Description
Supersingular isogeny graphs have been used in the Charles–Goren–Lauter cryptographic hash function and the supersingular isogeny Diffie–Hellman (SIDH) protocole of De\,Feo and Jao. A recently proposed alternative to SIDH is the commutative supersingular isogeny Diffie–Hellman (CSIDH) protocole, in which the isogeny graph is first restricted to $\FF_p$-rational curves $E$ and $\FF_p$-rational isogenies then oriented by the quadratic subring $\ZZ[\pi] \subset \End(E)$ generated by the Frobenius endomorphism $\pi: E \rightarrow E$. We introduce a general notion of orienting supersingular elliptic curves and their isogenies, and use this as the basis to construct a general oriented supersingular isogeny Diffie-Hellman (OSIDH) protocole.<br/> By imposing the data of an orientation by an imaginary quadratic ring $\OO$, we obtain an augmented category of supersingular curves on which the class group $\Cl(\OO)$ acts faithfully and transitively. This idea is already implicit in the CSIDH protocol, in which supersingular curves over $\FF_p$ are oriented by the Frobenius subring $\ZZ[\pi] \simeq \ZZ[\sqrt{-p}]$. In contrast we consider an elliptic curve $E_0$ oriented by a CM order $\OO_K$ of class number one. To obtain a nontrivial group action, we consider $\ell$-isogeny chains, on which the class group of an order $\OO$ of large index $\ell^n$ in $\OO_K$ acts, a structure we call a whirlpool. The map from $\ell$-isogeny chains to its terminus forgets the structure of the orientation, and the original base curve $E_0$, giving rise to a generic supersingular elliptic curve. Within this general framework we define a new oriented supersingular isogeny Diffie-Hellman (OSIDH) protocol, which has fewer restrictions on the proportion of supersingular curves covered and on the torsion group structure of the underlying curves. Moreover, the group action can be carried out effectively solely on the sequences of moduli points (such as $j$-invariants) on a modular curve, thereby avoiding expensive isogeny computations, and is further amenable to speedup by precomputations of endomorphisms on the base curve $E_0$.<br/> lien: http://desktop.visio.renater.fr/scopia?ID=721072***2120&autojoin
Next sessions
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Random lattices that are modules over the ring of integers
Speaker : Nihar Gargava - Institut de Mathématiques d'Orsay
We investigate the average number of lattice points within a ball where the lattice is chosen at random from the set of unit determinant ideal or modules lattices of some cyclotomic number field. The goal is to consider the space of such lattice as a probabilistic space and then study the distribution of lattice point counts. This is inspired by the connections of this problem to lattice-based[…]-
Cryptography
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Schéma de signature à clé publique : Frobénius-UOV
Speaker : Gilles Macario-Rat - Orange
L'exposé présente un schéma de signature à clé publique post-quantique inspiré du schéma UOV et introduisant un nouvel outil : les formes de Frobénius. L'accent est mis sur le rôle et les propriétés des formes de Frobénius dans ce nouveau schéma : la simplicité de description, la facilité de mise en oeuvre et le gain inédit sur les tailles de signature et de clé qui bat RSA-2048 au niveau de[…] -
Yoyo tricks with a BEANIE
Speaker : Xavier Bonnetain - Inria
TBD-
Cryptography
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Symmetrical primitive
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