Theoretical Foundations of Applied Discrete Mathematics

On linear translators and boomerang connectivity tables over Abelian groups

B. A. Pogorelov, M. A. Pudovkina


Abstract: In 2018 in order to study the resistance of a block cipher against boomerang attacks, a tool called the Boomerang Connectivity Table (BCT) for cryptographic transformations on $\mathbb{F}_{p^m}$ was introduced, where usually $p=2$. We generalize BCT $b(s)$ of a permutation $s$ over an arbitrary finite Abelian group $(X,+)$ and also its boomerang-uniformity ${\sigma^{(b)}}(s)$, which is equal to the maximal element of BCT $b(s)$. We study XSH-block ciphers whose round function consists of key-addition layer, $s\in S(X)$ and $h \in {\rm{Hol}}(X, + )$, where ${\rm{Hol}}(X, + )$ is the holomorph of $(X, + )$. If $X=\mathbb{F}^{n}_{2^m}$, then SPN-ciphers with a linear transformation are XSH-ciphers. We study influence of a linear translator of $s$ on properties of BCT and get boomerang-uniformity of XSH-ciphers estimates. For $X\in \{\mathbb{F}^{n}_{2^m}, \mathbb{Z}_{2^{n}} \}$, we provide examples.

Keywords: Boomerang Connectivity Table $($BCT$)$, holomorph, automorphism group, linear translator, boomerang-uniformity, boomerang attack.

UDC: 519.7}\maketitle

DOI: 10.17223/2226308X/18/9