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**2022/1068**

**Title:**

Evaluating isogenies in polylogarithmic time

**Authors:**
Damien Robert

**Abstract:**

Let π βΆ πΈ β πΈβ² be an N-isogeny between elliptic curves (or abelian varieties) over a finite field π½_π. We show that there always exist an efficient representation of π that takes polylogarithmic π(log^π(1) π log π) space and which can evaluate π at any point π β πΈ(π½_{π^π}) in polylogarithmic π(log^π(1) π) arithmetic operations in π½_{π^π}.

Furthermore, this efficient representation can be computed by evaluating π on π(log π) points defined over extensions of degree π(log π) over π½_π. In particular, if π is represented by the equation π»(π₯) = 0 of its kernel πΎ, then using VΓ©luβs formula the efficient representation can be computed in time π Μ(π log π + log^2 π).

**ePrint:**
https://eprint.iacr.org/2022/1068

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