In supersingular isogeny-based cryptography, the path-finding problem reduces to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism? It is known that a small endomorphism enables polynomial-time path-finding and endomorphism ring computation (Love-Boneh [36]). As this paper neared completion, it was shown that the endomorphism ring problem in the presence of one known endomorphism reduces to a vectorization problem (Wesolowski [54]). In this paper, we give explicit classical and quantum algorithms for path-finding to an initial curve using the knowledge of one endomorphism. An endomorphism gives an explicit orientation of a supersingular elliptic curve. We use the theory of oriented supersingular isogeny graphs and algorithms for taking ascending/descending/horizontal steps on such graphs. Although the most general runtimes are subexponential, we show that every supersingular elliptic curve has (potentially large) endomorphisms whose exposure would lead to a classical polynomial-time path-finding algorithm.

翻译：在超星系基于世系的加密法中,路由调查问题被降为内分形环状问题。 路由调查能否简化为只了解一个内分形论? 已知一个小内分形体可以使多米时间路由调查和内分形环计算( Love- Boneh [36] )。 本文接近完成时,文件显示,在已知内分形中存在的内分形环问题会降低为传导问题( Wesolowski [54] )。 在本文中,我们使用一个内分形学知识,将路径调查的明显古典和量量衡算算法推到初始曲线。 内分形论提供了超超表层椭圆形曲线的明确方向。 我们使用方向超离层异形图和算法理论在这种图形中进行升迁/ 脱色/ / 横向步骤。 虽然最普遍的运行时间是亚化的,但我们显示,每一个超上层的内分形曲线都有( 潜在大) 内分形反型的内分形分析法式反向后古体的磁体分析法系。