# General Linear Group Action on Tensors: A Candidate for Post-Quantum Cryptography

## TL;DR

This paper proposes the general linear group action on tensors as a promising post-quantum cryptographic candidate, analyzing its hardness, security, and potential for building quantum-resistant cryptographic primitives.

## Contribution

It introduces the tensor isomorphism problem as a hard problem for cryptography, and develops the concept of pseudorandom group actions for secure cryptographic schemes.

## Key findings

- Tensor isomorphism problem is among the hardest isomorphism problems.
- The proposed group action assumptions are supported by attack analyses.
- Cryptographic primitives based on these assumptions are quantum-secure.

## Abstract

Starting from the one-way group action framework of Brassard and Yung (Crypto '90), we revisit building cryptography based on group actions. Several previous candidates for one-way group actions no longer stand, due to progress both on classical algorithms (e.g., graph isomorphism) and quantum algorithms (e.g., discrete logarithm). We propose the general linear group action on tensors as a new candidate to build cryptography based on group actions. Recent works (Futorny--Grochow--Sergeichuk, Lin. Alg. Appl., 2019) suggest that the underlying algorithmic problem, the tensor isomorphism problem, is the hardest one among several isomorphism testing problems arising from areas including coding theory, computational group theory, and multivariate cryptography. We present evidence to justify the viability of this proposal from comprehensive study of the state-of-art heuristic algorithms, theoretical algorithms, and hardness results, as well as quantum algorithms. We then introduce a new notion called pseudorandom group actions to further develop group-action based cryptography. Briefly speaking, given a group $G$ acting on a set $S$, we assume that it is hard to distinguish two distributions of $(s, t)$ either uniformly chosen from $S\times S$, or where $s$ is randomly chosen from $S$ and $t$ is the result of applying a random group action of $g\in G$ on $s$. This subsumes the classical decisional Diffie-Hellman assumption when specialized to a particular group action. We carefully analyze various attack strategies that support the general linear group action on tensors as a candidate for this assumption. Finally, we establish the quantum security of several cryptographic primitives based on the one-way group action assumption and the pseudorandom group action assumption.