Forensic categories: a framework for SQIsign-like primitives

Using the language of categories, we introduce a novel framework abstracting the key algorithmic features of the Deuring correspondence between supersingular elliptic curves and quaternion orders and of the post-quantum signature scheme SQIsign based on it. We then show how to construct an interactive identification scheme and digital signature within this framework, and also instantiate more advanced primitives, such as a chameleon hash function. We present two distinct instantiations of the framework based on isogenies of supersingular elliptic curves: the first recovers (one-dimensional) SQIsign, while the second leads to the recently published SQInstructor.