Efficient computation of (2n , 2n)-isogenies

Elliptic curves are abelian varieties of dimension one; the two-dimensional analogues are abelian surfaces. In this work we present an algorithm to compute (2(n ), 2(n))-isogenies between abelian surfaces defined over finite fields. These isogenies are the natural generalization of 2(n)-isogenies of elliptic curves. The efficient computation of such isogeny chains gained a lot of attention as the runtime of the attacks on SIDH (Castryck-Decru, Maino-Martindale, Robert) depends on this computation. Different results deduced in the development of our algorithm are also interesting beyond these applications. For instance, we derive a formula for the evaluation of (2, 2)-isogenies. Given an element in Mumford coordinates, this formula outputs the (unreduced) Mumford coordinates of its image under the (2, 2)-isogeny. Furthermore, we study 4-torsion points on Jacobians of hyperelliptic curves and explain how to extract square roots of coefficients of 2-torsion points from these points.