Description
Additive polynomials over a field $ F$ of characteristic $ p>0$ have the form $ f(X)=\sum\limits^m_{k=0} a_k X^{p^k}$ with $ a_k \in F$. In case $ a_0 \neq 0$ they are Galois polynomials with an $ \mathbb{F}_p$-vector space of solutions, and any finite Galois extension $ E$ over $ F$ can be generated by such an additive polynomial.<br/> The Galois group of $ f(X)$ or $ E/F$ , respectively, acts linearly on the solution space and thus is a subgroup of the linear group $ \operatorname{GL}_m(\mathbb{F}_p)$. It can be computed via subgroup descent from $ \operatorname{GL}_m(\mathbb{F}_p)$ in analogy to the Stauduhar method. On the other hand, any additive polynomial can be obtained as a characteristic polynomial of a Frobenius module over $ F$, i.e., an $ F$-vector space $ M$ with a $ \phi$-semilinear Frobenius operator $ \Phi$, where $ \phi$ denotes the Frobenius endomorphism of $ F$. The smallest connected linear algebraic group in which the representing matrix of $ \Phi$ is contained gives an upper bound for the Galois group.<br/> Since lower bounds can be obtained by specialization of the matrix in analogy to the classical Dedekind criterion, this technique gives a useful tool for the construction of Galois extensions with given (connected) Galois group (in positive characteristic). This will be demonstrated by examples, among others the Dickson groups $ G_2(q)$. References:<br/> Goss, D.: Basic structures of function field arithmetic. Springer-Verlag 1996, Chapter I.<br/> Malle, G.: Explicit realization of the Dickson groups $ G_2(q)$ as Galois groups. Preprint, Kassel 2002.<br/> Matzat, B. H.: Frobenius modules and Galois groups. Preprint, Heidelberg 2002.
Next sessions
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Random lattices that are modules over the ring of integers
Speaker : Nihar Gargava - Institut de Mathématiques d'Orsay
We investigate the average number of lattice points within a ball where the lattice is chosen at random from the set of unit determinant ideal or modules lattices of some cyclotomic number field. The goal is to consider the space of such lattice as a probabilistic space and then study the distribution of lattice point counts. This is inspired by the connections of this problem to lattice-based[…]-
Cryptography
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Schéma de signature à clé publique : Frobénius-UOV
Speaker : Gilles Macario-Rat - Orange
L'exposé présente un schéma de signature à clé publique post-quantique inspiré du schéma UOV et introduisant un nouvel outil : les formes de Frobénius. L'accent est mis sur le rôle et les propriétés des formes de Frobénius dans ce nouveau schéma : la simplicité de description, la facilité de mise en oeuvre et le gain inédit sur les tailles de signature et de clé qui bat RSA-2048 au niveau de[…] -
Yoyo tricks with a BEANIE
Speaker : Xavier Bonnetain - Inria
TBD-
Cryptography
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Symmetrical primitive
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