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.
Prochains exposés
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SoK: Security of the Ascon Modes
Orateur : Charlotte Lefevre - Radboud University
The Ascon authenticated encryption scheme and hash function of Dobraunig et al (Journal of Cryptology 2021) were recently selected as winner of the NIST lightweight cryptography competition. The mode underlying Ascon authenticated encryption (Ascon-AE) resembles ideas of SpongeWrap, but not quite, and various works have investigated the generic security of Ascon-AE, all covering different attack[…] -
Comprehensive Modelling of Power Noise via Gaussian Processes with Applications to True Random Number Generators
Orateur : Maciej Skorski - Laboratoire Hubert Curien
The talk examines power noise modelling through Gaussian Processes for secure True Random Number Generators. While revisiting one-sided fractional Brownian motion, we obtain novel contributions by quantifying posterior uncertainty in exact analytical form, establishing quasi-stationary properties, and developing rigorous time-frequency analysis. These results are applied to model oscillator[…]-
Cryptography
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TRNG
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CryptoVerif: a computationally-sound security protocol verifier
Orateur : Bruno Blanchet - Inria
CryptoVerif is a security protocol verifier sound in the computational model of cryptography. It produces proofs by sequences of games, like those done manually by cryptographers. It has an automatic proof strategy and can also be guided by the user. It provides a generic method for specifying security assumptions on many cryptographic primitives, and can prove secrecy, authentication, and[…]-
Cryptography
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