1970-01-01

(2026-03) The principal ideal problem for endomorphism rings of superspecial abelian varieties

• [[Wouter Castryck]]
• [[Jonathan Komada Eriksen]]
• [[Riccardo Invernizzi]]
• [[Frederik Vercauteren]]

2026-03-04

• #cryptography
• #paper

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.