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]). An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this paper, we use the volcano structure of the oriented supersingular isogeny graph to take ascending/descending/horizontal steps on the graph and deduce path-finding algorithms to an initial curve. Each altitude of the volcano corresponds to a unique quadratic order, called the primitive order. We introduce a new hard problem of computing the primitive order given an arbitrary endomorphism on the curve, and we also provide a sub-exponential quantum algorithm for solving it. In concurrent work (Wesolowski [54]), it was shown that the endomorphism ring problem in the presence of one endomorphism with known primitive order reduces to a vectorization problem, implying path-finding algorithms. Our path-finding algorithms are more general in the sense that we don't assume the knowledge of the primitive order associated with the endomorphism.
翻译:在超星系基于外星系的密码学中, 路由调查问题会降低到内地貌问题。 路由调查能否简化为只了解一个内地貌论? 已知一个小内地貌主义可以使多米时间路由调查和内地貌论环计算( Love- Boneh [36] ) 。 内地貌主义会给超级外星椭圆曲线带来一个明确方向。 在本文中, 我们使用方向超星系外星图的火山结构来在图表上采取升/ 降/ 水平/ 横向步骤, 并将路径调查算法推导成一个初始曲线。 火山的每个高度都对应一个独特的四边秩序, 称为原始秩序。 我们引入了一个新的硬问题, 计算原始秩序, 在曲线上任意的内地貌形态论中, 我们还提供了一种亚异质量算法来解决这个问题。 与此同时( Wesolowski [54), 显示, 终地貌论将问题带一个已知的内地貌学和已知的原始秩序下, 我们的原始形态算算法会降低路径。