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

M. A. Goltvanitsa

Matematičeskie voprosy kriptografii, Tome 14 (2023), pp. 25-53

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 $\mathrm{AGL}(V)$. Every such subgroup determines new binary operation $\circ$ on the set $V$ and can be used in cryptanalysis. We investigate the structure properties of the group of linear maps associated with the group $\mathcal{T}$. The membership criterion for the right regular representation of group $(V, +)$ to belong to the normalizer of $\mathcal{T}$ in symmetric group $\mathrm{Sym}\,(V)$ is obtained. A practically realizable algorithm for testing whether given $\mathrm{s}$-box belongs to the normalizer of some group $\mathcal{T}$ in $\mathrm{Sym}\,(V)$ is proposed and investigated.