The SQInstructor: a guide to SQIsign and the Deuring Correspondence with level structures

We explore the use of level structures to generalize the SQIsign signature scheme. We give a general framework, where the challenge is a pair of level structures on the public key and the commitment curve, and the response is required to map one onto the other. 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, solve new constrained norm equations and study the distribution of odd norm vectors in lattices of isogenies.