Abstract
In this paper we give an exposition of supersingular isogeny graphs, quaternion ideal graphs and Bruhat–Tits trees, and of their connections. Bruhat–Tits trees are combinatorial objects whose vertices and edges have a very simple representation as two-by-two matrices, which, as we show, is useful for understanding certain aspects of the corresponding elliptic curves and isogenies. Moreover, Bruhat–Tits trees can be given an orientation and a notion of depth that we translate into the setting of supersingular isogeny graphs. We give some suggestions towards using Bruhat–Tits trees as a tool for cryptanalysis of certain cryptosystems based on supersingular isogeny graphs.
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Notes
- 1.
See e.g. [40, Theorem 4.1].
- 2.
Pizer [35] used this description of supersingular isogeny graphs to prove the Ramanujan property when p ≡ 1 (mod 12).
- 3.
- 4.
We are using the prime ℓ to be consistent with the isogeny graphs to which we want to connect this theory. In the Shimura curves literature, p is widely used as the chosen prime, so p-adic upper half-plane is more standard.
- 5.
Although we will focus at first on the SIKE parameters, it could be that the most interesting case occurs for a different parameter set within the SIDH family of protocols.
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We want to thank the organisers of the Women in Numbers Europe 3 conference for giving us the opportunity to work on this project. We also wish to thank the anonymous reviewers for their helpful comments.
Jana Sotáková, as well as a follow up visit of Laia Amorós, Annamaria Iezzi and Chloe Martindale at the CWI in Amsterdam, were supported by the Dutch Research Council (NWO) through Gravitation-grant Quantum Software Consortium—024.003.037. Laia Amorós was supported by Academy of Finland grant #282938 and by Helsinki Institute for Information Technology HIIT. Chloe Martindale was partially supported by CHIST-ERA USEIT (NWO project 651.002.004).
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Amorós, L., Iezzi, A., Lauter, K., Martindale, C., Sotáková, J. (2021). Explicit Connections Between Supersingular Isogeny Graphs and Bruhat–Tits Trees. In: Cojocaru, A.C., Ionica, S., García, E.L. (eds) Women in Numbers Europe III. Association for Women in Mathematics Series, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-030-77700-5_2
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