## Cryptology ePrint Archive: Report 2011/143

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

Romain Cosset and 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.

Category / Keywords: public-key cryptography / elliptic curve cryptosystem