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UID:6969@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20190926T130000
DTEND;TZID=Europe/Paris:20190926T140000
DTSTAMP:20241120T202638Z
URL:https://www.i2m.univ-amu.fr/evenements/orienting-supersingular-isogeny
 -graphs/
SUMMARY:Leonardo Colò (I2M\, Aix-Marseille Université): Orienting supersi
 ngular isogeny graphs
DESCRIPTION:Leonardo Colò: Supersingular isogeny graphs have been used in 
 the Charles–Goren–Lauter cryptographic hash function and the supersing
 ular isogeny Diffie–Hellman (SIDH) protocole of De Feo and Jao. A recent
 ly proposed alternative to SIDH is the commutative supersingular isogeny D
 iffie–Hellman (CSIDH) protocole\, in which the isogeny graph is first re
 stricted to F_p-rational curves E and  F_p-rational isogenies then or
 iented by the quadratic subring Z[\\pi] of  End(E) generated by the Fr
 obenius endomorphism \\pi on E.\n\n\n\n\nWe introduce a general notion o
 f orienting supersingular elliptic curves and their isogenies\, and use th
 is as the basis to construct a general oriented super- singular isogeny Di
 ffie-Hellman (OSIDH) protocole. By imposing the data of an orientation by
  an imaginary quadratic ring O\, we obtain an augmented category of super
 singular curves on which the class group Cl(O) acts faithfully and trans
 itively. This idea is already implicit in the CSIDH protocol\, in which su
 persingular curves over F_p are oriented by the Frobenius subring Z[\\p
 i] ≃ Z[\\sqrt{−p}]. In contrast\, we consider an elliptic curve E_0
  oriented by a CM order O_K of class number one. To obtain a nontrivial
  group action\, we consider l-isogeny chains\, on which the class group o
 f an order O of large index l^{n} in O_K acts\, a structure we call 
 a whirlpool. The map from l-isogeny chains to its terminus forgets the st
 ructure of the orientation\, and the original base curve E_0\, giving ris
 e to a generic supersingular elliptic curve. \n\nWithin this general frame
 work\, we define a new oriented supersingular isogeny Diffie-Hellman (OSID
 H) protocol\, which has fewer restrictions on the proportion of supersingu
 lar curves covered and on the torsion group structure of the underlying cu
 rves. 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 amenab
 le to speedup by precomputations of endomorphisms on the base curve E_0.\
 n This is joint work with David Kohel.\n\n\n\n
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/Leonardo_Colo-.jpg
CATEGORIES:Séminaire,Arithmétique et Théorie de l’Information
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