G. Bourgeois

Publications28
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Citations31
Highly Influential Citations0
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Algebraic attack on NTRU using Witt vectors and Gröbner bases
TLDR
We present an algebraic attack on NTRU (restricted to the case where the parameter q is a power of two) using the method of the Witt vectors proposed by Silverman, Smart and Vercauteren. Expand
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How to solve the matrix equation XA-AX=f(X)
Let f be an analytic function defined on a complex domain Omega and A be a (n,n) complex matrix. We assume that there exists a unique alpha satisfying f(alpha)=0. When f'(alpha)=0 and A is non… Expand
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The equations $A^2-2AB+B^2=0$ and $AX-XA=X^{\alpha}$ in matrix over fields or rings
Model Based System Engineering with Capella
PAIRS OF MATRICES, ONE OF WHICH COMMUTES WITH THEIR COMMUTATOR
Let A, B be n × n complex matrices such that C = AB BA and A commute. For n = 2, we prove that A, B are simultaneously triangularizable. For n � 3, we give an example of matrices A, B such that the… Expand
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THE MATRIX EQUATION XA −AX = f(X)
Let f be an analytic function defined on a complex domain and A 2 M n(C). We assume that there exists a uniquesatisfying f(�) = 0. When f 0 (�) = 0 and A is non derogatory, we solve completely the… Expand
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The matrix equations $XA-AX=X^{\alpha}g(X)$ over fields or rings
Let $n,\alpha\geq 2$. Let $K$ be an algebraically closed field with characteristic $0$ or greater than $n$. We show that the dimension of the variety of pairs $(A,B)\in {M_n(K)}^2$, with $B$… Expand
Algebraic systems of matrices and Grobner basis
One studies a particular algebraic system where the unknowns are matrices. We solve this system according to the parameters values thanks to the theory of Grobner basis.
DYNAMICAL SYSTEMS OF SIMPLICES IN DIMENSION 2 OR 3
Let T0 = (A 0 ··· A d) be a d-simplex, G0 its centroid, S its circumsphere, O the center of S. Let (A i) be the points where S intersects the lines (G0A i), T1 the d-simplex (A 0 ··· A d), and G1 its… Expand
Algebraic systems of matrices and Gröbner basis theory
Abstract The problem of finding all the n × n complex matrices A , B , C such that, for all real t , e tA + e tB + e tC is a scalar matrix reduces to the study of a symmetric system ( S ) in the… Expand
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