Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph - Archive ouverte HAL Access content directly
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Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

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Abstract

We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.
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Dates and versions

hal-03094375 , version 1 (04-01-2021)
hal-03094375 , version 2 (04-01-2021)
hal-03094375 , version 3 (18-01-2022)

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Enric Florit, Benjamin Smith. Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph. Arithmetic, Geometry, Cryptography, and Coding Theory 2021, May 2021, Luminy, France. ⟨hal-03094375v3⟩
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