Cryptology ePrint Archive: Report 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.

Category / Keywords: Isogenies; post-quantum cryptography; abelian varieties; hyperelliptic-curve cryptography

Date: received 4 Jan 2021

Contact author: smith at lix polytechnique fr

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Version: 20210106:214032 (All versions of this report)

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