A Guide to the Design of Digital Signatures based on Cryptographic Group Actions

Cryptography based on group actions has been studied since 1990. In recent years, however, the area has seen a revival, partially due to its role in post-quantum cryptography. For instance, several works have proposed signature schemes based on group actions, as well as a variety of techniques aimed at improving their performance and efficiency. Most of these techniques can be explained as transforming one Sigma protocol into another, while essentially preserving security. In this work, we present a unified taxonomy of such techniques. In particular, we describe all techniques in a single fashion, show how they impact the performance of the resulting protocols and analyse in detail how different techniques can be combined for optimal performance. Furthermore, to provide a tangible perspective, we apply the results of our analysis to the (group action-based) candidates in the current NIST call for digital signatures. This gives a full overview of the state of the art of signatures based on group actions, as well as a flexible tool which is easy to adapt and employ in the design of future schemes.