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
-
Dissecting CRAFT, a full-round attack
Orateur : Eran Lambooij - Inria
I will present the first full-round key recovery attack on CRAFT, a block cipher introduced at ToSC 2019. The attack builds on the previous observation (ToSC 2026) that the state of CRAFT can be decomposed into two parts that barely exchange information. We transform this property into a dissection attack on the full-round cipher. This shows that in some cases we can elevate the dissection attack[…]-
Cryptography
-
-
Key Attack on the ACDGV Matrix Encryption Scheme
Orateur : Anmoal Porwal - Technical University of Munich
I will present our key-recovery attack on the ACDGV public-key encryption scheme proposed at ASIACRYPT 2024 by Aragon, Couvreur, Dyseryn, Gaborit, and Vinçotte. The secret key is a Gabidulin code hidden by appending random rows and columns and by left- and right-multiplication with invertible matrices. Our attack exploits the resulting algebraic structure to recover an equivalent secret key. It[…]-
Cryptography
-
Asymmetric primitive
-
-
Module Learning With Errors and Structured Extrapolated Dihedral Cosets
Orateur : Jinwei Zheng - Télécom Paris
The Module Learning With Errors (MLWE) problem is the fundamental hardness assumption underlying the key encapsulation and signature schemes ML-KEM and ML-DSA, which have been selected by NIST for post-quantum cryptography standardization. Understanding its quantum hardness is crucial for assessing the security of these standardized schemes. Inspired by the equivalence between LWE and[…]-
Cryptography
-