We show how to efficiently evaluate functions on Jacobian varieties and their quotients. We deduce an algorithm to compute (l, l) isogenies between Jacobians of genus two curves in quasilinear time in the degree l 2 .

We propose a pseudo-primality test using cyclic extensions of $\mathbb Z/n \mathbb Z$. For every positive integer $k \leq \log n$, this test achieves the security of $k$ Miller-Rabin tests at the cost of $k^{1/2+o(1)}$ Miller-Rabin tests.Comment: Published in Rendiconti del Circolo Matematico di Palermo Journal, Springe

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We show how to efficiently evaluate functions on Jacobian varieties and their quotients. We deduce an algorithm to compute (l, l) isogenies between Jacobians of genus two curves in quasi-linear time in the degree l 2 .

International audienceWe construct extension rings with fast arithmetic using isogenies between elliptic curves. As an application, we give an elliptic version of the AKS primality criterion