Isogeny graphs of ordinary abelian varieties

Ernest Hunter Brooks; Dimitar Jetchev; Benjamin Wesolowski

Paper 2016/947

Isogeny graphs of ordinary abelian varieties

Ernest Hunter Brooks, Dimitar Jetchev, and Benjamin Wesolowski

Abstract

Fix a prime number $ℓ$ . Graphs of isogenies of degree a power of $ℓ$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called $l$ -isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(ℓ, ℓ)$ -isogenies: those whose kernels are maximal isotropic subgroups of the $ℓ$ -torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.

Metadata

Available format(s): PDF
Publication info: Preprint. MINOR revision.
Keywords: Jacobians of hyperelliptic curves genus 2 cryptography isogeny graphs $(\ell \ell)$-isogenies principally polarised abelian varieties
Contact author(s): benjamin wesolowski @ epfl ch
History: 2016-10-01: received
Short URL: https://ia.cr/2016/947
License: CC BY

BibTeX

@misc{cryptoeprint:2016/947,
      author = {Ernest Hunter Brooks and Dimitar Jetchev and Benjamin Wesolowski},
      title = {Isogeny graphs of ordinary abelian varieties},
      howpublished = {Cryptology {ePrint} Archive, Paper 2016/947},
      year = {2016},
      url = {https://eprint.iacr.org/2016/947}
}