Consider the problem of efficiently evaluating isogenies ϕ:E→E/H of
elliptic curves over a finite field Fq, where the kernel H=⟨G⟩ is a cyclic group of odd (prime) order: given E, G, and a
point (or several points) P on E, we want to compute ϕ(P). This
problem is at the heart of efficient implementations of group-action- and
isogeny-based post-quantum cryptosystems such as CSIDH. Algorithms based on
V{\'e}lu's formulae give an efficient solution to this problem when the kernel
generator G is defined over Fq. However, for general isogenies,
G is only defined over some extension Fqk, even though
⟨G⟩ as a whole (and thus ϕ) is defined over the base field
Fq; and the performance of V{\'e}lu-style algorithms degrades
rapidly as k grows. In this article we revisit the isogeny-evaluation problem
with a special focus on the case where 1≤k≤12. We improve
V{\'e}lu-style isogeny evaluation for many cases where k=1 using special
addition chains, and combine this with the action of Galois to give greater
improvements when k>1