Getting to the right root: radical isogenies of every degree

What it is

Thomas Decru's ePrint 2026/2595 (received 22 October 2026, approved 24 October 2026, preprint, CC BY) proves corrected radical isogeny formulae for every degree N≥4, odd and even, over fields of characteristic coprime to 2N. Earlier work had only conjectured compact linear-size formulae for odd N.

How it works

Each building block is written as a twisted sum of a discrete Kronecker function over Z/NZ. The formulae then follow from a distribution relation for the Kronecker theta function.

For even N, the Nth root over F_p is fixed only up to sign, and the sign decides whether a walk stays on the surface of the 2-isogeny volcano. The paper shows that the radical times a 2-torsion coordinate of the next curve is a square. This gives the horizontal choice for all even N. Before this, the choice was known only for N≤14.

What to do

For an implementation, the page gives these figures:

• One radical step costs about 12.5N multiplications and one exponentiation, with no inversions.
• For horizontal 2-isogeny walks, radical 2^m-isogenies with m in {5, 6, 7} are optimal for primes of 512 to 4096 bits. The abstract does not say which m goes with which size.
• Radical isogenies can be evaluated on points. A class group action can then be computed from two given points, one on the curve and one on its quadratic twist, with no torsion point sampling.

Caveats

The page is an abstract and metadata only. It gives no explicit formulae, pseudocode, benchmarks or code, and it does not say whether the work has been peer reviewed. The cost and optimality claims come from the abstract, so check them against the full text before relying on them.