Security Analysis of the Extended Grendel Permutation over $\mathbb{Z}_{pq}$

Abstract

We study the security of the Extended Grendel permutation defined over the ring $\mathbb{Z}_{pq}$, where $p \equiv q \equiv 3 \pmod{4}$. The construction combines an arithmetic S-box derived from the quadratic residue symbol through the CRT-based map $L_{pq}$, together with an MDS diffusion layer and round constants. This work complements our previous paper~\cite{Lkoaiza2025ExtendedGrendel}, which focused on the construction itself, by providing a systematic cryptographic analysis.

Using the wide-trail strategy together with a decomposition induced by the Chinese remainder theorem, we derive bounds against linear and differential cryptanalysis, examine resistance to integral distinguishers, and study algebraic attacks, including interpolation and preimage attacks. We also outline a polynomial-system modeling approach, supported by Gröbner basis techniques, to evaluate the complexity of symbolic attacks related to the $L_{pq}$ layer. Our results indicate that no low-complexity classical distinguishers arise in the considered attack models.