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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Polytopes in the Fiat-Shamir with Aborts Paradigm
Speaker : Hugo Beguinet - ENS Paris / Thales
The Fiat-Shamir with Aborts paradigm (FSwA) uses rejection sampling to remove a secret’s dependency on a given source distribution. Recent results revealed that unlike the uniform distribution in the hypercube, both the continuous Gaussian and the uniform distribution within the hypersphere minimise the rejection rate and the size of the proof of knowledge. However, in practice both these[…]-
Cryptography
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Mode and protocol
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Post-quantum Group-based Cryptography
Speaker : Delaram Kahrobaei - The City University of New York