The principal ideal problem for endomorphism rings of superspecial abelian varieties

Wouter Castryck; Jonathan Komada Eriksen; Riccardo Invernizzi; Frederik Vercauteren

Paper 2026/454

The principal ideal problem for endomorphism rings of superspecial abelian varieties

Wouter Castryck

, KU Leuven

Jonathan Komada Eriksen

, KU Leuven

Riccardo Invernizzi

, KU Leuven

Frederik Vercauteren

, KU Leuven

Abstract

We describe a Las Vegas algorithm for the principal ideal problem in matrix rings $M_g(O)$ for $g \geq 2$, over maximal orders $O$ in the rational quaternion algebra $B_{p, \infty}$ ramified at $\infty$ and a prime number $p$. Under plausible heuristic assumptions, the method has expected polynomial runtime. An implementation in SageMath shows that it runs very efficiently in practice, with compact output. Our main auxiliary result is a method for finding endomorphisms of superspecial abelian varieties (i.e., powers of supersingular elliptic curves) with a prescribed kernel.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Published elsewhere. Minor revision. ANTS XVII
Keywords: principal ideal problem quaternions maximal orders abelian varieties
Contact author(s): wouter castryck @ gmail com
jonathan eriksen97 @ gmail com
riccardo invernizzi @ esat kuleuven be
frederik vercauteren @ gmail com
History: 2026-06-11: revised; 2026-03-04: received; See all versions
Short URL: https://ia.cr/2026/454
License: CC BY

BibTeX

@misc{cryptoeprint:2026/454,
      author = {Wouter Castryck and Jonathan Komada Eriksen and Riccardo Invernizzi and Frederik Vercauteren},
      title = {The principal ideal problem for endomorphism rings of superspecial abelian varieties},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/454},
      year = {2026},
      url = {https://eprint.iacr.org/2026/454}
}