Cryptographic Group and Semigroup Actions

TL;DR

This paper explores cryptographic group and semigroup actions, introducing algorithms for the semigroup action problem, analyzing bounds, and examining attacks like Pohlig-Hellman, with relevance to isogeny-based cryptography.

Contribution

It presents generic algorithms for the semigroup action problem and analyzes bounds and attacks, extending cryptographic understanding of group and semigroup actions.

Findings

Introduces algorithms for the semigroup action problem

Provides bounds for the problem's complexity

Analyzes Pohlig-Hellman type attacks in this context

Abstract

We consider actions of a group or a semigroup on a set, which generalize the setup of discrete logarithm based cryptosystems. Such cryptographic group actions have gained increasing attention recently in the context of isogeny-based cryptography. We introduce generic algorithms for the semigroup action problem and discuss lower and upper bounds. Also, we investigate Pohlig-Hellman type attacks in a general sense. In particular, we consider reductions provided by non-invertible elements in a semigroup, and we deal with subgroups in the case of group actions.