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Title:
Further Connections Between Isogenies of Supersingular Curves and Bruhat-Tits Trees
Authors: Steven Galbraith, Valerie Gilchrist, Shai Levin, Ari Markowitz
Abstract:We further explore the explicit connections between supersingular curve isogenies and Bruhat-Tits trees. By identifying a supersingular elliptic curve E over \mathbb{F}_p as the root of the tree, and a basis for the Tate module T_\ell(E); our main result is that given a vertex M of the Bruhat-Tits tree one can write down a generator of the ideal I \subseteq \text{End}(E) directly, using simple linear algebra, that defines an isogeny corresponding to the path in the Bruhat-Tits tree from the root to the vertex M. In contrast to previous methods to go from a vertex in the Bruhat-Tits tree to an ideal, once a basis for the Tate module is set up and an explicit map \Phi : \text{End}(E) \otimes_{\mathbb{Z}_\ell} \to M_2( \mathbb{Z}_\ell ) is constructed, our method does not require any computations involving elliptic curves, isogenies, or discrete logs. This idea leads to simplifications and potential speedups to algorithms for converting between isogenies and ideals.
ePrint: https://eprint.iacr.org/2024/1971
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