Paper 2026/576

Radical 3-isogenies for the ideal class group actions on $(2, \varepsilon)$-structures

Abstract

Chenu and Smith introduced the notion of $(d,\varepsilon)$-structures, pairs consisting of an elliptic curve over $\mathbb{F}_{p^2}$ and an isogeny of degree $d$ from the curve to its Galois conjugate, and defined an ideal class group action on supersingular $(d,\varepsilon)$-structures, inherited from the action on oriented supersingular elliptic curves. They also outlined extensions of the CSIDH key exchange and of the Delfs–Galbraith algorithm for the supersingular isogeny problem. In particular, they suggested that the latter, called the generalized Delfs–Galbraith algorithm, may be faster than the original one by a constant factor for small $d$, provided that the class group action on $(d,\varepsilon)$-structures can be evaluated efficiently. In this paper, we focus on the case $d = 2$ and present explicit radical $3$-isogenies for evaluating the action of the class of a prime ideal above $3$. Note that $3$ is the norm of the acting ideal, not the degree $d$ of the isogeny defining a structure; a radical $3$-isogeny maps a $(2,\varepsilon)$-structure to another. Our approach relies on two representations of $(2,\varepsilon)$-structures, by reductions of degree-$2$ $\mathbb{Q}$-curves and by Montgomery curves. We show that any $(2,\varepsilon)$-structure can be represented as a pair of a curve coefficient and a single sign, and from these representations we derive radical $3$-isogenies that efficiently implement the action. As an application, we give an explicit meet-in-the-middle algorithm for finding an ideal class connecting two given $(2,\varepsilon)$-structures, which is a part of the generalized Delfs–Galbraith algorithm.