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

TL;DR

This paper generalizes the SQIsign signature scheme using level structures, introducing a framework involving isogenies and a new explicit Deuring correspondence for supersingular elliptic curves.

Contribution

It presents a novel framework for SQIsign with level structures and provides new explicit Deuring correspondence and solutions to constrained norm equations.

Findings

Framework for SQIsign with level structures established

New explicit Deuring correspondence for supersingular elliptic curves

Solutions to constrained norm equations provided

Abstract

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.