Paper 2024/878

Radical Vélu Isogeny Formulae

Abstract

We provide explicit radical $N$-isogeny formulae for all odd integers $N$. The formulae are compact closed-form expressions which require one $N$th root computation and $\mathcal{O}(N)$ basic field operations. The formulae are highly efficient to compute a long chain of $N$-isogenies, and have the potential to be extremely beneficial for speeding up certain cryptographic protocols such as CSIDH. Unfortunately, the formulae are conjectured, but we provide ample supporting evidence which strongly suggests their correctness. For CSIDH-512, we notice an additional 35% speed-up when using radical isogenies up to $N=199$, compared to the work by Castryck, Decru, Houben and Vercauteren, which uses radical isogenies up to $N=19$ only. The addition of our radical isogenies also speeds up the computation of larger class group actions in a comparable fashion.