Description
La question est la suivante. Soit X une variété algébrique sur Q, et Qbar une clôture algébrique de Q. Pour tout entier n>0 on a alors le groupe de cohomologie étale V_n de X sur Qbar à coefficients dans Z/nZ. Ce V_n est un Z/nZ-module de type fini, muni d'une action du groupe de Galois Gal(Qbar/Q). Concrètement, cela veut dire qu'on a des extensions galoisiennes finies K_n de Q, avec Gal(K_n/Q) agissant fidèlement sur V_n. Peut-on calculer ces extensions K_n, et V_n en tant que Gal(K_n/Q)-module, en temps polynomial en n ? Les résultats dans l'article `On the computation of coefficients of a modular form' (http://www.arxiv.org/abs/math.NT/0605244) montrent que ce genre de calcul peut se faire en temps polynomial dans d'autres cas que celui des points de torsion des courbes elliptiques, par exemple, pour le motif associé à la fonction tau de Ramanujan. Dans l'exposé il sera expliqué ce qui a été fait, et ce qu'on pourrait espérer faire, peut-être.
Prochains exposés
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Lie algebras and the security of cryptosystems based on classical varieties in disguise
Orateur : Mingjie Chen - KU Leuven
In 2006, de Graaf et al. proposed a strategy based on Lie algebras for finding a linear transformation in the projective linear group that connects two linearly equivalent projective varieties defined over the rational numbers. Their method succeeds for several families of “classical” varieties, such as Veronese varieties, which are known to have large automorphism groups. In this talk, we[…]-
Cryptography
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Some applications of linear programming to Dilithium
Orateur : Paco AZEVEDO OLIVEIRA - Thales & UVSQ
Dilithium is a signature algorithm, considered post-quantum, and recently standardized under the name ML-DSA by NIST. Due to its security and performance, it is recommended in most use cases. During this presentation, I will outline the main ideas behind two studies, conducted in collaboration with Andersson Calle-Vierra, Benoît Cogliati, and Louis Goubin, which provide a better understanding of[…] -
Wagner’s Algorithm Provably Runs in Subexponential Time for SIS^∞
Orateur : Johanna Loyer - Inria Saclay
At CRYPTO 2015, Kirchner and Fouque claimed that a carefully tuned variant of the Blum-Kalai-Wasserman (BKW) algorithm (JACM 2003) should solve the Learning with Errors problem (LWE) in slightly subexponential time for modulus q = poly(n) and narrow error distribution, when given enough LWE samples. Taking a modular view, one may regard BKW as a combination of Wagner’s algorithm (CRYPTO 2002), run[…]-
Cryptography
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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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Structured-Seed Local Pseudorandom Generators and their Applications
Orateur : Nikolas Melissaris - IRIF
We introduce structured‑seed local pseudorandom generators (SSL-PRGs), pseudorandom generators whose seed is drawn from an efficiently sampleable, structured distribution rather than uniformly. This seemingly modest relaxation turns out to capture many known applications of local PRGs, yet it can be realized from a broader family of hardness assumptions. Our main technical contribution is a[…]-
Cryptography
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