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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On the average hardness of SIVP for module lattices of fixed rank
Speaker : 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
Speaker : 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
Speaker : 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
Speaker : 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
Speaker : 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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