Description
Résumé : La cryptanalyse de schémas de cryptographie à clé publique repose sur un ensemble de techniques algorithmiques et algébriques en théorie des nombres. Dans une première partie de cette thèse, nous présentons des améliorations de l’algorithme LLL, dû à Lenstra, Lenstra et Lovasz pour réduire un réseau euclidien, c’est-à-dire réduire la norme et orthogonaliser le plus possible les vecteurs de la base. Nous montrons aussi comment utiliser cet algorithme pour réduire des réseaux modules en rang 2 dans un corps de nombres cyclotomique ayant des sous-corps. En effet, certains schémas comme NTRU ou Falcon, dont la sécurité repose sur ce problème difficile, ont été proposés en cryptographie post-quantique et pour du chiffrement homomorphe. Nous améliorons aussi les techniques d’algèbre linéaire creuse et proposons de meilleurs algorithmes lorsque la matrice est à diagonale dominante. Ces avancées nous permettent de réaliser de nouveaux records de calculs de corps de nombres : nombre de classes, générateurs du groupe des unités, générateur d’un idéal principal. Dans une seconde partie, nous étudions différents problèmes classiques en théorie des nombres : nous améliorons différents algorithmes pour tester la primalité d’un entier et en particulier, le test cyclotomique initialement proposé par Adleman et dernièrement développé par Mihailescu. Puis, nous étudions différents algorithmes dans un modèle dit de l’anneau boîte noire, c’est-à-dire que nous étudions le nombre d’additions et de multiplications dans l’anneau, sans nous intéresser à la façon de représenter et de faire les calculs dans cet anneau. Ceci nous permet dans le dernier chapitre, d’instancier ces algorithmes en fonction de différents anneaux pour proposer des algorithmes efficaces en cryptanalyse. Ce faisant, nous sommes capables de distribuer plus facilement les calculs de tout l’algorithme, alors que les algorithmes de calcul d’indice utilisent une étape d’algèbre linéaire qu’il est difficile de paralléliser.
Abstract: Cryptanalysis of public-key cryptosystems is based on algorithmic and algebraic techniques in number theory. In the first part of this thesis, we present improvements to the LLL algorithm, originally designed by Lenstra, Lenstra, and Lovász in the eighties to reduce a Euclidean lattice, i.e. to reduce the norm and orthogonalize the basis vectors as possible. We also show how to use this algorithm to reduce rank-2 module lattices in a cyclotomic number field with subfields using symplectic matrices. Indeed, some schemes such as NTRU or Falcon, whose security relies on this difficult problem, have been proposed in post-quantum cryptography and for homomorphic encryption. We also improve linear algebra techniques and propose better algorithms when the matrix is diagonally dominant. These advanced methods allow us to set new records in the computation of number fields class number, generators of the unit group and generator of a principal ideal. In the second part, we study various classical problems in number theory: we improve algorithms for testing the primality of an integer, and in particular, the cyclotomic test initially proposed by Adleman and developed by Mihailescu. Then, we study different algorithms in the black-box ring model, i.e. we study the number of additions and multiplications in the ring, without looking at how the elements are represented and how the computations are performed in the ring. This allows us in the final chapter to instantiate these algorithms in different rings in order to propose efficient algorithms for cryptanalysis. In doing so, we are able to more easily distribute the computations of the entire algorithm, while index-calculus algorithms use a linear algebra step which is difficult to parallelize.
Next sessions
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Combining Partial Sums and FFT for the Fastest Known Attack on 6‑Round AES
Speaker : Shibam Ghosh - Inria
The partial-sums technique introduced by Ferguson et al. (2000) achieved a 6‑round AES attack with time complexity 2^{52} S‑box evaluations, a benchmark that has stood since. In 2014, Todo and Aoki proposed a comparable approach based on the Fast Fourier Transform (FFT). In this talk, I will show how to combine partial sums with FFT to get "the best of both worlds". The resulting attack on 6[…]-
Cryptography
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Lie algebras and the security of cryptosystems based on classical varieties in disguise
Speaker : 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
Speaker : 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^∞
Speaker : 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
Speaker : 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
Speaker : 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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