Description
We here investigate the hardness of one of the most relevant problems in multivariate cryptography, namely MinRank: given non-negative intgers n,k,r, and matrices M_0,...,M_k, of size n with entries in F_q, decide whether there exists an F_q-linear combination of those matrices which has rank less than or equal to r. Our starting point is the Kipnis-Shamir modeling of the problem. We first prove new properties satisfed by this modeling. Then, we propose a practical resolution of it - based on a Groebner basis approach - that permits us to efficiently solve two challenges proposed by Courtois for his zero-knowledge authentication scheme, built upon MinRank.<br/> Next we turn to the theoretical complexity of the problem: we exhibit a multi-homogeneous structure of the algebraic system modeling the probem, that yields a theoretical bound on its hardness, reflecting the practical behaviour of our approach. Our main result is that, when the size of the matrices involved minus the target rank is constant, we can solve MinRank in polynomial time.<br/> This is a joint work with Jean-Charles Faugères and Ludovic Perret.
Prochains exposés
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Verification of Rust Cryptographic Implementations with Aeneas
Orateur : 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
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[…] -
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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