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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On the average hardness of SIVP for module lattices of fixed rank
Orateur : Radu Toma - Sorbonne Université
In joint work with Koen de Boer, Aurel Page, and Benjamin Wesolowski, we study the hardness of the approximate Shortest Independent Vectors Problem (SIVP) for random module lattices. We use here a natural notion of randomness as defined originally by Siegel through Haar measures. By proving a reduction, we show it is essentially as hard as the problem for arbitrary instances. While this was[…] -
Attacks and Remedies for Randomness in AI: Cryptanalysis of PHILOX and THREEFRY
Orateur : Yevhen Perehuda - Ruhr-University Bochum
In this work, we address the critical yet understudied question of the security of the most widely deployed pseudorandom number generators (PRNGs) in AI applications. We show that these generators are vulnerable to practical and low-cost attacks. With this in mind, we conduct an extensive survey of randomness usage in current applications to understand the efficiency requirements imposed in[…]-
Cryptography
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Lightweight (AND, XOR) Implementations of Large-Degree S-boxes
Orateur : Marie Bolzer - LORIA
The problem of finding a minimal circuit to implement a given function is one of the oldest in electronics. In cryptography, the focus is on small functions, especially on S-boxes which are classically the only non-linear functions in iterated block ciphers. In this work, we propose new ad-hoc automatic tools to look for lightweight implementations of non-linear functions on up to 5 variables for[…]-
Cryptography
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Symmetrical primitive
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Implementation of cryptographic algorithm
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Algorithms for post-quantum commutative group actions
Orateur : Marc Houben - Inria Bordeaux
At the historical foundation of isogeny-based cryptography lies a scheme known as CRS; a key exchange protocol based on class group actions on elliptic curves. Along with more efficient variants, such as CSIDH, this framework has emerged as a powerful building block for the construction of advanced post-quantum cryptographic primitives. Unfortunately, all protocols in this line of work are[…] -
Endomorphisms via Splittings
Orateur : Min-Yi Shen - No Affiliation
One of the fundamental hardness assumptions underlying isogeny-based cryptography is the problem of finding a non-trivial endomorphism of a given supersingular elliptic curve. In this talk, we show that the problem is related to the problem of finding a splitting of a principally polarised superspecial abelian surface. In particular, we provide formal security reductions and a proof-of-concept[…]-
Cryptography
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