On the primitivity of the AES-128 key-schedule

TL;DR

This paper proves that the group generated by the AES-128 key-scheduling algorithm acts primitively on the message space, implying it has no non-trivial invariant subspaces, which enhances understanding of its algebraic structure.

Contribution

It establishes the primitivity of the group generated by AES-128's key-scheduling, a novel algebraic property relevant for cryptanalysis and security analysis.

Findings

The group generated by AES-128 key-schedule is primitive.

No proper subspace remains invariant under the group's action.

Supports the robustness of AES-128 against certain algebraic attacks.

Abstract

The key-scheduling algorithm in the AES is the component responsible for selecting from the master key the sequence of round keys to be xor-ed to the partially encrypted state at each iteration. We consider here the group $\Gamma$ generated by the action of the AES-128 key-scheduling operation, and we prove that the smallest group containing $\Gamma$ and all the translations of the message space is primitive. As a consequence, we obtain that no proper and non-trivial subspace can be invariant under its action.