论文标题
未能散布到超大型等源图中
Failing to hash into supersingular isogeny graphs
论文作者
论文摘要
基于超级同学的密码学中，一个重要的开放问题是，如果没有信任的权威，“硬超单向曲线”的具体示例，即在计算内态曲线的近似曲线方程中，计算内态曲线的方程式很难，就像随机超级曲线一样困难。一个相关的开放问题是在没有揭示内态词环或通往已知内态曲线曲线的途径的超级$ \ ell $发育图的顶点中产生哈希函数。这样的哈希功能将打开有趣的加密应用程序。在本文中，我们记录了一些（迄今为止）解决这个问题的尝试，希望我们能够刺激进一步的研究，并阐明这项工作的挑战和障碍。本文中包含的数学方法包括：（i）超级多项式的迭代根发现； （ii）专业模块化多项式的GCD； （iii）使用分区多项式来创建小型方程系统； （iv）在阿贝尔表面的同等基因图中随机步行； （v）使用量子随机步行。
An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of "hard supersingular curves" that is, equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular $\ell$-isogeny graph which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd's of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces; and (v) using quantum random walks.
