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**2011/143**

**Title:**

Computing (\ell,\ell)-isogenies in polynomial time on Jacobians of genus~2 curves

**Authors:**
Romain Cosset, Damien Robert

**Abstract:**

In this paper, we compute \ell-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in \ell, when \ell is an odd prime which is coprime to the characteristic. We use level~n symmetric theta structure where n=2 or n=4. In a second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus~2 hyperelliptic curves to theta coordinates of level~2 or 4. Combined with the preceding algorithm, this gives a method to compute (\ell,\ell)-isogenies in polynomial time on Jacobians of genus~2 curves.

**ePrint:**
https://eprint.iacr.org/2011/143

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