Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem
This work addresses cryptographic security concerns by providing theoretical foundations for isogeny-based cryptography, though it builds incrementally on prior methods.
The authors tackled the problem of analyzing isogeny graphs of ordinary abelian varieties to establish random self-reducibility of the discrete logarithm problem for certain abelian surfaces, achieving this by computing bounds on prime ideal norms under the Generalized Riemann Hypothesis and extending existing algorithms to genus 2.
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.