Isogenies of certain abelian varieties over finite fields with p-ranks zero

TL;DR

This paper investigates isogenies between specific abelian varieties over finite fields with p-rank zero, revealing that such varieties with maximal endomorphism rings are connected by prime-degree cyclic isogenies, with implications for cryptography.

Contribution

It demonstrates that abelian varieties with maximal endomorphism rings are linked by prime-degree cyclic isogenies, advancing understanding in isogeny-based cryptography.

Findings

Any two such abelian varieties are connected by a cyclic isogeny of prime degree.

Endomorphism rings are maximal orders in the endomorphism algebra.

Results have potential applications in cryptography.

Abstract

We study the isogenies of certain abelian varieties over finite fields with non-commutative endomorphism algebras with a view to potential use in isogeny-based cryptography. In particular, we show that any two such abelian varieties with endomorphism rings maximal orders in the endomorphism algebra are linked by a cyclic isogeny of prime degree.