On permutations perfectly diffusing classes of partitions of $V_n^l(2^m)$

B. A. Pogorelov ; M. A. Pudovkina

Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 16-19

Résumé

Let $V_n (2^m)$ be an $n$-dimensional vector space over $\mathbb{F}_{2^{m}}$ and ${\overline{V}}_n^l(2^m)$ consists of all pairwise different elements from the Cartesian product $V_n^l(2^m)$, $l,n,m \in \mathbb{N}$, $n,l \geq 2$. We consider permutations from the symmetrical group $S(V_n (2^m))$ acting coordinate-wise on vectors from $V_n^l(2^m)$, and partitions of ${\overline{V}}_n^l(2^m)$, which are generalizations of classical differential partitions ($l = 2$) and are used in high order differential, truncated differential, impossible differential, polytopic and multiple differential techniques. For a partition ${\mathbf{W}}^{(n,l)}$ of ${\overline{V}}_n^l(2^m)$, we study the minimum Hamming distance $\text{d}_{{\mathbf{W}}^{(n,l)}}(s)$ between a permutation $s$ and the set $\mathrm{IG}_{\mathbf{W}}$ consisting of all permutations from $S(V_n (2^m))$ preserving ${\mathbf{W}}^{(n,l)}$. We describe properties of permutations $s$ with the maximum distance $\text{d}_{{\mathbf{W}}^{(n,l)}}(s)$, which perfectly diffuse ${\mathbf{W}}^{(n,l)}$. We get a criterion of perfect diffusion of ${\mathbf{W}}^{(n,l)}$ for any $l\in \mathbb{N}$. We show the connection between permutations perfectly diffusing ${\mathbf{W}}$, APN-permutations, AB-permutations, and differentially $2r$-uniform permutations, $r \ge 1$.

Détail

Mots-clés : perfect diffusion, imprimitive group, APN-permutation, AB-permutation, polytopic technique.
Keywords: wreath product, differentially $d$-uniform permutation, differential technique