Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

Enric Florit; Benjamin Smith

Paper 2021/012

Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph

Enric Florit and Benjamin Smith

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.

Metadata

Available format(s): PDF
Publication info: Published elsewhere. Proceedings of AGC2T 18
Keywords: Isogenies post-quantum cryptography abelian varieties hyperelliptic-curve cryptography
Contact author(s): smith @ lix polytechnique fr
History: 2022-01-17: revised; 2021-01-06: received; See all versions
Short URL: https://ia.cr/2021/012
License: CC BY

BibTeX

@misc{cryptoeprint:2021/012,
      author = {Enric Florit and Benjamin Smith},
      title = {Automorphisms and isogeny graphs of abelian varieties, with applications to the superspecial Richelot isogeny graph},
      howpublished = {Cryptology {ePrint} Archive, Paper 2021/012},
      year = {2021},
      url = {https://eprint.iacr.org/2021/012}
}