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.

椭圆曲线是一维阿贝尔曲线；二维类似物是阿贝尔曲面。在这项工作中，我们提出了一种算法来计算有限域上定义的阿贝尔曲面之间的\((2^n,2^n)\) -同源性。这些同源是椭圆曲线的\(2^n\)同源的自然推广。这种同源链的高效计算引起了广泛关注，因为 SIDH（Castryck-Decru、Maino-Martindale、Robert）攻击的运行时间取决于这种计算。除了这些应用之外，我们的算法开发过程中推导出的不同结果也很有趣。例如，我们推导出一个评估 (2, 2)-同基因的公式。给定 Mumford 坐标中的元素，此公式输出其图像在 (2, 2)-同构下的（未简化的）Mumford 坐标。此外，我们研究了超椭圆曲线雅可比行列式上的 4 扭转点，并解释了如何从这些点中提取 2 扭转点系数的平方根。