A Note on the Hardness of Problems from Cryptographic Group Actions

TL;DR

This paper demonstrates that the hardness of problems based on cryptographic group actions is unlikely to be NP-hard unless the Polynomial Hierarchy collapses, linking their complexity to well-studied complexity classes.

Contribution

It establishes that the Group Action Inverse Problem cannot be NP-hard unless major complexity class collapses occur, providing insights into the problem's computational difficulty.

Findings

GAIP cannot be NP-hard unless the Polynomial Hierarchy collapses.

The paper links cryptographic group action problems to Graph Isomorphism and NP intermediate problems.

Provides upper bounds on the worst-case complexity of cryptographic assumptions.

Abstract

Given a cryptographic group action, we show that the Group Action Inverse Problem (GAIP) and other related problems cannot be NP-hard unless the Polynomial Hierarchy collapses. We show this via random self-reductions and the design of interactive proofs. Since cryptographic group actions are the building block of many security protocols, this result serves both as an upper bound on the worst-case complexity of some cryptographic assumptions and as proof that the hardness in the worst and in the average case coincide. We also point out the link with Graph Isomorphism and other related NP intermediate problems.