Computing isogenies between abelian varieties

We describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let $A$ be an abelian variety of dimension $g$ defined over a field of odd characteristic. Our algorithm decomposes in two principal steps. First, given a theta null point for $A$ and a subgroup $K$ isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety $A/K$. Then, from the knowledge of a theta null point of $A/K$, we give an algorithm to obtain a rational expression for an isogeny from $A$ to $A/K$. The algorithm resulting as the combination of these two steps can be viewed as a higher dimensional analog of the well known algorithm of V\'elu to compute isogenies between elliptic curves. In the case that $K$ is isomorphic to $(\Z / \ell \Z)^g$ for $\ell \in \N^*$, the overall time complexity of this algorithm is equivalent to $O(\log \ell)$ additions in $A$ and a constant number of $\ell^{th}$ root extractions in the base field of $A$. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows to encode a point of level $4\ell$ of a $g$ dimensional abelian variety using only $g(g+1)/2\cdot 4^g$ coordinates. We also give formulas to compute the Weil and commutator pairings given input points in theta coordinates.