We explore the use of level structures to generalize the SQIsign signature scheme. We give a general framework where, given the public key and the commitment, the challenge is to exhibit an isogeny between them with an additional requirement, namely to map a chosen level structure to another. We then instantiate the framework using 1-dimensional and 2-dimensional isogenies. In doing that we provide a new explicit Deuring correspondence for supersingular elliptic curves with level structures and solve new constrained norm equations.

翻译：本文探讨了利用层级结构对 SQIsign 签名方案进行推广的方法。我们提出了一个通用框架，在该框架中，给定公钥与承诺值，挑战在于展示两者之间的同源映射，并满足一项附加要求：将选定的层级结构映射至另一指定结构。随后，我们分别基于一维与二维同源映射对该框架进行了具体实例化。在此过程中，我们为带层级结构的超奇异椭圆曲线建立了新的显式 Deuring 对应关系，并求解了新型约束范数方程。