Computing separable isogenies in quasi-optimal time

Abstract

Let $A$ be an abelian variety of dimension $g$ together with a principal polarization $\phi: A \rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of $A[\ell]$ which is maximal isotropic for the Riemann form associated to $\phi$. We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient abelian variety together with a polarization compatible with $\phi$. Then $B$, as a polarized abelian variety, and the isogeny $f:A\rightarrow B$ are also defined over $k$. In this paper, we describe an algorithm that takes as input a theta null point of $A$ and a polynomial system defining $K$ and outputs a theta null point of $B$ as well as formulas for the isogeny $f$. We obtain a complexity of $\tilde{O}(\ell^{\frac{rg}{2}})$ operations in $k$ where $r=2$ (resp. $r=4$) if $\ell$ is a sum of two squares (resp. if $\ell$ is a sum of four squares) which constitutes an improvement over the algorithm described in [7]. We note that the algorithm is quasi-optimal if $\ell$ is a sum of two squares since its complexity is quasi-linear in the degree of $f$.