Searcharxiv⌕ Search

arXiv subjects

Guénaël Renault

Publications and source records attributed to Guénaël Renault.

4 recordsLinked to original sources
arXiv2017

A Probabilistic Analysis on a Lattice Attack against DSA

Analyzing the security of cryptosystems under attacks based on the malicious modification of memory registers is a research topic of high importance. This type of attacks may affect the randomness of the secret parameters by forcing a limited number of bits to a certain value which can be unknown to the attacker. In this context, we revisit the attack on DSA presented by Faugère, Goyet and Renault during the conference SAC 2012: we simplify their method and we provide a probabilistic approach in opposition to the heuristic proposed in the former to measure the limits of the attack. More precisely, the main problem is formulated as the search for a closest vector to a lattice, then we study the distribution of the vectors with bounded norms in a this family of lattices and we apply the result to predict the behavior of the attack. We validated this approach by computational experiments.

cs.CR↗
arXiv2016

On the p-adic stability of the FGLM algorithm

Nowadays, many strategies to solve polynomial systems use the computation of a Gr{ö}bner basis for the graded reverse lexicographical ordering, followed by a change of ordering algorithm to obtain a Gr{ö}bner basis for the lexicographical ordering. The change of ordering algorithm is crucial for these strategies. We study the p-adic stability of the main change of ordering algorithm, FGLM. We show that FGLM is stable and give explicit upper bound on the loss of precision occuring in its execution. The variant of FGLM designed to pass from the grevlex ordering to a Gr{ö}bner basis in shape position is also stable. Our study relies on the application of Smith Normal Form computations for linear algebra.

cs.SC↗
arXiv2013

Polynomial Systems Solving by Fast Linear Algebra

Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to stick to the most general case, we consider a representation of the solutions from which one can easily recover the exact solutions or a certified approximation of them. Under generic assumption, such a representation is given by the lexicographical Gröbner basis of the system and consists of a set of univariate polynomials. The best known algorithm for computing the lexicographical Gröbner basis is in $\widetilde{O}(d^{3n})$ arithmetic operations where $n$ is the number of variables and $d$ is the maximal degree of the equations in the input system. The notation $\widetilde{O}$ means that we neglect polynomial factors in $n$. We show that this complexity can be decreased to $\widetilde{O}(d^{ωn})$ where $2 \leq ω< 2.3727$ is the exponent in the complexity of multiplying two dense matrices. Consequently, when the input polynomial system is either generic or reaches the Bézout bound, the complexity of solving a polynomial system is decreased from $\widetilde{O}(D^3)$ to $\widetilde{O}(D^ω)$ where $D$ is the number of solutions of the system. To achieve this result we propose new algorithms which rely on fast linear algebra. When the degree of the equations are bounded uniformly by a constant we propose a deterministic algorithm. In the unbounded case we present a Las Vegas algorithm.

cs.SC↗