Elementary Abelian regular subgroups of vector space affine group related to cryptanalysis.~II

M. A. Goltvanitsa

Matematičeskie voprosy kriptografii, Tome 15 (2024), pp. 9-47

Résumé

Let $p$ be a prime number, $(V,+)$ be a finite-dimensional vector space over finite field $\mathbb{F}_p$ of cardinality $p$. We investigate elementary Abelian regular subgroups $\mathcal{T}$ of affine group $\mathrm{AGL}(V)$. Each such subgroup defines new binary operation $\circ$ on the set $V$ and can be utilized in cryptanalysis, especially in cryptanalysis of block ciphers constructed as SP-networks. In the previous paper we propose the first practical algorithm for testing whether given s-box preserving zero belong to the normalizer of some group $\mathcal{T}$ in $\mathrm{Sym}(V)$. In this paper we generalize this algorithm for an arbitrary s-box. We find some arithmetic properties of linear groups associated with groups $\mathcal{T}$. Basing on utilizing automorphisms of direct sums of commutative algebras we suggest the first practical method for construction of $\circ$-affine SP-networks with an arbitrary block size.