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SUMMARY:Kirsten Eisenträger (Pennsylvania State)\, Sean Hallgren (Pennsyl
 vania State)\, Chris Leonardi (University of Waterloo)\, Travis Morrison (
 Pennsylvania State)\, and Jennifer Park (University of Michigan)
DTSTART:20200629T214500Z
DTEND:20200629T221500Z
DTSTAMP:20260423T195908Z
UID:ANTS14/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ANTS14/2/">C
 omputing endomorphism rings of supersingular elliptic curves and connectio
 ns to pathfinding in isogeny graphs</a>\nby Kirsten Eisenträger (Pennsylv
 ania State)\, Sean Hallgren (Pennsylvania State)\, Chris Leonardi (Univers
 ity of Waterloo)\, Travis Morrison (Pennsylvania State)\, and Jennifer Par
 k (University of Michigan) as part of Algorithmic Number Theory Symposium 
 (ANTS XIV)\n\n\nAbstract\nComputing endomorphism rings of supersingular el
 liptic curves is an important problem in computational number theory\, and
  it is also closely connected to the security of some of the recently prop
 osed isogeny-based cryptosystems. In this paper we give a new algorithm fo
 r computing the endomorphism ring of a supersingular elliptic curve $E$ th
 at runs\, under certain heuristics\, in time $O((\\log p)^2p^{1/2})$. The 
 algorithm works by first finding two cycles of a certain form in the super
 singular $\\ell$-isogeny graph $G(p\,\\ell)$\, generating an order $\\Lamb
 da \\subseteq \\mathrm{End}(E)$. Then all maximal orders containing $\\Lam
 bda$ are computed\, extending work of Voight. The final step is to determi
 ne which of these maximal orders is the endomorphism ring. As part of the 
 cycle finding algorithm\, we give a lower bound on the set of all $j$-inva
 riants $j$ that are adjacent to $j^p$ in $G(p\,\\ell)$\, answering a quest
 ion in <a href="https://arxiv.org/abs/1909.07779">arXiv:1909.07779</a>.\n\
 nThe slides used in the pre-recorded video can be found <a href="https://m
 ath.mit.edu/~drew/ANTSXIV/ComputingEndomorphismRingsVideoSlides.pdf">here<
 /a>.\n\nChairs: Steven Galbraith and Christophe Petit\n
LOCATION:https://researchseminars.org/talk/ANTS14/2/
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