Computing isogenies between abelian varieties

TL;DR

This paper presents an efficient algorithm for computing separable isogenies between abelian varieties using algebraic theta functions, extending elliptic curve isogeny methods to higher dimensions with improved encoding techniques.

Contribution

It introduces a higher-dimensional analog of Vélu's algorithm for abelian varieties, including a compressed point representation and formulas for pairings, enhancing computational efficiency.

Findings

Algorithm complexity is O(log ℓ) for isogeny computation.

Introduces a compressed representation encoding points with fewer coordinates.

Provides formulas for Weil and commutator pairings in theta coordinates.

Abstract

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.