Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem

TL;DR

This paper explores the structure of isogeny graphs of ordinary abelian varieties over finite fields, providing bounds on prime ideals, analyzing expansion properties, and applying these results to the discrete logarithm problem in genus 2.

Contribution

It introduces bounds on prime ideals generating isogeny graphs with good expansion, and demonstrates their application to the discrete logarithm problem in genus 2 abelian varieties.

Findings

Bounds on norms of prime ideals for good expansion

Proved random self-reducibility of DLP in certain abelian surfaces

Extended isogeny computation algorithms to genus 2 without heuristics

Abstract

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class group, and generating set thereof, induces an isogeny graph on the orbit of the variety for this subgroup. We compute (under the Generalized Riemann Hypothesis) some bounds on the norms of prime ideals generating it, such that the associated graph has good expansion properties. We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and Robert for computing explicit isogenies in genus 2, to prove random self-reducibility of the discrete logarithm problem within the subclasses of principally polarizable ordinary abelian surfaces with fixed endomorphism ring. In addition, we remove the heuristics in the complexity analysis of an algorithm of Galbraith for explicitly computing isogenies between two elliptic curves in the same isogeny class, and extend it to a more general setting including genus 2.