Diffusion of systems of blocks for the translation group ${\mathbb{F}_2^m}^+$ of the vector space $\mathbb{F}_2^m$ by permutations with nontrivial automorphism group

D. A. Burov^a, I. M. Karabeynikov<sup>b

a</sup> TVP Laboratories
^b Foundation for Assistance for Secure Information Technologies Development

Abstract: In this paper we study the diffusion of systems of blocks for the translation group ${\mathbb{F}_2^m}^+$ of the additive group of a vector space $\mathbb{F}_2^m$ by permutations with nontrivial automorphism group. It is shown that nontrivial automorphism group implies restrictions on system of blocks of the group ${\mathbb{F}_2^m}^+$ whose images by permutation equal systems of blocks of the group ${\mathbb{F}_2^m}^+$. We investigate the diffusion of systems of blocks for the translation group ${\mathbb{F}_2^m}^+$ by picewise-monomial permutations on subgroups of index 3 and 5. We prove sufficient conditions on differential $\delta$-uniformity for non-existence of systems of blocks for the group ${\mathbb{F}_2^m}^+$ whose images by picewise-monomial permutation equal systems of blocks of the group ${\mathbb{F}_2^m}^+$. These conditions are more weaker than conditions for arbitrary permutations.

Keywords: automorphism group of mapping, system of blocks, picewise-monomial permutation, s-box, partitioning cryptanalysis.

UDC: 519.719.2

Received: 01.06.2025

DOI: 10.4213/dm1881