Computing isogenies between abelian varieties

Abstract

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 comprises 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 present an algorithm to obtain a rational expression for an isogeny from A to A / K . The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (ℤ/ ℓ ℤ) g for ℓ ∈ℕ * , the overall time complexity of this algorithm is equivalent to O (log ℓ ) additions in A and a constant number of ℓth root extractions in the base field of A . In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4 ℓ of a g -dimensional abelian variety using only g ( g +1)/2⋅4 g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.