Description
Isogenies are an essential tool in Elliptic Curves cryptography, where they are used in a wide variety of area: fast point counting, complex multiplication methods... Velu's formulas give an efficient method for computing such isogenies, but there are no formula known for curves of higher genera or general abelian varieties (except some special case for isogenies of degree 2). In this talk we will present the framework of the theta structure on an abelian variety, developped by Mumford in 1967, which allows us to compute isogenies. For this we lift a theta null point of level $l$, corresponding to an abelian variety B, to the modular space of theta null points of level $lk$. We use a specialized Groebner algorithm that considerably speed-up this phase, and we show how to detect degenerate solutions. To each lifted point corresponds an isogeny of degree $\pi: A \to B$ of degree $k$. We then explain how to compute their dual efficiently.<br/> This is a joint work with Jean-Charles Faugere and David Lubicz.
Next sessions
-
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[…] -
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
-