing monomial code contains a large group called lower triangular affine group. Furthermore, the codewords of minimum weight correspond exactly to the orbits of the minimum weight codewords that are obtained from (evaluations) of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum weight codewords of a decreasing monomial code and henceforth of a polar code.

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arXiv2014

On the Complexity of the F5 Gröbner basis Algorithm

We study the complexity of Gröbner 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 $d$ in a Gröbner basis computed by Faugère's $F_5$ algorithm~(Fau02) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Gröbner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Gröbner bases with signatures that $F_5$ computes and use it to bound the complexity of the algorithm. Our estimates show that the version of~$F_5$ 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.

cs.SC↗

arXiv2013

On the degree of the polynomial defining a planar algebraic curves of constant width

In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$ represents the support function of this set. Our aim is to examine properties of the degree of the defining polynomial of this family of curves in terms of the degree of $p$. Thanks to the theory of elimination, we compute the total degree and the partial degrees of this polynomial, and we solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest degree polynomial whose graph is a non-circular curve of constant width. Computations of partial degrees of the defining polynomial of algebraic surfaces of constant width are also provided in the same way.

math.AG↗

arXiv2012

On the Complexity of Solving Quadratic Boolean Systems

A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in $4\log_2 n\,2^n$ 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 $O(2^{0.841n})$ when $m=n$, while a probabilistic variant of the Las Vegas type has expected complexity $O(2^{0.792n})$. 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.

cs.SC↗

arXiv2010

Complexity reduction of C-algorithm

The C-Algorithm introduced in [Chouikha2007] is designed to determine isochronous centers for Lienard-type differential systems, in the general real analytic case. However, it has a large complexity that prevents computations, even in the quartic polynomial case. The main result of this paper is an efficient algorithmic implementation of C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of it is proposed in the rational case. It is called RCA (Rational C-Algorithm) and is widely used in [BardetBoussaadaChouikhaStrelcyn2010] and [BoussaadaChouikhaStrelcyn2010] to find many new examples of isochronous centers for the Liénard type equation.

math.CA↗

arXiv2010

Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$ where $A,\;B\in \mathbb{R}[x,y]$, which can be reduced to the Liénard type equation. When $deg(A)\leq 4$ and $deg(B) \leq 4$, using the so-called C-algorithm we found $36$ new families of isochronous centers. When the Urabe function $h=0$ we provide an explicit general formula for linearization. This paper is a direct continuation of \cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly.

math.CA↗

arXiv2007

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.

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