Minimal degrees of invariants of (super)groups - a connection to cryptology

TL;DR

This paper explores the minimal degrees of invariants in various groups, linking these mathematical properties to cryptographic security and analyzing supergroup invariants beyond monomials.

Contribution

It provides new results on minimal degrees of invariants for finite, abelian, algebraic groups, and supergroups, connecting these to cryptology and revealing non-monomial bases.

Findings

Derived minimal degrees for finite, abelian, and algebraic groups.

Linked minimal degrees to cryptographic security considerations.

Identified non-monomial bases for certain supergroup invariants.

Abstract

We investigate questions related to the minimal degree of invariants of finitely generated diagonalizable groups. These questions were raised in connection to security of a public key cryptosystem based on invariants of diagonalizable groups. We derive results for minimal degrees of invariants of finite groups, abelian groups and algebraic groups. For algebraic groups we relate the minimal degree of the group to the minimal degrees of its tori. Finally, we investigate invariants of certain supergroups that are superanalogs of tori. It is interesting to note that a basis of these invariants is not given by monomials.