We investigate special structures due to automorphisms in isogeny graphs of principally polarized abelian varieties, and abelian surfaces in particular. We give theoretical and experimental results on the spectral and statistical properties of (2, 2)-isogeny graphs of superspecial abelian surfaces, including stationary distributions for random walks, bounds on eigenvalues and diameters, and a proof of the connectivity of the Jacobian subgraph of the (2, 2)-isogeny graph. Our results improve our understanding of the performance and security of some recently-proposed cryptosystems, and are also a concrete step towards a better understanding of general superspecial isogeny graphs in arbitrary dimension.
翻译:我们调查了主要为两极化的亚虫品种的异形图中,特别是亚虫表面的异形图中因自制而形成的特殊结构。我们给出了超特殊亚虫表面的光谱和统计特性(2,2)异种图的理论和实验结果,包括随机行走的固定分布、易基因值和直径的界限,以及雅各戈比亚2,2,2-异种图子图的连接性。我们的结果提高了我们对最近提出的一些加密系统的性能和安全性的理解,同时也是朝着更深入了解任意尺寸的一般超特殊异种图而迈出的具体步骤。