Here is a discussion thread for the 7th session of The Isogeny Club, where Marc Houben presents his talk titled: Horizontal Racewalking using Radical Isogenies

**Abstract**:

Radical isogeny formulae are equations that can be used to efficiently compute long chains of isogenies of small degree. Basically, they express the coefficients of the next curve in a chain of N-isogenies explicitly in terms of some expression involving the N-th root of a quantity depending on the Weierstrass coefficients of the input curve. One can prove that such an expression always exists, but finding it is a nontrivial task. We present a new method for finding radical isogeny formulae that extends the range for which we know them from N ≤ 13 to N ≤ 37.

We rewrite the existing and new formulae to optimize for fast evaluation. For even N, we present a conjecture that determines which N-th root must be taken in order to stay on the surface of the CSIDH isogeny graph, and we prove this conjecture for N ≤ 14. The combination of the above results in a speed up of a factor 3 for long chains of 2-isogenies over 512 bit prime fields, and we gain 12% over the previous implementation of CSIDH with radical isogenies.

**Video**:

(we post a link to the video when available)