In symmetric cryptography, vectorial Boolean functions over the finite field \(\mathbb {F}_{2^n}\) are used to construct strong S-boxes. A strong S-box must meet various criteria to resist known attacks, including differential, linear, boomerang, and their variants. To evaluate an S-box’s resistance, several tables are utilized, such as the Difference Distribution Table (DDT) and the Boomerang Connectivity Table (BCT). Recent developments in boomerang attacks have revisited the concept of the boomerang switch effect, illustrating the effectiveness of this technique. As a result, a new tool called the Boomerang Difference Table (BDT) was introduced as an alternative to the traditional BCT. Additionally, two novel tables have been proposed: the Upper Boomerang Connectivity Table (UBCT) and the Lower Boomerang Connectivity Table (LBCT). These tables are enhancements over the BCT and facilitate a systematic evaluation of boomerangs that can return over multiple rounds. This paper focuses on the new tools for measuring the revisited version of boomerang attacks and the related tables \(\texttt {UBCT}\) , \(\texttt {LBCT}\) , as well as the so-called Extended Boomerang Connectivity Table ( \(\texttt {EBCT}\) ). Specifically, we examine the properties of these novel tools and investigate the corresponding tables. We also study their interconnections, their links to the DDT, and their values for affine equivalent vectorial functions and compositional inverses of permutations of \(\mathbb {F}_{2^n}\) . Moreover, we introduce the concept of the nontrivial boomerang connectivity uniformity and determine the explicit values of all the entries of the \(\texttt {UBCT}\) , \(\texttt {LBCT}\) , and \(\texttt {EBCT}\) for the important cryptographic case of the inverse function.
