Description
La distance d'une fonction booléenne f de m variables au code de Reed-Muller est une mesure la non-linearité de f. Il s'agit d'une notion importante en cryptographie. L'analyse de Fourier est une méthode d'approche normale de cette question. En particulier, la non-linéarité de f est égale à [ 2^m - R(f) ] /2, où R(f) est l'amplitude spectrale de f i.e. le module maximal de ses coefficients de Fourier.<br/> Un des problèmes majeurs consiste à déterminer la valeur minimale (rayon spectral) de R(f) lorsque f décrit l'ensemble de toutes les fonctions booléennes, équivalent à déterminer le rayon de recouvrement du code de Reed-Muller du premier ordre.<br/> Le rayon spectral est facile à calculer lorsque m est pair, il est réalisé par les fonctions quadratiques non-dégénérée. Le cas impair est encore ouvert, sauf pour les petites valeurs de m : 3, 5 et 7. Dans les années 80, Paterson-Wiedemann ont découvert une fonction booléenne de 15 variables moins linéaire que les fonctions quadratiques : contre-exemple à une conjecture formulée dans les années 70 par Mykkelveit.<br/> Au cours de mon exposé, je montrerai d'autres contre-exemples plus linéaire mais mieux structurés construits à partir des sommes de Gauss. Enfin, je discuterai de la conjecture de Patterson et Wiedemann qui affirme l'équivalence asympototique entre le rayon spectral R(m) et 2^{m/2}.<br/> Philippe Langevin, janvier 2003.
Next sessions
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Verification of Rust Cryptographic Implementations with Aeneas
Speaker : Aymeric Fromherz - Inria
From secure communications to online banking, cryptography is the cornerstone of most modern secure applications. Unfortunately, cryptographic design and implementation is notoriously error-prone, with a long history of design flaws, implementation bugs, and high-profile attacks. To address this issue, several projects proposed the use of formal verification techniques to statically ensure the[…] -
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[…] -
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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