Fast and Frobenius: Rational Isogeny Evaluation over Finite Fields

Consider the problem of efficiently evaluating isogenies $$\phi : \mathcal {E}\rightarrow {E}/H$$ elliptic curves over a finite field $$\mathbb {F}_q$$ , where kernel $$H = \langle {G}\rangle $$ is cyclic group odd (prime) order: given $$\mathcal {E}$$ $$G$$ and point (or several points) P on we want to compute (P)$$ . This at heart efficient implementations group-action- isogeny-based post-quantum cryptosystems such as CSIDH. Algorithms based Vélu’s formulæ give an solution when generator G defined but for general only some extension {F}_{q^k}$$ even though $$\langle whole (and thus ) base ; performance Vélu-style algorithms degrades rapidly k grows. In this article revisit isogeny evaluation with special focus case $$1 \le 12$$ We improve many cases $$k 1$$ using addition chains, combine action Galois greater improvements >