General properties of boomerang tables over abelian groups

B. A. Pogorelov^a, M. A. Pudovkina<sup>b

a</sup> Academy of Cryptography of the Russian Federation, Moscow
^b National Research Nuclear University (MEPhI)

Abstract: In this paper we study block ciphers on a finite abelian group $(X, + )$. In order to find out alternative key-addition groups of block ciphers, we consider finite abelian groups $(X, + )$ and their regular permutation representations. For a permutation $s \in S(X)$ on $(X, + )$, we introduce the boomerang connectivity table, the Feistel boomerang table, the Feistel boomerang difference table over $(X, + )$, which are natural generalization of corresponding matrices over the additive group of a vector space $\left( {{V_n}({p^m}), + } \right)$. We describe general properties of the matrices which hold for any finite abelian group $(X, + )$. So we get sufficient conditions for permutations on $(X, + )$ to have the maximum value of the boomerang uniformity and its variations, which is equal to $\left| X \right|$. We get evaluation of the boomerang uniformity of permutations from ${S_q} \uparrow {S_n}$. We introduce holomorph-equivalence of permutations on $(X, + )$, which generalize affine-equivalence of vectorial boolean functions, and get properties of boomerang tables such permutations. We describe group properties of permutations on $(X, + )$ with a linear translator. We characterize properties of boomerang tables of such permutations. To show applications our results, we consider a class of block ciphers on $(X, + )$ and get their boomerang uniformities.

Key words: boomerang connectivity table, Feistel boomerang table, Feistel boomerang difference table, boomerang uniformity, linear translator, holomorph, imprimitive group, permutation representation.

UDC: 519.7

Received 21.V.2025

DOI: 10.4213/mvk509